# A priori estimates for the motion of charged liquid drop: A dynamic approach
via free boundary Euler equations

Vesa Julin  and  Domenico Angelo La Manna

###### Abstract.

We study the motion of charged liquid drop in three dimensions where the
equations of motions are given by the Euler equations with free boundary with
an electric field. This is a well-known problem in physics going back to the
famous work by Rayleigh. Due to experiments and numerical simulations one may
expect the charged drop to form conical singularities called Taylor cones,
which we interpret as singularities of the flow. In this paper, we study the
well-posedness and regularity of the solution. Our main theorem is in the
spirit of the Beale-Kato-Majda criterion and roughly states that if the flow
remains $C^{1,\alpha}$-regular in shape and the velocity remains Lipschitz-
continuous, then the flow remains smooth, i.e., $C^{\infty}$ in time and
space, assuming that the initial data is smooth. Due to the appearance of
Taylor cones we expect the $C^{1,\alpha}$-regularity assumption to be optimal,
while the Lipschitz-regularity assumption on the velocity is standard in the
classical theory of Euler equations. We also quantify the
$C^{\infty}$-estimate via high order energy estimates. This result is new also
for the Euler equations with free boundary without the electric field. We
point out that we do not consider the problem of existence in this paper. It
will be studied in forthcoming work.

###### Contents

  1. 1 Introduction and the main result
    1. 1.1 State-of-the-art
    2. 1.2 Statement of the Main Theorem
    3. 1.3 Overview of the proof and the structure of the paper
  2. 2 Notation and Preliminary results
    1. 2.1 Half-integer Sobolev spaces
    2. 2.2 Half-integer Sobolev spaces on a surfaces
    3. 2.3 Geometric Preliminaries
    4. 2.4 Functional and geometric inequalities
  3. 3 Elliptic estimates for vector fields and functions
    1. 3.1 Regularity estimates for vector fields
    2. 3.2 Regularity estimates for functions
  4. 4 Useful formulas
  5. 5 Estimation of the error terms
  6. 6 First regularity estimates
  7. 7 Energy estimates
  8. 8 Higher regularity estimates
  9. 9 Proof of the Main Theorem

##  1\. Introduction and the main result

###  1.1. State-of-the-art

In this paper we study the problem of charged liquid drop from rigorous
mathematical point of view. In the model the two effecting forces are the
surface tension, which prefers to keep the drop spherical, and the repulsive
electrostatic force which both act on the boundary of the drop. The problem is
well-known and goes back to Rayleigh [47] who studied the linear stability of
the sphere and showed that the sphere becomes unstable when the total electric
charge is above a given treshhold. When the total electric charge is above
this Rayleigh treshhold, the drop begins to elongate and may eventually form a
conical singularity at the tip with a certain opening angle. Such
singularities are called _Taylor cones_ due to the work by Taylor [50] and the
numerical and experimental evidence suggest that the charged drop typically
forms such a singularity [24, 44, 50, 56]. In this paper our goal is to study
the well-posedness of the problem and the regularity of the solution. We refer
to [45] and [29] for an introduction to the topic.

The static problem of charged liquid drop can be seen as a nonlocal
isoperimetric problem and it has been studied from the point of view of
Calculus of Variations in recent years [27, 28, 36, 45]. The main issue is
that the associated minimization problem, formulated in the framework of
Calculus of Variations, is not well-posed, in the sense that the problem does
not have a minimizer [27, 45]. Even more surprising is that the results in
[27, 45] show that even if the total electric charge is below the Rayleigh
treshhold, the sphere is not a local minimizer of the associated energy. This
means that the electrostatic term is not lower order with respect to the
surface tension, which makes the problem mathematically challenging. In order
to make the variational problem well-posed one may restrict the problem to
convex sets [28] or regularize the functional by adding a curvature term [29]
which could lead to the existence of minimizer as the result in [26] suggests.

Here we study this problem from the point of view of fluid-dynamics, which is
the framework studied e.g. in [24], where the authors derive the PDE system in
the irrotational case (see also [5]). Indeed, as it is observed in [24] the
problem is by nature evolutionary, where the drop deforms as a function of
time given by the Euler equations with the surface of the drop being the free
boundary, which law of motions is coupled with the system which we give in
(1.3) below. The problem can thus be seen as the Euler equations for
incompressible fluids with free boundary with an additional term given by the
electric field. While we are not aware of any previous mathematical research
on this particular topic, the Euler equations with free boundary without the
electric field has been studied rather extensively in recent years. We give
only a brief overview on this challenging problem and refer to [12, 43] for
more detailed introduction to the topic. We also mention the work [20], where
the authors study the Stokes flow associated with the charged liquid drop near
the sphere and show that under smallness assumption the flow is well defined.
Our point of view is thus somewhat similar to [20] in the sense that our main
goal is to show that the evolutionary equation related to the liquid drop is
well-posed.

We stress that in our case it is crucial to include the surface tension in the
model since otherwise the problem might be ill-posed. For the problem without
the electric field one may study the Euler equations also without the surface
tension, when one assumes the so called Rayleigh-Taylor sign condition [17],
which one should not confuse with the Rayleigh treshhold mentioned above. For
the water-wave problem the well-posedness is proven by Wu [53, 54] and the
general case is due to Linblad [39], see also [3, 37]. Concerning the problem
with surface tension, which is closer to ours, the short time existence of
solution in the irrotational case for starshaped sets is due to Beyer-Günther
[6, 7] and the general case is proven by Coutand-Shkoller [11]. We also
mention the earlier works concerning the well-posedness of the problem in the
planar case [2, 33, 55]. The works that are closest to ours are Shatah-Zheng
[49] and Schweizer [48], where the authors prove regularity estimates for the
free boundary Euler equations with surface tension. Our work is also inspired
by Masmoudi-Rousset [43], where the authors prove similar estimates for the
Euler equations without the surface tension.

As usual with geometric evolution equations, the Euler equations with free
boundary may develop singularities in finite time. In [13] the authors
construct an example where the equations develop singularities where the drop
changes its topology. We stress that in the absence of the electric field, we
do not expect the flow to develop conical singularities predicted by Taylor
[50], where both the curvature and the velocity become singular. Indeed,
Taylor cones are special type of singularities as the evolving sets
$\Omega_{t}$ do not change their topology, but only lose their regularity. We
also point out that the analysis in [50] does not give a rigorous mathematical
proof for the fact that the Taylor cone is a singularity of the associated
flow. Indeed, since there is no monotonicity formula for the Euler equations,
similar to the one by Huisken [32, 42] for the mean curvature flow, there is
little hope to have general classification of the singularities at the moment.
We refer to [21] and [23] which both study critical points of energy
functionals, which are very much similar to ours, with conical or cusp-like
singularities. Then again, as we interpret the Taylor cone as a singularity of
the flow given by the Euler equations (1.3), it is not clear why the
singularity is a critical point of the potential energy.

###  1.2. Statement of the Main Theorem

We study the motion of an incompressible charged drop in vacuum in
$\mathbb{R}^{3}$ and denote the fluid domain by $\Omega_{t}$. We assume that
we have an initial smooth and compact set $\Omega_{0}\subset\mathbb{R}^{3}$
and a smooth initial velocity field $v_{0}:\Omega_{0}\to\mathbb{R}^{3}$ which
evolve to a smooth family of sets and vector fields
$(\Omega_{t},v(\cdot,t))_{t\in[0,T)}$. The total energy is given by

(1.1)
$J_{t}(\Omega_{t},v)=\frac{1}{2}\int_{\Omega_{t}}|v(x,t)|^{2}\,dx+\mathcal{H}^{2}(\partial\Omega_{t})+\frac{Q}{\text{Cap}(\Omega_{t})},$

where $\text{Cap}(\Omega)$ is the electrostatic capacity given by

$\text{Cap}(\Omega):=\inf\big{\\{}\int_{\mathbb{R}^{3}}\frac{1}{2}|\nabla
u|^{2}\,dx:\,u(x)\geq 1\,\,\text{for all
}\,x\in\Omega\,,u\in\dot{H}^{1}(\mathbb{R}^{3})\big{\\}},$

where $\|u\|_{\dot{H}^{1}(\mathbb{R}^{3})}=\|\nabla
u\|_{L^{2}(\mathbb{R}^{3})}+\|u\|_{L^{6}(\mathbb{R}^{3})}$. We denote the
capacitary potential as $U_{\Omega}\in\dot{H}^{1}(\mathbb{R}^{3})$ which is
the function for which
$\text{Cap}(\Omega)=\int_{\mathbb{R}^{3}}\frac{1}{2}|\nabla
U_{\Omega}|^{2}\,dx$. This is equivalent to say that
$U_{\Omega}\in\dot{H}^{1}(\mathbb{R}^{3})$ satisfies

(1.2) $\begin{cases}-\Delta U_{\Omega}=0\qquad\text{in
}\,\mathbb{R}^{3}\setminus\Omega\\\ U_{\Omega}=1\qquad\text{on
}\,\partial\Omega.\end{cases}$

We denote the mean curvature of $\Sigma_{t}=\partial\Omega_{t}$ by
$H_{\Sigma_{t}}$, which for us is the sum of the principle curvatures given by
orientation via the outer normal $\nu_{\Sigma_{t}}$. With this convention
convex sets have positive mean curvature. As usual we denote the material
derivative of a vector field $F$ by

$\mathcal{D}_{t}F=\partial_{t}F+(v\cdot\nabla)F.$

The equations of motion are given by the Euler equations with free boundary
(for the derivation see [24])

(1.3) $\displaystyle\begin{cases}\mathcal{D}_{t}v+\nabla p=0\qquad\text{in
}\,\Omega_{t}\\\ \operatorname{div}v=0\qquad\text{in }\,\Omega_{t}\\\
v_{n}=V_{t}\qquad\text{on }\,\Sigma_{t}=\partial\Omega_{t}\\\
p=H_{\Sigma_{t}}-\frac{Q}{2C_{t}^{2}}|\nabla U_{\Omega_{t}}|^{2}\qquad\text{on
}\Sigma_{t},\end{cases}$

where $C_{t}=\text{Cap}(\Omega_{t})$, $V_{t}$ is the normal velocity,
$v_{n}=v\cdot\nu$ and $p$ is the pressure. We say that the system (1.3) has a
smooth solution in time-interval $(0,T)$ with initial data
$(\Omega_{0},v_{0})$, if there is a family of $C^{\infty}$-difeomorphisms
$(\Phi_{t})_{t\in[0,T)}$, which depend smoothly on $t$, such that
$\Phi_{0}=id$ and $\Phi_{t}(\Omega_{0})=\Omega_{t}$, the functions
$v(t,\Phi(t,x))$ and $p(t,\Phi(t,x))$ are smooth and the equations hold in the
classical sense. Moreover we require that $v(t,\Phi(t,\cdot))\to v_{0}$ as
$t\to 0$, where $v_{0}:\Omega_{0}\to\mathbb{R}^{3}$ is the initial velocity
field. When the total electric charge is zero, i.e. $Q=0$, the system reduces
to the more familiar Euler equations with free boundary with surface tension.

The characteristic property of the solution of (1.3) is the conservation of
the energy (1.1), i.e.,

$\frac{d}{dt}J(\Omega_{t},v)=0,$

which follows from straightforward calculation. Therefore one could guess, and
we will prove this in our main theorem, that assuming that the flow given by
the system (1.3) does not develop singularities then it preserves the
regularity of the initial data $(\Omega_{0},v_{0})$ at least in sense of
certain Sobolev norm. In particular, we point out that, unlike the mean
curvature flow [42], the flow given by the system (1.3) is not smoothing.

We parametrize the moving boundary $\Sigma_{t}=\partial\Omega_{t}$ by using a
fixed reference surface $\Gamma$ which we assume to be smooth and compact. We
use the height function parametrization which means that for every $t$ we
associate function the $h(\cdot,t):\Gamma\to\mathbb{R}$ with the moving
boundary $\Sigma_{t}$ as

$\Sigma_{t}=\\{x+h(x,t)\nu_{\Gamma}(x):x\in\Gamma\\}.$

We assume that $\Gamma$ satisfies the interior and exterior ball condition
with radius $\eta>0$ and note that $\eta$ is not necessarily small. For
example, from application point of view a relevant case is when the initial
set is star-shaped in which case it is natural to choose the reference
manifold to be a sphere in which case $\eta$ is its radius. It is clear that
the height-function parametrization is well defined as long as

$\sup_{t\in[0,T)}\|h(\cdot,t)\|_{L^{\infty}(\Gamma)}<\eta.$

Therefore we define the quantity

(1.4) $\sigma_{T}:=\eta-\sup_{t\in[0,T)}\|h(\cdot,t)\|_{L^{\infty}(\Gamma)}$

and the above condition reads as $\sigma_{T}>0$.

As in [43, 49, 48] we note that we do not consider the existence in this
paper, but study it in forthcoming work. Instead, as in [48] we assume that
the following qualitative short time existence result holds.

_Throughout the paper we assume that for every smooth initial set and smooth
initial velocity field the system ( 1.3) has a smooth solution which exists a
short interval of time. _

In this paper we are interested in finding a priori estimates which guarantee
that the system (1.3) does not develop singularities. To this aim we fix a
small $\alpha>0$ and define

(1.5)
$\Lambda_{T}:=\sup_{t\in[0,T)}\left(\|h(\cdot,t)\|_{C^{1,\alpha}(\Gamma)}+\|\nabla
v(\cdot,t)\|_{L^{\infty}(\Omega_{t})}+\|v_{n}(\cdot,t)\|_{H^{2}(\Sigma_{t})}\right).$

We note that $\alpha$ can be any positive number. Our goal is to show that if
the quantity $\Lambda_{T}$ is bounded then the flow can be extended beyond
time $T$ and is smooth if the initial data is smooth. We will prove this in a
quantitative way and define an energy quantity of order $l\geq 1$ as

(1.6)
$\hat{E}_{l}(t):=\sum_{k=0}^{l}\|\mathcal{D}_{t}^{l+1-k}v\|_{H^{\frac{3}{2}k}(\Omega_{t})}^{2}+\|v(\cdot,t)\|_{H^{\lfloor\frac{3}{2}(l+1)\rfloor}(\Omega_{t})}^{2},$

where $\lfloor\frac{3}{2}(l+1)\rfloor$ denotes the integer part of
$\frac{3}{2}(l+1)$. We define the Hilbert space for half-integers
$H^{\frac{3}{2}k}(\Omega_{t})$ via extension in Section 2. In the last term we
use a Hilbert space of integer order since it simplifies the calculations. The
fact that the boundedness of $\tilde{E}_{l}(t)$ for every $l$ implies the
smoothness of the flow will be clear from the results in Section 8. Indeed, we
first show that the bound on $\hat{E}_{l}(t)$ implies a bound for the pressure
$p$. By the a priori estimate we know that the fluid domain remains
$C^{1,\alpha}$-regular. We use this and estimates for harmonic functions to
conclude that the bound on the pressure implies bound on the curvature (see
Lemma 5.2), which then gives the regularity of the fluid domain $\Omega_{t}$.

Our main result reads as follows. Recall that we assume that the reference
surface $\Gamma\subset\mathbb{R}^{3}$ satisfies the interior and exterior ball
condition with radius $\eta$.

######  Main Theorem.

Assume that $\Omega_{0}$ is a smooth initial set which boundary satisfies
$\partial\Omega_{0}=\\{x+h_{0}(x)\nu_{\Gamma}(x):x\in\Gamma\\}$ with

$\|h_{0}\|_{L^{\infty}(\Gamma)}<\eta$

and let $v_{0}\in C^{\infty}(\Omega_{0};\mathbb{R}^{3})$ be the initial
velocity field. Assume that the system (1.3) has a smooth solution in time-
interval $[0,T)$ and the parametrization satisfies

(1.7) $\Lambda_{T}<\infty\quad\text{and}\quad\sigma_{T}>0,$

where $\sigma_{T}$ is defined in (1.4) and $\Lambda_{T}$ in (1.5). Then there
is $\delta>0$, which depends on $T$, $\sigma_{T}$, $\Lambda_{T}$,
$\|v_{0}\|_{H^{3}(\Omega_{0})}$ and on
$\|H_{\Sigma_{0}}\|_{H^{2}(\Sigma_{0})}$, such that the solution exists for a
time interval $[0,T+\delta)$. We also have the following quantitative
regularity estimate for every $l\geq 1$

$\sup_{t\in[0,T+\delta)}\hat{E}_{l}(t)\leq C_{l},$

where $\hat{E}_{l}(t)$ is defined in (1.6). The constant $C_{l}$ depends on
$l,T$, $\sigma_{T},\Lambda_{T}$ and on $\hat{E}_{l}(0)$, i.e., on
$\hat{E}_{l}(t)$ at time $t=0$. In particular, the system (1.3) has a smooth
solution as long as the a priori estimates (1.7) hold.

Moreover, there is $T_{0}>0$, which depends on $\sigma_{0}$, i.e.,
$\sigma_{T}$ at $T=0$, $\|H_{\Sigma_{0}}\|_{H^{2}(\Sigma_{0})}$ and on
$\|v\|_{H^{3}(\Omega_{0})}$ such that the a priori estimates (1.7) hold for
$T=T_{0}$.

Let us make a few comments on the Main Theorem. First, from the point of view
of the shape of the drop, the result says that if the parametrization of the
flow remains $C^{1,\alpha}$-regular then the flow does not develop
singularities. We expect this to be optimal in the sense that, we cannot relax
the $C^{1,\alpha}$-regularity to Lipschitz regularity as the flow may create
conical singularities as discussed before.

From the point of view of the velocity, the assumption on Lipschitz regularity
of $v$, which is slightly stronger than the boundedness of the
$\operatorname{curl}v$, is similar to the Beale-Kato-Majda criterion and thus
natural in the theory of the Euler equations [40]. Indeed, in the case when
the drop does not chance its shape, i.e., $\Omega_{t}=\Omega_{0}$ the
condition (1.7) reduces to

$\sup_{t\in[0,T)}\|\nabla v(\cdot,t)\|_{L^{\infty}(\Omega_{0})}<\infty,$

which guarantees that the equations do not develop singularities by standard
results for the Euler-equations [40]. Whether one may remove this condition is
beyond our reach at the moment as it is already a fundamental problem in the
theory of the Euler equations without the free boundary. The condition on the
$H^{2}$-integrability of the normal component of the velocity $v$ on the other
hand is related to the fact that the boundary $\Sigma_{t}$ is moving. We do
not expect this to be optimal but again this problem is too involved for us to
solve at the moment. Our main contribution to the problem is to find lower
order sufficient conditions which guarantee that the flow is well-defined and
provide the regularity estimates of all order $l$, which is new even in the
absence of the electric field.

Finally the last statement of the Main Theorem says that the first statement
is not empty, i.e., that the a priori estimates stay bounded up to time
$T_{0}$, which depends on the initial data by requiring that
$\|H_{\Sigma_{0}}\|_{H^{2}(\Sigma_{0})}$ and $\|v_{0}\|_{H^{3}(\Omega_{0})}$
are bounded. Thus we improve the result in [49] in the absence of the electric
field, i.e., in the case $Q=0$. We also note that since the regularity
estimates in Main Theorem are quantitative, the result can be applied for non-
smooth initial data by standard approximation.

###  1.3. Overview of the proof and the structure of the paper

As the paper is long we give a brief overview of the proof of the Main Theorem
and of the structure of the paper. The proof is based on energy estimates
which are standard in this context [11, 48, 49]. We define the energy
functional of order $l\geq 1$ as

$\begin{split}\mathcal{E}_{l}(t)=\frac{1}{2}\int_{\Omega_{t}}&|\mathcal{D}_{t}^{l+1}v|^{2}\,dx+\frac{1}{2}\int_{\Sigma_{t}}|\nabla_{\tau}(\mathcal{D}_{t}^{l}v\cdot\nu)|^{2}\,d\mathcal{H}^{2}\\\
&-\frac{Q}{2C_{t}^{2}}\int_{\Omega_{t}^{c}}|\nabla(\partial_{t}^{l+1}U_{\Omega_{t}})|^{2}\,dx+\int_{\Omega_{t}}|\nabla^{\lfloor\frac{1}{2}(3l+1)\rfloor}(\operatorname{curl}v)|^{2}\,dx,\end{split}$

which is similar to the quantity in [48] defined on graphs. Here $v$ is the
velocity, $\mathcal{D}_{t}^{l}v$ is the material derivative of order $l$ and
$\lfloor\frac{1}{2}(3l+1)\rfloor$ denotes the largest integer smaller than
$\frac{1}{2}(3l+1)$. Note that we need an additional term involving the time
derivative of the capacitary potential $U_{\Omega_{t}}$, as it appears as a
high order term in the linearization of the pressure (see Lemma 4.7). This
additional term causes problems as it is not immediately clear why the energy
is positive or even bounded from below. We also define the associated energy
quantity, where we include the spatial regularity

$E_{l}(t)=\sum_{k=0}^{l}\|\mathcal{D}_{t}^{l+1-k}v\|_{H^{\frac{3}{2}k}(\Omega_{t})}^{2}+\|v\|_{H^{\lfloor\frac{3}{2}(1+1)\rfloor}(\Omega_{t})}^{2}+\|\mathcal{D}_{t}^{l}v\cdot\nu\|_{H^{1}(\Sigma_{t})}^{2}+1.$

Note that this quantity takes into account the natural scaling of the system
(1.3), where time scales of order $\frac{3}{2}$ with respect to space as
observed e.g. in [49].

We prove high order energy estimates by first showing that if
$\Lambda_{T},\sigma_{T}$ satisfy (1.7) then for all $t<T$ it holds

(1.8) $\frac{d}{dt}\mathcal{E}_{l}(t)\leq CE_{l}(t)$

for $l\geq 2$. The novelty of (1.8) is that the RHS has linear and not
polynomial dependence on $E_{l}(t)$, which is crucial in order to show that
the flow remains smooth as long as (1.7) holds. This makes the proof
challenging as we need to estimate all nonlinear error terms in
$\frac{d}{dt}\mathcal{E}_{l}(t)$ in an optimal way. We complete the argument
by proving

(1.9) $E_{l}(t)\leq C(C_{0}+\mathcal{E}_{l}(t))$

which holds for $l\geq 1$. The inequalities (1.8) and (1.9) then imply the
energy estimates for $l\geq 2$ and the quantitative $C^{\infty}$-regularity of
the flow.

We prove (1.8) and (1.9) by an induction argument over $l$, where the
constants depend on $\sup_{t<T}E_{l-1}(t)$ which is bounded by the previous
step. Therefore the first challenge is to start the argument and to bound
$E_{1}(t)$. The issue is that (1.8) does not hold for $l=1$. Instead, we show
a weaker estimate

(1.10) $\frac{d}{dt}\mathcal{E}_{1}(t)\leq
C(1+\|p\|_{H^{2}(\Omega_{t})}^{2})E_{1}(t),$

which we expect to be sharp. Therefore in order to start the induction
argument we use an ad-hoc argument to show

(1.11) $\int_{0}^{T}\|p\|_{H^{2}(\Omega_{t})}^{2}\,dt\leq C.$

The inequalities (1.9), (1.10) and (1.11) then imply the first order energy
estimate.

We show (1.11) by studying the function

$\Phi(t)=-\int_{\Sigma_{t}}p\,\Delta_{\Sigma_{t}}v_{n}\,d\mathcal{H}^{2},$

where $p$ is the pressure and $\Delta_{\Sigma_{t}}$ the Laplace-Beltrami
operator, and prove that it holds

$\frac{d}{dt}\Phi(t)\leq-\frac{1}{3}\|p\|_{H^{2}(\Omega_{t})}^{2}+\text{lower
order terms}.$

We show that the a priori estimates (1.7) imply that $\Phi$ is bounded and
thus we obtain (1.11) by integrating the above inequality over $(0,T)$. We
point out that the low order energy estimate is the most challenging part of
the proof as we have to work with domains with low regularity. We need rather
deep results from differential geometry, boundary regularity for harmonic
functions and elliptic regularity in order to overcome this problem. Let us
finally outline the structure of the paper.

In Section 2 we introduce our notation. Due to the presence of the surface
tension, the problem is geometrically involved and we need notation and tools
from differential geometry to overcome these issues. We also define the
function spaces that we need which include the Hilbert spaces in the domain
$H^{k}(\Omega)$ and on the boundary $H^{k}(\Sigma)$ for half-integers
$k=0,\frac{1}{2},1,\dots$. We also recall functional inequalities such as
interpolation inequality and Kato-Ponce inequality.

In Section 3 we prove div-curl type estimates in order to transform the high
order energy estimates into regularity for the shape and the velocity. We
first recall the result from [10] and prove its lower order version in Theorem
3.6. In Theorem 3.9 we prove sharp boundary regularity estimates for harmonic
functions with Dirichlet boundary data by using methods from [22]. We believe
that these two results are of independent interest.

In Section 4 we derive commutation formulas as in [49] and formula for the
material derivatives of the pressure on the moving boundary. These formulas
include four different error terms which we bound in Section 5. All the error
terms have different structure and therefore we need to treat them one by one,
which makes the Section 5 long. The further difficulty is due to the fact that
the time and space derivatives have different scaling.

The core of the proof of the Main Theorem is in the next three sections. In
Section 6 we prove (1.11), in Section 7 we prove (1.8) and (1.10), and in
Section 8 we prove (1.9). The short final section then contains the proof of
the Main Theorem.

##  2\. Notation and Preliminary results

In this section we introduce our notation and recall some basic results on
function spaces and geometric inequalities. Many of these results are well-
known for experts but we include them since they might be difficult to find,
while some results we did not find at all in the existing literature.

We first introduce notation related to Riemannian geometry. As an introduction
to the topic we refer to [38]. We will always deal with compact hypersurfaces
$\Sigma\subset\mathbb{R}^{3}$, which then can be seen as boundaries of sets
$\Omega$, i.e., $\partial\Omega=\Sigma$. We denote its outer unit normal by
$\nu_{\Omega}$ and denote it sometimes by $\nu_{\Sigma}$ or merely $\nu$ when
its meaning is obvious from the context. We use the outward orientation and
denote the second fundamental form by $B_{\Sigma}$ and the mean curvature by
$H_{\Sigma}$, which is defined as the sum of the principle curvatures. Again
we write simply $B$ and $H$ when the meaning is clear from the context. We
note that we use the convention in our notation that $\Sigma=\partial\Omega$
denotes a generic surface, $\Sigma_{t}=\partial\Omega_{t}$ denotes the
evolving surface given by the equations (1.3) and $\Gamma=\partial G$ is our
reference surface which we introduce later.

Since $\Sigma$ is embedded in $\mathbb{R}^{3}$ it has natural metric $g$
induced by the Euclidian metric. Then $(\Sigma,g)$ is a Riemannian manifold
and we denote the inner product on each tangent space $X,Y\in T_{x}\Sigma$ by
$\langle X,Y\rangle$, which we may write in local coordinates as

$\langle X,Y\rangle=g(X,Y)=g_{ij}X^{i}Y^{j}.$

We extend the inner product in a natural way for tensors. We denote smooth
vector fields on $\Sigma$ by $\mathscr{T}(\Sigma)$ and by a slight abuse of
notation we denote smooth $k$:th order tensor fields on $\Sigma$ by
$\mathscr{T}^{k}(\Sigma)$. We write $X^{i}$ for vectors and $Z_{i}$ for
covectors in local coordinates.

We denote the Riemannian connection on $\Sigma$ by $\bar{\nabla}$ and recall
that for a function $u\in C^{\infty}(\Sigma)$ the covariant derivative
$\bar{\nabla}u$ is a $1$-tensor field defined for $X\in\mathscr{T}(\Sigma)$ as

$\bar{\nabla}u(X)=\bar{\nabla}_{X}u=Xu,$

i.e., the derivative of $u$ in the direction of $X$. The covariant derivative
of a smooth $k$-tensor field $F\in\mathscr{T}^{k}(\Sigma)$, denoted by
$\bar{\nabla}F$, is a $(k+1)$-tensor field and we have the following recursive
formula for $Y_{1},\dots,Y_{k},X\in\mathscr{T}(\Sigma)$

$\bar{\nabla}F(Y_{1},\dots,Y_{k},X)=(\bar{\nabla}_{X}F)(Y_{1},\dots,Y_{k}),$

where

$(\bar{\nabla}_{X}F)(Y_{1},\dots,Y_{k})=XF(Y_{1},\dots,Y_{k})-\sum_{i=1}^{k}F(Y_{1},\dots,\bar{\nabla}_{X}Y_{i},\dots,Y_{k}).$

Here $\bar{\nabla}_{X}Y$ is the covariant derivative of $Y$ in the direction
of $X$ (see [38]) and since $\bar{\nabla}$ is the Riemannian connection it
holds $\bar{\nabla}_{X}Y=\bar{\nabla}_{Y}X+[X,Y]$ for every
$X,Y\in\mathscr{T}(\Sigma)$. We denote the $k$:th order covariant derivative
of a function $u$ on $\Sigma$ by
$\bar{\nabla}^{k}u\in\mathscr{T}^{k}(\Sigma)$. The notation
$\bar{\nabla}_{i_{1}}\cdots\bar{\nabla}_{i_{k}}u$ means a coefficient of
$\bar{\nabla}^{k}u$ in local coordinates. We may raise the index of
$\bar{\nabla}_{i}u$ by using the inverse of the metric tensor $g^{ij}$ as
$\bar{\nabla}^{i}u=g^{ij}\bar{\nabla}_{j}u$. We denote the divergence of a
vector field $X\in\mathscr{T}(\Sigma)$ by $\operatorname{div}_{\Sigma}X$ and
the Laplace-Beltrami operator for a function $u:\Sigma\to\mathbb{R}$ by
$\Delta_{\Sigma}u$. We recall that by the divergence theorem

$\int_{\Sigma}\operatorname{div}_{\Sigma}X\,d\mathcal{H}^{2}=0.$

We will first fix our reference surface which we denote by $\Gamma$ which is a
boundary of a smooth, compact set $G$, i.e., $\Gamma=\partial G$. Since $G$ is
smooth it satisfies the interior and exterior ball condition with radius
$\eta$, and we denote the tubular neighborhood of $\Gamma$ by
$\mathcal{N}_{\eta}(\Gamma)$ which is defined as

$\mathcal{N}_{\eta}(\Gamma)=\\{x\in\mathbb{R}^{3}:\text{dist}(x,\Gamma)<\eta\\}.$

Then the map $\Psi:\Gamma\times(-\eta,\eta)\to\mathcal{N}_{\eta}(\Gamma)$
defined as $\Psi(x,s)=x+s\nu_{\Gamma}(x)$ is a diffeomorphism. We say that a
hypersurface $\Sigma$, or a domain $\Omega$ with $\partial\Omega=\Sigma$, is
$C^{1,\alpha}(\Gamma)$-regular for some small $\alpha>0$, when it can be
written as

$\Sigma=\\{x+h(x)\nu_{\Gamma}(x):x\in\Gamma\\},$

for a $C^{1,\alpha}(\Gamma)$-regular function $h:\Gamma\to\mathbb{R}$ with
$\|h\|_{L^{\infty}}<\eta$. In particular, all $C^{1,\alpha}(\Gamma)$-regular
sets are diffeomorphic. We say that a set $\Sigma$ is uniformly
$C^{1,\alpha}(\Gamma)$-regular if the height-function satisfies
$\|h\|_{C^{1,\alpha}(\Gamma)}\leq C$ and $\|h\|_{L^{\infty}}\leq c\eta$ for
constants $C$ and $c<1$. Finally we say that $\Sigma$ is uniformly
$C^{1}$-regular if $\|h\|_{C^{1}(\Gamma)}\leq C$.

Let us next fix our notation in the ambient space $\mathbb{R}^{3}$. We denote
the $k$:th order differential of a vector field
$F:\mathbb{R}^{3}\to\mathbb{R}^{m}$ by $\nabla^{k}F$, the divergence of
$F:\mathbb{R}^{3}\to\mathbb{R}^{3}$ by $\operatorname{div}F$ and the Laplace
operator in $\mathbb{R}^{3}$ by $\Delta$. The notation $(\nabla F)^{T}$ stands
for the transpose of $\nabla F$. When we restrict
$F:\mathbb{R}^{3}\to\mathbb{R}^{3}$ on $\Sigma$, we define its normal and
tangential part as

$F_{n}:=F\cdot\nu_{\Sigma}\qquad\text{and}\qquad
F_{\tau}=F-F_{n}\,\nu_{\Sigma}.$

We use the notation $x\cdot y$ for the inner product of two vectors in
$\mathbb{R}^{n}$.

Since $\Sigma$ is a smooth hypersurface we may extend every function and
vector field defined on $\Sigma$ to $\mathbb{R}^{3}$. We may thus define a
tangential differential of a vector field $F:\Sigma\to\mathbb{R}^{m}$ by

$\nabla_{\tau}F=\nabla F-(\nabla F\nu_{\Sigma})\otimes\nu_{\Sigma}$

where we have extended $F$ to $\mathbb{R}^{3}$. We may then extend the
definition of $\operatorname{div}_{\Sigma}$ to fields
$F:\Sigma\to\mathbb{R}^{3}$ by
$\operatorname{div}_{\Sigma}F=\text{Tr}(\nabla_{\tau}F)$ and the divergence
theorem generalizes to

$\int_{\Sigma}\operatorname{div}_{\Sigma}F\,d\mathcal{H}^{2}=\int_{\Sigma}H_{\Sigma}(F\cdot\nu_{\Sigma})\,d\mathcal{H}^{2}.$

We note that the tangential gradient of $u\in C^{\infty}(\Sigma)$ is
equivalent to its covariant derivative in the sense that for every vector
field $X\in\mathscr{T}(\Sigma)$ we find a vector field
$\tilde{X}:\Sigma\to\mathbb{R}^{3}$ which satisfies
$\tilde{X}\cdot\nu_{\Sigma}=0$ and

$\bar{\nabla}_{X}u=\nabla_{\tau}u\cdot\tilde{X}.$

Let us comments briefly on the notation related to the equations (1.3). We
denote the derivative with respect to time by $\partial_{t}F$ and the material
derivative as

$\mathcal{D}_{t}F:=\partial_{t}F+(v\cdot\nabla)F.$

The material derivative does not commute with the spatial derivative and we
denote the commutation

$[\mathcal{D}_{t},\nabla]u=\mathcal{D}_{t}\nabla u-\nabla\mathcal{D}_{t}u.$

We denote by $U_{\Omega}$ the capacitary potential defined in (1.2) and denote
$U_{t}=U_{\Omega_{t}}$, $H_{t}=H_{\Sigma_{t}}$ etc… when the meaning is clear
from the context. To shorten further the notation we denote

(2.1) $Q(t):=\frac{Q}{(\text{Cap}(\Omega_{t}))^{2}}.$

We may thus write the pressure in (1.3) as

$p=H_{t}-\frac{Q(t)}{2}|\nabla U_{t}|^{2}.$

Let us next fix the notation for the function spaces. We define the Sobolev
space $W^{l,p}(\Sigma)$ in a standard way for $p\in[1,\infty]$, see e.g. [4],
denote the Hilbert space $H^{l}(\Sigma)=W^{l,2}(\Sigma)$ and define the
associated norm for $u\in W^{l,p}(\Sigma)$ as

$\|u\|_{W^{l,p}(\Sigma)}^{p}=\sum_{k=0}^{l}\int_{\Sigma}|\bar{\nabla}^{k}u|^{p}\,d\mathcal{H}^{2}$

and for $p=\infty$

$\|u\|_{W^{l,\infty}(\Sigma)}=\sum_{k=0}^{l}\sup_{x\in\Sigma}|\bar{\nabla}^{k}u|.$

We often denote
$\|u\|_{C^{0}(\Sigma)}=\|u\|_{L^{\infty}(\Sigma)}=\sup_{x\in\Sigma}|u(x)|$ for
continuous function $u:\Sigma\to\mathbb{R}$ and
$\|u\|_{C^{m}(\Sigma)}=\|u\|_{W^{m,\infty}(\Sigma)}$ . We define the Hölder
norm of a continuous function $u:\Sigma\to\mathbb{R}$ by

$\|u\|_{C^{\alpha}(\Sigma)}=\|u\|_{L^{\infty}(\Sigma)}+\sup_{\begin{subarray}{c}x\neq
y\\\ x,y\in\Sigma\end{subarray}}\frac{|u(y)-u(x)|}{|y-x|^{\alpha}}.$

We define the Hölder norm for a tensor field $F\in\mathscr{T}^{k}(\Sigma)$ as
in [35]

$\|F\|_{C^{\alpha}(\Sigma)}=\sup\\{\|F(X_{1},\dots,X_{k})\|_{C^{\alpha}(\Sigma)}:X_{i}\in\mathscr{T}(\Sigma)\,\text{
with }\,\|X_{i}\|_{C^{1}(\Sigma)}\leq 1\\}.$

Finally we define the $H^{-1}(\Sigma)$-norm by duality, i.e.,

$\|u\|_{H^{-1}(\Sigma)}:=\sup\Big{\\{}\int_{\Sigma}u\,g\,d\mathcal{H}^{2}:\|g\|_{H^{1}(\Sigma)}\leq
1\Big{\\}}.$

For functions defined in the domain $u:\Omega\to\mathbb{R}$ we define the
Sobolev space $W^{l,p}(\Omega)$ as functions which have $k$:th order weak
derivative in $\Omega$ and the corresponding norm is bounded

$\|u\|_{W^{l,p}(\Omega)}^{p}:=\sum_{k=0}^{l}\int_{\Omega}|\nabla^{k}u|^{p}\,dx<\infty.$

As before we denote the Hilbert space as $H^{l}(\Omega)=W^{l,2}(\Omega)$ and
define $H^{-1}(\Omega)$ by duality. Finally given an index vector
$\alpha=(\alpha)_{i=1}^{k}\in\mathbb{N}^{k}$ we define its norm by

$|\alpha|=\sum_{i=1}^{k}\alpha_{i}.$

Throughout the paper we use the notation $S\star T$ from [31, 41] to denote a
tensor formed by contraction on some indexes of tensors $S$ and $T$, using the
coefficients of the metric tensor $g_{ij}$ if $S$ and $T$ are defined on the
boundary $\Sigma$. We also use the convention that
$\bar{\nabla}^{k}u\star\bar{\nabla}^{l}v$ denotes contraction of some indexes
of tensors $\bar{\nabla}^{i}u$ and $\bar{\nabla}^{j}v$ for any $i\leq k$ and
$j\leq l$. In other words, we include also the lower order covariant
derivatives.

Following the notation from [52], we first introduce the real interpolation
method and then the interpolation spaces. Let $X$ and $Y$ be Banach spaces
endowed respectively with the norm $\|\cdot\|_{X}$ and $\|\cdot\|_{Y}$. The
couple $(X,Y)$ is said to be an interpolation couple if both $X$ and $Y$ are
embedded in a Hausdorff topological vector space $V$. In this case we have
that $X\cap Y$ endowed with the norm $\|v\|_{X\cap
Y}=\max\\{\|v\|_{X},\|v\|_{Y}\\}$ is a Banach space. Moreover, we also have
that $X+Y=\\{z=x+y,\,x\in X,\,y\in Y\\}$ endowed with the norm

$\|z\|_{X+Y}=\inf_{x\in X,\,y\in Y}\\{\|x\|_{X}+\|y\|_{Y},\,z=x+y\\}$

is a Banach space and it is immediate to check that

$X\cap Y\subset X,Y\subset X+Y.$

For $z\in X+Y$ and $t>0$ we introduce the $K$ functional

$K(t,z,X,Y)=\inf_{x\in X,\,y\in Y}\\{\|x\|_{X}+t\|y\|_{Y},\,x+y=z\\}.$

For $\theta\in(0,1)$, $p\in[1,\infty)$ and $z\in X+Y$ we let

$\|z\|^{p}_{\theta,p}=\int_{0}^{\infty}\left(\frac{K(t,z,X,Y)}{t^{\theta}}\right)^{p}\frac{dt}{t}$

and define

$(X,Y)_{\theta,p}=\\{z\in X+Y:\|z\|_{\theta,p}<\infty\\}.$

We note that $(X,Y)_{\theta,p}$ is a Banach space.

Finally, we recall that if $(X_{1},X_{2})$ and $(Y_{1},Y_{2})$ are
interpolation couples and $\mathcal{F}:X_{1}+X_{2}\to Y_{1}+Y_{2}$ is a linear
operator which is bounded $X_{i}\to Y_{i}$ by $M_{i}$. Then for
$\theta\in(0,1)$ and $p\in[1,\infty)$ the operator

(2.2) $\mathcal{F}:(X_{1},X_{2})_{\theta,p}\to(Y_{1},Y_{2})_{\theta,p}$

is bounded and we may estimate its norm by $M_{1}^{1-\theta}M_{2}^{\theta}$.

###  2.1. Half-integer Sobolev spaces

Before giving the definition of half-integer Sobolev space in a domain, we
exploit the extension properties of Sobolev functions. Throughout the paper we
assume that the boundary $\Sigma=\partial\Omega$ is
$C^{1,\alpha}(\Gamma)$-regular and thus it is $H^{1}$ extension domain, i.e.,
there is a linear operator $T:H^{1}(\Omega)\to H^{1}(\mathbb{R}^{3})$ such
that

$\|T(u)\|_{H^{1}(\mathbb{R}^{3})}\leq C\|u\|_{H^{1}(\Omega)}.$

We refer to [9] for the study of Sobolev spaces under Lipschitz-regularity and
the references therein.

We need more regularity for the boundary for higher order Sobolev extension
$m\geq 2$, although we do not need the optimal condition. Instead, we assume
the following for the second fundamental form

(Hm) $\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq C\quad\text{ if
}\,m=2,\qquad\,\|B_{\Sigma}\|_{L^{\infty}(\Sigma)}+\|B_{\Sigma}\|_{H^{m-2}(\Sigma)}\leq
C\quad\text{ if }\,m>2,$

which guarantees that we may extend a given function $u\in H^{m}(\Omega)$ to
the whole space. Note that for $m\geq 4$ the condition (Hm) reduces to
$\|B_{\Sigma}\|_{H^{m-2}(\Sigma)}\leq C$ by the Sobolev-embedding, which
agrees with the assumption e.g. in [10].

######  Proposition 2.1.

Let $m\in\mathbb{N}$, with $m\geq 2$, and let $\Omega$ be a smooth domain
which is uniformly $C^{1,\alpha}(\Gamma)$-regular and satisfies (Hm). Then
there is an extension operator $T:H^{m}(\Omega)\to H_{0}^{m}(\mathbb{R}^{3})$
such that

$\|T(u)\|_{H^{m}(\mathbb{R}^{3})}\leq C_{m}\|u\|_{H^{m}(\Omega)}.$

###### Proof.

Let $x_{0}\in\partial\Omega$. There exist $\delta>0$ and a diffeomorphism
$\Psi:\mathbb{R}^{3}\to\mathbb{R}^{3}$ such that $\Psi^{-1}(\Omega\cap
B_{\delta}(x_{0}))=B_{1}^{+}=B_{1}\cap\mathbb{R}_{+}^{3}$. We note that since
$\Omega$ is $C^{1,\alpha}(\Gamma)$-regular and satisfies (Hm) we may choose
the diffeomorphism such that it satisfies
$\|\Psi\|_{C^{1,\alpha}(\mathbb{R}^{3})}\leq C$ and

(2.3) $\|\nabla^{2}\Psi\|_{L^{4}(\mathbb{R}^{3})}\leq C\quad\text{ if
}\,m=2,\qquad\|\Psi\|_{H^{m}(\mathbb{R}^{3})}\leq C\quad\text{ if }\,m>2.$

For $u:\Omega\to\mathbb{R}$ smooth we let $u^{\prime}=u\circ\Psi$. We may
extend $u^{\prime}$ to a function $T(u^{\prime})\in H^{m}(B_{1})$ such that

$\|T(u^{\prime})\|_{H^{m}(B_{1})}\leq C\|u^{\prime}\|_{H^{m}(B_{1}^{+})}.$

The construction of $T(u^{\prime})\in H^{m}(B_{1})$ is classical but we recall
it for the reader’s convenience. We define

$T(u^{\prime})(x^{\prime},x_{n})=\begin{cases}u^{\prime}(x^{\prime},x_{n})\,\,\,\,\
&x_{n}\geq 0\\\
\sum_{j=1}^{m+1}\lambda_{j}u^{\prime}(x^{\prime},-j^{-1}x_{n})\,\,\,\,\,&x_{n}<0\end{cases}$

where $\lambda=(\lambda_{1},\dots,\lambda_{m+1})$ solve the system

$\begin{cases}\sum_{j}\lambda_{j}=1\\\ \sum_{j}(-j)^{-1}\lambda_{j}=1\\\
\vdots\\\ \sum_{j}(-j)^{-m}\lambda_{j}=1.\end{cases}$

This system, known as Vandermonde system, has a unique solution, hence
$Tu^{\prime}$ is well defined. Finally we define the extension operator as

$T(u):=T(u^{\prime})\circ\Psi^{-1}.$

Let us show that $T$ is a bounded operator.

It is straightforward to check that

$\|Tu^{\prime}\|_{H^{m}(B_{1})}\leq C\|u^{\prime}\|_{H^{m}(B_{1}^{+})}.$

Let us then show that

(2.4) $\|u^{\prime}\|_{H^{m}(B_{1}^{+})}\leq C\|u\|_{H^{m}(\Omega)}.$

We first note that

$\nabla u^{\prime}=\nabla u\star\nabla\Psi$

and for $m\geq 2$

$\nabla^{m}u^{\prime}=\sum_{|\alpha|\leq
m-1}\nabla^{1+\alpha_{1}}\Psi\star\cdots\star\nabla^{1+\alpha_{m}}\Psi\star\nabla^{1+\alpha_{m+1}}u.$

For $m=2$ we have then by Hölder’s inequality, by (2.3) and by the Sobolev
embedding

$\|\nabla^{2}u^{\prime}\|_{L^{2}(B_{1})}\leq
C\|\Psi\|_{C^{1}(\mathbb{R}^{3})}\|\nabla^{2}u\|_{L^{2}(\Omega)}+\|\nabla^{2}\Psi\|_{L^{4}(\mathbb{R}^{3})}\|\nabla
u\|_{L^{4}(\Omega)}\leq C\|u\|_{H^{2}(\Omega)}.$

To treat the case $m\geq 3$ we first observe that by Sobolev embedding it
holds

$\|\nabla^{m-2}u\|_{L^{\infty}(B_{1}^{+})}\leq\|u\|_{H^{m}(\Omega)}\quad\text{and}\quad\|\nabla^{m-2}\Psi\|_{L^{\infty}(B_{1}^{+})}\leq\|\Psi\|_{H^{m}(\mathbb{R}^{3})}.$

Hence for $m\geq 3$ we have by Hölder’s inequality, by (2.3) and by the
Sobolev embedding

$\begin{split}\|\nabla^{m}u^{\prime}\|_{L^{2}(B_{1})}&\leq
C\big{(}(1+\|\Psi\|_{H^{m}(\mathbb{R}^{3})}^{m})\|u\|_{H^{m}(\Omega)}+\|\nabla^{2}\Psi\|_{L^{4}(\mathbb{R}^{3})}\|\nabla^{m-1}u\|_{L^{4}(\Omega)}\\\
&\,\,\,\,\,\,\,+\|\nabla^{m-1}\Psi\|_{L^{4}(\mathbb{R}^{3})}\|\nabla^{2}u\|_{L^{4}(\Omega)}\big{)}\\\
&\leq C(1+\|\Psi\|_{H^{m}(\mathbb{R}^{3})}^{m})\|u\|_{H^{m}(\Omega)}\\\ &\leq
C\|u\|_{H^{m}(\Omega)}.\end{split}$

Thus we have (2.4).

Similarly we show that

$\|T(u^{\prime})\circ\Psi^{-1}\|_{H^{m}(\Omega\cap B_{\delta}(x_{0}))}\leq
C\|Tu^{\prime}\|_{H^{m}(B_{1})}.$

The claim then follows from standard covering argument. ∎

Throughout the paper we will refer to the operator $T$ as the canonical
extension operator or simply as the extension operator. We define the half-
integer Sobolev space in the domain $\Omega$ using the canonical extension
operator.

######  Definition 2.2.

We say that a function $u\in L^{2}(\Omega)$ is in $H^{\frac{1}{2}}(\Omega)$ if

$\|u\|_{H^{\frac{1}{2}}(\Omega)}:=\|u\|_{L^{2}(\Omega)}+\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{|Tu(x)-Tu(y)|^{2}}{|x-y|^{3}}\,dxdy<\infty.$

For $k\geq 1$ we say that $u\in H^{k+\frac{1}{2}}(\Omega)$ if $u\in
H^{k}(\Omega)$ and

$\|u\|_{H^{k+\frac{1}{2}}(\Omega)}:=\|u\|_{H^{k}(\Omega)}+\|T(\nabla^{k}u)\|_{H^{\frac{1}{2}}(\Omega)}<\infty.$

Finally we define the $H^{-\frac{1}{2}}(\Omega)$-norm by duality.

We define $H^{\frac{1}{2}}_{\star}(\Omega)$ as the space of functions via
interpolation such that $u\in H^{\frac{1}{2}}_{\star}(\Omega)$ if
$T(u)\in(L^{2}(\mathbb{R}^{3}),H^{1}(\mathbb{R}^{3}))_{\frac{1}{2},2}$ and
endow it with the norm

(2.5)
$\|u\|_{H_{\star}^{\frac{1}{2}}(\Omega)}^{2}:=\|u\|_{L^{2}(\Omega)}^{2}+\int_{0}^{\infty}\left(\frac{K(t,T(u),L^{2}(\mathbb{R}^{3}),H^{1}(\mathbb{R}^{3}))}{t^{1/2}}\right)^{2}\,\frac{dt}{t}.$

This gives an equivalent definition for the half-integer Sobolev space.

######  Proposition 2.3.

Let $m\in\mathbb{N}$, with $m\geq 2$, and let $\Omega$ be a smooth domain
which is uniformly $C^{1,\alpha}(\Gamma)$-regular and satisfies (Hm). The
norms in Definition 2.2 and (2.5) are equivalent, i.e.,

$\|u\|_{H^{\frac{1}{2}}(\Omega)}\simeq\|u\|_{H_{\star}^{\frac{1}{2}}(\Omega)}.$

We do not give the details of the proof, but only refer to [9] and mention
that it follows from the fact that
$H^{\frac{1}{2}}(\mathbb{R}^{3})=(L^{2}(\mathbb{R}^{3}),H^{1}(\mathbb{R}^{3}))_{\frac{1}{2},2}$,
see [52].

###  2.2. Half-integer Sobolev spaces on a surfaces

We begin by defining the space $H^{\frac{1}{2}}(\Sigma)$. Again there are many
ways to do this. We choose the definition via harmonic extension.

######  Definition 2.4.

Let $\Sigma=\partial\Omega$ be uniformly $C^{1,\alpha}(\Gamma)$-regular. We
say that $u\in H^{\frac{1}{2}}(\Sigma)$ if $u\in L^{2}(\Sigma)$ and

$\|u\|_{H^{\frac{1}{2}}(\Sigma)}=\|u\|_{L^{2}(\Sigma)}+\inf\\{\|\nabla
v\|_{L^{2}(\Omega)}:\,v-u\in H^{1}_{0}(\Omega)\\}<\infty.$

We define the space $H^{-\frac{1}{2}}(\Sigma)$ and its norm by duality.

By standard theory the $C^{1,\alpha}(\Gamma)$-regularity of $\Sigma$ ensures
that Definition 2.4 is equivalent to the definition via Gagliardo seminorm

$\|u\|_{H^{\frac{1}{2}}(\Sigma)}^{2}\simeq\|u\|_{L^{2}(\Sigma)}^{2}+\int_{\Sigma}\int_{\Sigma}\frac{|u(x)-u(y)|^{2}}{|x-y|^{3}}\,d\mathcal{H}^{2}_{x}d\mathcal{H}^{2}_{y}.$

Moreover, this norm is also equivalent to the norm obtained via interpolation.
Indeed, let us define the interpolation space (see beginning of Section 2)

$H^{\frac{1}{2}}_{\star}(\Sigma)=(L^{2}(\Sigma),H^{1}(\Sigma))_{\frac{1}{2},2}.$

Let us show that

(2.6)
$\|u\|_{H^{\frac{1}{2}}(\Sigma)}\simeq\|u\|_{H_{\star}^{\frac{1}{2}}(\Sigma)}.$

Due to the non-local nature of the problem we give the proof of (2.6) in
detail.

We fix a small $\delta>0$ and cover $\Sigma$ with finitely many balls of
radius $\delta$ centered at $x_{i}\in\Sigma$, i.e.,

$\Sigma\subset\bigcup_{i=1}^{N}B_{\delta}(x_{i}).$

Since $\Sigma$ is $C^{1,\alpha}(\Gamma)$, there are $C^{1,\alpha}$-regular
functions $\phi_{i}$ such that $\Sigma\cap B_{2\delta}(x_{i})$ is contained in
the graph of $\phi_{i}$ for every $i=1,\dots,N$, when $\delta$ is small
enough. Let $\\{\eta_{i}\\}_{i=1,\dots,N}$ be a partition of unity
subordinated to the open covering $B_{\delta}(x_{i})$. Then it holds

$\|u\|_{H^{\frac{1}{2}}(\Sigma)}\leq\sum_{i=1}^{N}\|\eta_{i}u\|_{H^{\frac{1}{2}}(\Sigma)}.$

Let us fix $i=1,\dots,N$ and by rotating and translating the coordinates we
may assume that $x_{i}=0$ and $\Sigma\cap
B_{2\delta}\subset\\{(x^{\prime},\phi_{i}(x^{\prime})):x^{\prime}\in\mathbb{R}^{2}\\}$
with $\phi_{i}(0)=0$ and $\nabla\phi_{i}(0)=0$. Denote $u_{i}=\eta_{i}u$ and
$v_{i}(x^{\prime})=u_{i}(x^{\prime},\phi_{i}(x^{\prime}))$. Note that then
$v_{i}:\mathbb{R}^{2}\to\mathbb{R}$, $\text{supp}\,v_{i}\subset
B_{\delta^{\prime}}\subset\mathbb{R}^{2}$ and $\text{supp}\,u_{i}\subset
B_{\delta}\subset\mathbb{R}^{3}$ with $\delta/2\leq\delta^{\prime}\leq\delta$.
Therefore we deduce by the $C^{1,\alpha}$-regularity of $\phi_{i}$ that

$\begin{split}&\int_{\Sigma}\int_{\Sigma}\frac{|u_{i}(x)-u_{i}(y)|^{2}}{|x-y|^{3}}\,d\mathcal{H}^{2}_{x}\,d\mathcal{H}^{2}_{y}\\\
&=\int_{\Sigma\cap B_{2\delta}}\int_{\Sigma\cap
B_{2\delta}}\frac{|u_{i}(x)-u_{i}(y)|^{2}}{|x-y|^{3}}\,d\mathcal{H}_{x}\,d\mathcal{H}_{y}+2\int_{\Sigma\setminus
B_{2\delta}}\int_{\Sigma\cap
B_{2\delta}}\frac{|u_{i}(x)-u_{i}(y)|^{2}}{|x-y|^{3}}\,d\mathcal{H}_{x}\,d\mathcal{H}_{y}\\\
&\leq
C\int_{B_{2\delta^{\prime}}}\int_{B_{2\delta^{\prime}}}\frac{|u_{i}(x^{\prime},\phi_{i}(x^{\prime}))-u_{i}(y^{\prime},\phi_{i}(y^{\prime}))|^{2}}{(|x^{\prime}-y^{\prime}|^{2}+(\phi_{i}(x^{\prime})-\phi_{i}(y^{\prime})^{2})^{3/2}}\,dx^{\prime}\,dy^{\prime}+C\|u\|_{L^{2}(\Sigma)}^{2}\\\
&\leq
C\|v_{i}\|_{H^{\frac{1}{2}}(\mathbb{R}^{2})}^{2}+C\|u\|_{L^{2}(\Sigma)}^{2}\\\
&\leq
C\|v_{i}\|_{H_{\star}^{\frac{1}{2}}(\mathbb{R}^{2})}^{2}+C\|u\|_{L^{2}(\Sigma)}^{2}.\end{split}$

This implies $\|u_{i}\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C\|v_{i}\|_{H_{\star}^{\frac{1}{2}}(\mathbb{R}^{2}))}+C\|u\|_{L^{2}(\Sigma)}$.
Let us denote by $L_{0}^{2}(B_{2\delta^{\prime}})$ and
$H_{0}^{1}(B_{2\delta^{\prime}})$ for functions $f\in L^{2}(\mathbb{R}^{2})$,
and respectively $f\in H^{1}(\mathbb{R}^{2})$, with $\text{supp}f\subset
B_{2\delta^{\prime}}$. Denote also
$\Psi:B_{2\delta^{\prime}}\to\Psi(B_{2\delta^{\prime}})\subset\mathbb{R}^{3}$,
$\Psi(x^{\prime})=(x^{\prime},\phi_{i}(x^{\prime}))$. We may estimate

$\begin{split}&K(t,v_{i},L^{2}(\mathbb{R}^{2}),H^{1}(\mathbb{R}^{2}))\\\
&=\inf_{f+g=v_{i}}\\{\|f\|_{L^{2}(\mathbb{R}^{2})}+t\|g\|_{H^{1}(\mathbb{R}^{2})},\,f\in
L^{2}(\mathbb{R}^{2}),\,g\in H^{1}(\mathbb{R}^{2})\\}\\\
&\leq\inf_{f+g=v_{i}}\\{\|f\|_{L^{2}(\mathbb{R}^{2})}+t\|g\|_{H^{1}(\mathbb{R}^{2})},\,f\in
L_{0}^{2}(B_{2\delta^{\prime}}),\,g\in H_{0}^{1}(B_{2\delta^{\prime}})\\}\\\
&\leq C\inf_{f+g=v_{i}}\\{\|f\circ\Psi^{-1}\|_{L^{2}(\Sigma\cap
B_{2\delta})}+t\|g\circ\Psi^{-1}\|_{H^{1}(\Sigma\cap B_{2\delta})},\,f\in
L_{0}^{2}(B_{2\delta^{\prime}}),\,g\in H_{0}^{1}(B_{2\delta^{\prime}})\\}\\\
&\leq C\inf_{\tilde{f}+\tilde{g}=u_{i}}\\{\|\tilde{f}\|_{L^{2}(\Sigma\cap
B_{2\delta})}+t\|\tilde{g}\|_{H^{1}(\Sigma\cap B_{2\delta})},\,\tilde{f}\in
L_{0}^{2}(\Sigma\cap B_{2\delta}),\,\tilde{g}\in H_{0}^{1}(\Sigma\cap
B_{2\delta})\\}\\\ &=C\,K(t,u_{i},L_{0}^{2}(\Sigma\cap
B_{2\delta}),H_{0}^{1}(\Sigma\cap B_{2\delta})).\end{split}$

Since $\text{supp}\,u_{i}\subset\Sigma\cap B_{\delta}$, it is easy to see that

$K(t,u_{i},L_{0}^{2}(\Sigma\cap B_{2\delta}),H_{0}^{1}(\Sigma\cap
B_{2\delta}))\leq C\,K(t,u_{i},L^{2}(\Sigma),H^{1}(\Sigma)).$

Therefore, we deduce

$\|v_{i}\|_{H_{\star}^{\frac{1}{2}}(\mathbb{R}^{2})}\leq
C\|u_{i}\|_{H_{\star}^{\frac{1}{2}}(\Sigma)}\leq
C\|u\|_{H_{\star}^{\frac{1}{2}}(\Sigma)}.$

Repeating the argument for every $i=1,\dots,N$ yields

$\|u\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C\|u\|_{H_{\star}^{\frac{1}{2}}(\Sigma)}.$

The opposite inequality can be proved in a similar way.

We also note that we may interpolate between $H^{\frac{1}{2}}(\Sigma)$ and its
dual and obtain by using [9, Theorem 4.1] and by a localization argument that

$(H^{-\frac{1}{2}}(\Sigma),H^{\frac{1}{2}}(\Sigma))_{\frac{1}{2},2}=L^{2}(\Sigma).$

In order to define higher order half-integer Sobolev spaces on the boundary we
use the fact that in our setting the boundary $\partial\Omega=\Sigma$ is given
by the parametrization $\Psi_{\Sigma}:\Gamma\to\Sigma$,
$\Psi_{\Sigma}(x)=x+h(x)\nu_{\Gamma}(x)$, where $\Gamma$ is the reference
surface. In our case $\Gamma$ is the boundary of a smooth set $G$ and the map
$\Psi:\Gamma\times(-\eta,\eta)\to\mathcal{N}_{\eta}(\Gamma)$,
$\Psi(x,s)=x+s\nu_{\Gamma}(x)$, is a diffeomorphism. Here
$\mathcal{N}_{\eta}(\Gamma)$ is the tubular neighborhood of $\Gamma$.
Therefore the projection map
$\pi_{\Gamma}:\mathcal{N}_{\eta}(\Gamma)\to\Gamma$ is well defined as

(2.7) $\pi_{\Gamma}(y)=x\qquad\text{where
}\,y=x+s\nu_{\Gamma}(x)\,\,\,\text{for some }\,\,s\in(-\eta,\eta).$

We extend $\pi_{\Gamma}$ to whole $\mathbb{R}^{3}$ and thus we may extend a
given function $u:\Gamma\to\mathbb{R}$ to $\mathbb{R}^{3}$ as
$(u\circ\pi_{\Gamma}):\mathbb{R}^{3}\to\mathbb{R}$. In particular, the $k$:th
order derivative $\nabla^{k}(u\circ\pi_{\Gamma})(x)$ is well defined for all
$x\in\Gamma$, and for $x\in\Gamma$ the function
$s\mapsto(u\circ\pi_{\Gamma})(x+s\nu_{\Gamma}(x))$ is constant for $|s|$
small. We use this extension to define the half-integer Sobolev norm on the
reference surface.

######  Definition 2.5.

For $m\geq 2$ we say that $u\in H^{m-\frac{1}{2}}(\Gamma)$ if $u\in
H^{m-1}(\Gamma)$ and the norm

$\|u\|_{H^{m-\frac{1}{2}}(\Gamma)}:=\|\nabla^{m-1}(u\circ\pi_{\Gamma})\|_{H^{\frac{1}{2}}(\Gamma)}+\|u\|_{H^{m-1}(\Gamma)}$

is bounded.

We define the half-integer Sobolev spaces on $\Sigma$ by mapping a function
$u\in C^{\infty}(\Sigma)$ back to $\Gamma$ by using the parametrization
$\Psi_{\Sigma}:\Gamma\to\Sigma$, $\Psi_{\Sigma}(x)=x+h(x)\nu_{\Gamma}(x)$. Let
us fix $m\geq 2$ and recall that

$\bar{\nabla}^{m}(u\circ\Psi_{\Sigma})=\sum_{|\alpha|\leq
m-1}\bar{\nabla}^{1+\alpha_{1}}\Psi_{\Sigma}\star\cdots\star\bar{\nabla}^{1+\alpha_{k}}\Psi_{\Sigma}\star\bar{\nabla}^{1+\alpha_{m+1}}u.$

If $\Sigma$ satisfies (Hm), then arguing as in the proof of Proposition 2.1 we
deduce

$\|u\|_{H^{m}(\Sigma)}\simeq\|u\circ\Psi_{\Sigma}\|_{H^{m}(\Gamma)}.$

Based on this we define the half-integer Sobolev space of order $m-1/2$ on
$\Sigma$ in the following way.

######  Definition 2.6.

Let $m\geq 2$ be an integer and assume $\Sigma$ is
$C^{1,\alpha}(\Gamma)$-regular. We say that $u$ is in the space
$H^{m-\frac{1}{2}}(\Sigma)$ if $(u\circ\Psi_{\Sigma})\in
H^{m-\frac{1}{2}}(\Gamma)$ and define the norm as

$\|u\|_{H^{m-\frac{1}{2}}(\Sigma)}:=\|u\circ\Psi_{\Sigma}\|_{H^{m-\frac{1}{2}}(\Gamma)},$

where $\Psi_{\Sigma}:\Gamma\to\Sigma$ is the parametrization
$\Psi_{\Sigma}(x)=x+h(x)\nu_{\Gamma}(x)$.

We define the space $H^{m-\frac{1}{2}}_{\star}(\Sigma)$ via interpolation as
the functions $u\in H^{m-1}(\Sigma)$ such that

(2.8)
$\|u\|_{H_{\star}^{m-\frac{1}{2}}(\Sigma)}^{2}:=\|u\|_{H^{m-1}(\Sigma)}^{2}+\int_{0}^{\infty}\left(\frac{K(t,u,H^{m-1}(\Sigma),H^{m}(\Sigma))}{t^{1/2}}\right)^{2}\,\frac{dt}{t}<\infty.$

We note that if $\Sigma$ satisfies the assumption (Hm) for $m\geq 2$ the norm
$H^{m-\frac{1}{2}}(\Sigma)$ in Definition 2.6 is equivalent with the
interpolation norm $\|u\|_{H_{\star}^{m-\frac{1}{2}}(\Sigma)}$ in (2.8). We
state this in the next proposition. The proof is similar to the argument for
(2.6) and we omit it.

######  Proposition 2.7.

Let $m\geq 2$ and assume that $\Sigma$ is $C^{1,\alpha}(\Gamma)$-regular and
satisfies the assumption (Hm). Then it holds

$H^{m-\frac{1}{2}}(\Sigma)=H_{\star}^{m-\frac{1}{2}}(\Sigma)\qquad\text{and}\qquad\|u\|_{H^{m-\frac{1}{2}}(\Sigma)}\simeq\|u\|_{H_{\star}^{m-\frac{1}{2}}(\Sigma)}.$

###  2.3. Geometric Preliminaries

We begin by recalling basic results from differential geometry. We define the
Riemann curvature tensor $R\in\mathscr{T}^{4}(\Sigma)$ [38, 42] via
interchange of covariant derivatives of a vector field $Y^{i}$ and a covector
field $Z_{i}$ as

(2.9)
$\begin{split}&\bar{\nabla}_{i}\bar{\nabla}_{j}Y^{s}-\bar{\nabla}_{j}\bar{\nabla}_{i}Y^{s}=R_{ijkl}g^{ks}Y^{l},\\\
&\bar{\nabla}_{i}\bar{\nabla}_{j}Z_{k}-\bar{\nabla}_{j}\bar{\nabla}_{i}Z_{k}=R_{ijkl}g^{ls}Z_{s},\end{split}$

where we have used the Einstein summation convention. We may write the Riemann
tensor in local coordinates by using the second fundamental form $B$ as

(2.10) $R_{ijkl}=B_{ik}B_{jl}-B_{il}B_{jk}.$

We will also need the Simon’s identity which reads as

(2.11)
$\Delta_{\Sigma}B_{ij}=\bar{\nabla}_{i}\bar{\nabla}_{j}H+HB_{il}g^{ls}B_{sj}-|B|^{2}B_{ij}.$

Let us recall that the interpolation inequality holds for smooth compact
$n$-dimensional hypersurface $\Sigma\subset\mathbb{R}^{n+1}$, see e.g. [4],

(2.12) $\|\bar{\nabla}^{k}u\|_{L^{p}(\Sigma)}\leq
C_{\Sigma}\|u\|_{W^{l,r}(\Sigma)}^{\theta}\|u\|_{L^{q}(\Sigma)}^{(1-\theta)},$

where

$\frac{1}{p}=\frac{k}{n}+\theta\left(\frac{1}{r}-\frac{l}{n}\right)+\frac{1}{q}(1-\theta).$

In particular, (2.12) holds on the reference surface
$\Gamma\subset\mathbb{R}^{3}$ and in $\mathbb{R}^{n}$ for functions with
compact support $\text{supp}\,u\subset B_{R}$.

In order to have the interpolation inequality for a general surface
$\Sigma\subset\mathbb{R}^{n+1}$ with control on the constant $C_{\Sigma}$, we
use the result in [41], which states that once the mean curvature $H_{\Sigma}$
satisfies the bound $\|H_{\Sigma}\|_{L^{n+\delta}(\Sigma)}\leq C$, then the
above interpolation inequality holds on $\Sigma$ with uniform bound on the
constant. We state this for our purpose, where $\Sigma$ is $2$-dimensional
surfaces that is $C^{1,\alpha}(\Gamma)$-regular and satisfies the bound
$\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq C$. The reason for the $L^{4}$-curvature
bound will be clear from the results in Section 6. The following interpolation
inequality follows from [41, Proposition 6.5].

######  Proposition 2.8.

Assume $\Sigma\subset\mathbb{R}^{3}$ is a compact $2$-dimensional hypersurface
which is $C^{1,\alpha}(\Gamma)$-regular and satisfies the bound
$\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq C$. Then for integers $k,l$, $0\leq k<l$
and numbers $p,r\in[1,\infty)$ and $q\in[1,\infty]$ we have for all tensor
fields $T$ that

$\|\bar{\nabla}^{k}T\|_{L^{p}(\Sigma)}\leq
C\|T\|_{W^{l,r}(\Sigma)}^{\theta}\|T\|_{L^{q}(\Sigma)}^{(1-\theta)},$

where $p$ and $\theta\in[0,1]$ are given by

$\frac{1}{p}=\frac{k}{2}+\theta\left(\frac{1}{r}-\frac{l}{2}\right)+\frac{1}{q}(1-\theta).$

In particular, we have the Sobolev embedding, i.e., for $p\in[1,n)$ it holds
$\|u\|_{L^{p^{*}}(\Sigma)}\leq C\|u\|_{W^{1,p}(\Sigma)}$ with
$p^{*}=\frac{np}{n-p}$, for $p=n$ it holds $\|u\|_{L^{q}(\Sigma)}\leq
C\|u\|_{W^{1,p}(\Sigma)}$ for all $q<\infty$ and for $p>n$ it holds
$\|u\|_{C^{\alpha}(\Sigma)}\leq C\|u\|_{W^{1,p}(\Sigma)}$ for
$\alpha=1-\frac{n}{p}$.

There is danger for confusion in terminology when we use interpolation of
function spaces and interpolation inequality. We use the term ’interpolation
argument’, when we interpolate between two function spaces, and ’interpolation
inequality’ or merely ’interpolation’ when we refer to Proposition 2.8.

Let $\Sigma=\partial\Omega\subset\mathbb{R}^{3}$ be a compact hypersurface in
$\mathbb{R}^{3}$ such that $\Sigma=\partial\Omega$ which is
$C^{1,\alpha}(\Gamma)$-regular. Then the Sobolev embedding extends to half-
integers, i.e., it holds

$\|u\|_{L^{p}(\Sigma)}\leq C\|u\|_{H^{\frac{1}{2}}(\Sigma)},\qquad\text{for
}\,p\leq 4$

and

$\|u\|_{L^{p}(\Omega)}\leq C\|u\|_{H^{\frac{1}{2}}(\Omega)},\qquad\text{for
}\,p\leq 3.$

We need the above interpolation inequality also for half-integers and for
functions defined in $\Omega$. To this aim we need to assume that $\Sigma$
satisfies the condition (Hm).

######  Corollary 2.9.

Let $m\in\mathbb{N}$ and $\Sigma\subset\mathbb{R}^{3}$ is compact
$2$-dimensional hypersurface which is $C^{1,\alpha}(\Gamma)$-regular such that
$\Sigma=\partial\Omega$ and satisfies the condition (Hm). Then for all half-
integers $k$ and $l$ with $k<l\leq m$ and for $q\in[1,\infty]$ it holds

$\|u\|_{H^{k}(\Sigma)}\leq
C\|u\|_{H^{l}(\Sigma)}^{\theta}\|u\|_{L^{q}(\Sigma)}^{1-\theta},$

where $\theta\in[0,1]$ is given by

$1=k-\theta(l-1)+\frac{2}{q}(1-\theta).$

In addition, it holds

$\|u\|_{H^{k}(\Omega)}\leq
C\|u\|_{H^{l}(\Omega)}^{\theta}\|u\|_{L^{q}(\Omega)}^{1-\theta},$

where $\theta\in[0,1]$ is given by

$\frac{1}{2}=\frac{k}{3}+\theta\left(\frac{1}{2}-\frac{l}{3}\right)+\frac{1}{q}(1-\theta).$

Moreover, the inequality (2.12) holds on $\Omega\subset\mathbb{R}^{3}$ with
$r=2$ and integers $k<l\leq m$.

###### Proof.

We sketch the proof only for the first claim when $k=\tilde{k}-\frac{1}{2}$
for $\tilde{k}\in\mathbb{N}$ and $l$ is an integer. By Proposition 2.7 and by
the classical interpolation theory stated in (2.2) we have

$\|u\|_{H^{k}(\Sigma)}\leq C\|u\|_{H_{\star}^{k}(\Sigma)}\leq
C\|u\|_{H^{\tilde{k}}(\Sigma)}^{\frac{1}{2}}\|u\|_{H^{\tilde{k}-1}(\Sigma)}^{\frac{1}{2}}.$

Proposition 2.8 yields

$\|u\|_{H^{\tilde{k}}(\Sigma)}\leq
C\|u\|_{H^{l}(E)}^{\theta_{1}}\|u\|_{L^{q}(E)}^{1-\theta_{1}},$

where $\theta_{1}$ is given by
$1=\tilde{k}-\theta_{1}(l-1)+\frac{2}{q}(1-\theta_{1})$, and

$\|u\|_{H^{\tilde{k}-1}(\Sigma)}\leq
C\|u\|_{H^{l}(E)}^{\theta_{2}}\|u\|_{L^{q}(E)}^{1-\theta_{2}},$

where $\theta_{2}$ is given by
$1=(\tilde{k}-1)-\theta_{2}(l-1)+\frac{2}{q}(1-\theta_{2})$. This implies the
claim. The case when $l$ is half-integer follows from the same argument.
Finally the second interpolation inequality follows by extending $u$ to whole
$\mathbb{R}^{3}$, where the inequality is well-known, and using Proposition
2.1. ∎

###  2.4. Functional and geometric inequalities

We begin by recalling the extension of the interpolation inequality (2.12), or
the Gagliardo-Nirenberg inequality, in $\mathbb{R}^{n}$ for fractional Sobolev
spaces [8]. We state the result in the setting that we need, where for all
$f\in C_{0}^{\infty}(B_{R})$ it holds

(2.13) $\|f\|_{W^{s,p}(B_{R})}\leq
C\|f\|_{W^{s_{1},p_{1}}(B_{R})}^{\theta}\|f\|_{L^{p_{2}}(B_{R})}^{1-\theta},$

for $0\leq s\leq s_{1}$, $p_{2}\in(1,\infty)$ and $\theta\in(0,1)$ which
satisfy

$s=\theta
s_{1}\qquad\text{and}\qquad\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{2}}.$

Next we recall the Kato-Ponce inequality, or the fractional Leibniz rule, in
$\mathbb{R}^{n}$ which is proven e.g. in [30]. We may define the norm
$\|f\|_{W^{k,p}(\mathbb{R}^{n})}$ for half-integer $k\geq 0$ and
$p\in(1,\infty)$ by using Bessel potentials $\langle D\rangle^{k}$ as

$\|f\|_{W^{k,p}(\mathbb{R}^{n})}=\|\langle
D\rangle^{k}f\|_{L^{p}(\mathbb{R}^{n})}.$

The Kato-Ponce inequality, in the form we are interested in, states that for
$f,g\in C_{0}^{\infty}(\mathbb{R}^{n})$ and for numbers $2\leq
p_{1},q_{2}<\infty$ and $2\leq p_{2},q_{1}\leq\infty$ with

(2.14)
$\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}=\frac{1}{2}$

it holds

(2.15) $\|fg\|_{H^{k}(\mathbb{R}^{n})}\leq
C\|f\|_{W^{k,p_{1}}(\mathbb{R}^{n})}\|g\|_{L^{q_{1}}(\mathbb{R}^{n})}+C\|f\|_{L^{p_{2}}(\mathbb{R}^{n})}\|g\|_{W^{k,q_{2}}(\mathbb{R}^{n})}.$

We need the following generalization of the Kato-Ponce inequality both on the
boundary $\Sigma$ and in the domain $\Omega$.

######  Proposition 2.10.

Let $m\geq 1$ be an integer and assume $\Sigma$ is
$C^{1,\alpha}(\Gamma)$-regular and satisfies the condition (Hm). Then for all
half-integers $k\leq m$ it holds

$\|fg\|_{H^{k}(\Sigma)}\leq
C\|f\|_{H^{k}(\Sigma)}\|g\|_{L^{\infty}(\Sigma)}+C\|f\|_{L^{\infty}(\Sigma)}\|g\|_{H^{k}(\Sigma)}$

and

$\|fg\|_{H^{k}(\Omega)}\leq
C\|f\|_{H^{k}(\Omega)}\|g\|_{L^{\infty}(\Omega)}+C\|f\|_{L^{\infty}(\Omega)}\|g\|_{H^{k}(\Omega)}.$

Moreover, assume that $\|B\|_{L^{4}}\leq C$ and let $k\in\mathbb{N}$. Then for
$p_{1},p_{2},q_{1},q_{2}\in[2,\infty]$ with $p_{1},q_{2}<\infty$ which
satisfies (2.14) it holds

$\|fg\|_{H^{k}(\Sigma)}\leq
C\|f\|_{W^{k,p_{1}}(\Sigma)}\|g\|_{L^{q_{1}}(\Sigma)}+C\|f\|_{L^{p_{2}}(\Sigma)}\|g\|_{W^{k,q_{2}}(\Sigma)}.$

###### Proof.

The second inequality follows immediately from the property of the extension
operator given by Proposition 2.1 and by the classical Kato-Ponce inequality
(2.15), see e.g. [10]. Also the first inequality follows from a similar
localization argument as we used in (2.6).

We prove the third inequality, since we will use the argument also later.
First by Leibniz formula we may write

$\bar{\nabla}^{k}(fg)=\sum_{i+j=k}\bar{\nabla}^{i}f\star\bar{\nabla}^{j}g.$

The claim thus follows once we prove

(2.16)
$\sum_{i+j=k}\|\bar{\nabla}^{i}f\star\bar{\nabla}^{j}g\|_{L^{2}(\Sigma)}\leq
C\|f\|_{W^{k,p_{1}}(\Sigma)}\|g\|_{L^{q_{1}}(\Sigma)}+\|f\|_{L^{p_{2}}(\Sigma)}\|g\|_{W^{k,q_{2}}(\Sigma)}.$

To this aim we use Hölder’s inequality as

$\sum_{i+j=k}\|\bar{\nabla}^{i}f\star\bar{\nabla}^{j}g\|_{L^{2}(\Sigma)}\leq\sum_{i+j=k}\|\bar{\nabla}^{i}f\|_{L^{4}(\Sigma)}\|\bar{\nabla}^{j}g\|_{L^{4}(\Sigma)}.$

By interpolation inequality in Proposition 2.8 we have

$\|\bar{\nabla}^{i}f\|_{L^{4}(\Sigma)}\leq
C\|f\|_{W^{k,p_{1}}(\Sigma)}^{\theta_{i}}\|f\|_{L^{p_{2}}(\Sigma)}^{1-\theta_{i}}\qquad\text{
with
}\qquad\theta_{i}=\frac{\frac{i}{2}-\frac{1}{4}+\frac{1}{p_{2}}}{\frac{k}{2}+\frac{1}{p_{2}}-\frac{1}{p_{1}}}$

and recalling that
$\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}=\frac{1}{2}$
we have

$\|\bar{\nabla}^{j}g\|_{L^{4}(\Sigma)}\leq
C\|g\|_{W^{k,q_{2}}(\Sigma)}^{\theta_{j}}\|g\|_{L^{q_{1}}(\Sigma)}^{1-\theta_{j}}\qquad\text{
with
}\qquad\theta_{j}=\frac{\frac{j}{2}-\frac{1}{4}+\frac{1}{q_{1}}}{\frac{k}{2}+\frac{1}{p_{2}}-\frac{1}{p_{1}}}.$

In particular, $i+j=k$ implies $\theta_{i}+\theta_{j}=1$. Therefore we have by
Young’s inequality
$a^{\theta_{i}}b^{\theta_{j}}\leq\theta_{i}a+\theta_{j}b\leq a+b$ that

$\begin{split}\sum_{i+j=k}\|\bar{\nabla}^{i}f\star\bar{\nabla}^{j}g\|_{L^{2}}&\leq
C\|f\|_{L^{p_{2}}}\|g\|_{L^{q_{1}}}\sum_{i+j=k}\|f\|_{W^{k,p_{1}}}^{\theta_{i}}\|f\|_{L^{p_{2}}}^{-\theta_{i}}\|g\|_{W^{k,q_{2}}}^{\theta_{j}}\|g\|_{L^{q_{1}}}^{-\theta_{j}}\\\
&\leq
C\|f\|_{L^{p_{2}}}\|g\|_{L^{q_{1}}}\left(\frac{\|f\|_{W^{k,p_{1}}}}{\|f\|_{L^{p_{2}}}}+\frac{\|g\|_{W^{k,q_{2}}}}{\|g\|_{L^{q_{1}}}}\right)\end{split}$

and the claim follows. ∎

We remark that we do not generalize the last inequality in Proposition 2.10
for half-integers $k$ since we do not define the space $W^{k,p}(\Sigma)$ for
$p\neq 2$, when $k$ is not an integer. However, under the assumption of
Proposition 2.10, we obtain a weaker version which reads as follows

(2.17) $\|fg\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C\|f\|_{H^{\frac{1}{2}}(\Sigma)}\|g\|_{L^{\infty}(\Sigma)}+\|f\|_{L^{p}(\Sigma)}\|g\|_{W^{1,q}(\Sigma)},$

for $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$. Again, since the proof is similar
to the argument we used in (2.6) we leave it for the reader, but refer to [15,
Lemma 4.3] for the proof of the case $p=q=4$. In particular, when $g$ is
Lipschitz, we may estimate the product simply by

$\|fg\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C\|f\|_{H^{\frac{1}{2}}(\Sigma)}\|g\|_{C^{1}(\Sigma)}.$

Next we recall (see e.g. [22]) that it holds $\|u\|_{H^{k+2}(\Sigma)}\leq
C_{\Sigma}(\|\Delta_{\Sigma}u\|_{H^{k}(\Sigma)}+\|u\|_{L^{2}(\Sigma)})$.
However, the constant depends on the curvature of $\Sigma$ and we need to
quantify this dependence.

######  Proposition 2.11.

Assume that $\Sigma$ is $C^{1,\alpha}(\Gamma)$-regular and satisfies
$\|B_{\Sigma}\|_{L^{4}}\leq C$. Then for $k=0,\frac{1}{2},1$ and
$\varepsilon>0$ it holds

$\|u\|_{H^{k+2}(\Sigma)}\leq(1+\varepsilon)\|\Delta_{\Sigma}u\|_{H^{k}(\Sigma)}+C_{\varepsilon}\|u\|_{L^{2}(\Sigma)}.$

Let $m$ be an integer with $m\geq 3$ and assume that $\Sigma$ satisfies in
addition the condition (Hm). Then for every half-integer $2\leq k\leq m$ it
holds

$\|u\|_{H^{k}(\Sigma)}\leq(1+\varepsilon)\|\Delta_{\Sigma}u\|_{H^{k-2}(\Sigma)}+C_{\varepsilon}\|u\|_{L^{2}(\Sigma)}.$

###### Proof.

The case $k=0$ follows from [15, Lemma 4.11] but we give the proof for the
reader’s convenience. We recall that the Riemann tensor $R$ satisfies by
(2.10) $|R|\leq C|B|^{2}$ and deduce by [22, Remark 2.4] (see also [4]) that

$\|\bar{\nabla}^{2}u\|_{L^{2}(\Sigma)}^{2}\leq\|\Delta_{\Sigma}u\|_{L^{2}(\Sigma)}^{2}+C\int_{\Sigma}|B|^{2}|\bar{\nabla}u|^{2}\,d\mathcal{H}^{2}.$

By Proposition 2.8 there is $\theta\in(0,1)$ such that

$\int_{\Sigma}|B|^{2}|\bar{\nabla}u|^{2}\,d\mathcal{H}^{2}\leq\|B\|_{L^{4}}^{2}\|\bar{\nabla}u\|_{L^{4}}^{2}\leq
C\|u\|_{H^{2}}^{2\theta}\|u\|_{L^{2}}^{2(1-\theta)}\leq\varepsilon\|u\|_{H^{2}}+C_{\varepsilon}\|u\|_{L^{2}}^{2}.$

This implies the claim for $k=0$.

For the case $k=1$ we use (2.9) and integration by parts

$\begin{split}\|\bar{\nabla}\Delta_{\Sigma}u\|_{L^{2}(\Sigma)}^{2}&=\int_{\Sigma}\bar{\nabla}^{k}\bar{\nabla}_{i}\bar{\nabla}^{i}u\,\bar{\nabla}_{k}\bar{\nabla}^{j}\bar{\nabla}_{j}u\,d\mathcal{H}^{2}\\\
&=\int_{\Sigma}\bar{\nabla}_{i}\bar{\nabla}^{k}\bar{\nabla}^{i}u\,\bar{\nabla}_{k}\bar{\nabla}^{j}\bar{\nabla}_{j}u\,d\mathcal{H}^{2}+\int_{\Sigma}(R\star\bar{\nabla}u\star\bar{\nabla}^{3}u)\,d\mathcal{H}^{2}\\\
&\geq-\int_{\Sigma}\bar{\nabla}^{k}\bar{\nabla}^{i}u\,\bar{\nabla}_{i}\bar{\nabla}_{k}\bar{\nabla}^{j}\bar{\nabla}_{j}u\,d\mathcal{H}^{2}-\|R\star\bar{\nabla}u\|_{L^{2}}\,\|\bar{\nabla}^{3}u\|_{L^{2}}.\end{split}$

As before we have by Proposition 2.8 and by $|R|\leq C|B|^{2}$ that

(2.18) $\|R\star\bar{\nabla}u\|_{L^{2}}\leq
C\|B\|_{L^{4}}^{2}\|\bar{\nabla}u\|_{L^{\infty}}\leq\varepsilon\|u\|_{H^{3}}+C_{\varepsilon}\|u\|_{L^{2}}.$

We proceed by using (2.9) and by integrating by parts

$\begin{split}-\int_{\Sigma}&\bar{\nabla}^{k}\bar{\nabla}^{i}u\,\bar{\nabla}_{i}\bar{\nabla}_{k}\bar{\nabla}^{j}\bar{\nabla}_{j}u\,d\mathcal{H}^{2}\\\
&\geq-\int_{\Sigma}\bar{\nabla}^{k}\bar{\nabla}^{i}u\,\bar{\nabla}_{k}\bar{\nabla}_{i}\bar{\nabla}^{j}\bar{\nabla}_{j}u\,d\mathcal{H}^{2}+\int_{\Sigma}\bar{\nabla}^{2}u\star
R\star\bar{\nabla}^{2}u\,d\mathcal{H}^{2}\\\
&\geq-\int_{\Sigma}\bar{\nabla}^{k}\bar{\nabla}^{i}u\,\bar{\nabla}_{k}\bar{\nabla}^{j}\bar{\nabla}_{i}\bar{\nabla}_{j}u\,d\mathcal{H}^{2}-\int_{\Sigma}\bar{\nabla}^{k}\bar{\nabla}^{i}u\,\bar{\nabla}_{k}\bar{[}\bar{\nabla}_{i}\nabla^{j}-\bar{\nabla}_{j}\nabla^{i}]\bar{\nabla}_{j}u\,d\mathcal{H}^{2}\\\
&\,\,\,\,\,\,\,\,-C\|B\|_{L^{4}}^{2}\|\bar{\nabla}^{2}u\|_{L^{4}}^{2}\\\
&\geq\int_{\Sigma}\bar{\nabla}^{j}\bar{\nabla}^{k}\bar{\nabla}^{i}u\,\bar{\nabla}_{k}\bar{\nabla}_{i}\bar{\nabla}_{j}u\,d\mathcal{H}^{2}+\int_{\Sigma}\bar{\nabla}_{k}\bar{\nabla}^{k}\bar{\nabla}^{i}u\,\bar{[}\bar{\nabla}_{i}\nabla^{j}-\bar{\nabla}_{j}\nabla^{i}]\bar{\nabla}_{j}u\,d\mathcal{H}^{2}\\\
&\,\,\,\,\,\,\,\,-C\|B\|_{L^{4}}^{2}\|\bar{\nabla}^{2}u\|_{L^{4}}^{2}\\\
&\geq\|\bar{\nabla}^{3}u\|_{L^{2}}^{2}-C\|B\|_{L^{4}}^{2}\|\bar{\nabla}^{2}u\|_{L^{4}}^{2}-\|R\star\bar{\nabla}u\|_{L^{2}}\,\|\bar{\nabla}^{3}u\|_{L^{2}}.\end{split}$

The inequality for $k=1$ then follows from (2.18) and from Proposition 2.8
which yields

$\|\bar{\nabla}^{2}u\|_{L^{4}}\leq\varepsilon\|u\|_{H^{3}}+C_{\varepsilon}\|u\|_{L^{2}}.$

The case $k=1/2$ follows from the previous two estimates and Proposition 2.7
with standard interpolation argument which we briefly sketch here for the
reader’s convenience. We define a linear operator
$\mathcal{F}:\tilde{H}^{k}(\Sigma)\to\tilde{H}^{k+2}(\Sigma)$ such that
$\mathcal{F}(g)=u$, where $u$ is the solution of

$\Delta_{\Sigma}u=g\qquad\text{on }\,\Sigma,$

and $\tilde{H}^{k}(\Sigma)=\\{f\in
H^{k}(\Sigma):\int_{\Sigma}f\,d\mathcal{H}^{2}=0\\}$. The operator
$\mathcal{F}$ is well-defined and by the previous estimates it satisfies

$\|\mathcal{F}\|_{\mathcal{L}(L^{2},H^{2})}\leq
C\qquad\text{and}\qquad\|\mathcal{F}\|_{\mathcal{L}(H^{1},H^{3})}\leq C$

By the interpolation theory discussed in (2.2) it holds

$\|\mathcal{F}(g)\|_{H_{\star}^{\frac{5}{2}}(\Sigma)}\leq
C\|g\|_{H_{\star}^{\frac{1}{2}}(\Sigma)}.$

Proposition 2.7 then yields

$\|\mathcal{F}(g)\|_{H^{\frac{5}{2}}(\Sigma)}\leq
C\|g\|_{H^{\frac{1}{2}}(\Sigma)}.$

We apply this to $\tilde{u}=u-\bar{u}$, where
$\bar{u}=\fint_{\Sigma}u\,d\mathcal{H}^{2}$ and the claim follows.

The argument for higher $m$ and $k$ is similar and we merely sketch it. Let
$k$ be an integer with $2\leq k\leq m$. Using (2.9) and arguing as above we
obtain after long but straightforward calculations that

$\|\bar{\nabla}^{k}u\|_{L^{2}(\Sigma)}^{2}\leq\|\bar{\nabla}^{k-2}\Delta_{\Sigma}u\|_{L^{2}(\Sigma)}^{2}+C\sum_{\alpha+\beta\leq
k-2}\|\bar{\nabla}^{\alpha}R\star\bar{\nabla}^{1+\beta}u\|_{L^{2}(\Sigma)}^{2}.$

Then by (2.10), (2.16), Proposition 2.8 and by the assumption
$\|B\|_{L^{\infty}},\|B\|_{H^{k-2}}\leq C$ we have

$\begin{split}\sum_{\alpha+\beta\leq
k-2}\|\bar{\nabla}^{\alpha}(R\star\bar{\nabla}^{1+\beta}u)\|_{L^{2}(\Sigma)}&\leq
C\|B\|_{L^{\infty}}^{2}\|u\|_{H^{k-1}}+C\|B\|_{L^{\infty}}\|B\|_{H^{k-2}}\|\bar{\nabla}u\|_{L^{\infty}}\\\
&\leq\varepsilon\|u\|_{H^{k}(\Sigma)}+C_{\varepsilon}\|u\|_{L^{2}(\Sigma)}.\end{split}$

This yields the claim for integers $2\leq k\leq l$.

If $k\leq m-\frac{1}{2}$ is an half-integer but not an integer, then we may
use the previous argument for integer $l=k+\frac{1}{2}\leq m$ and deduce

$\|u\|_{H^{l}}\leq(1+\varepsilon)\|\Delta_{\Sigma}u\|_{H^{l-2}(\Sigma)}+C_{\varepsilon}\|u\|_{L^{2}(\Sigma)}.$

The same holds for $l-1$. Hence, the claim follows by Proposition 2.7 and by
the same interpolation argument we used above. ∎

By using the previous proposition and the Simon’s identity (2.11) we deduce
that we may bound the second fundamental form by the mean curvature.

######  Proposition 2.12.

Assume that $\Sigma$ is $C^{1,\alpha}(\Gamma)$-regular. Then for every
$p\in(1,\infty)$ it holds

$\|B_{\Sigma}\|_{L^{p}(\Sigma)}\leq C(1+\|H_{\Sigma}\|_{L^{p}(\Sigma)}).$

If in addition $\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq C$, then for
$k=\frac{1}{2},1,2$ it holds

$\|B_{\Sigma}\|_{H^{k}(\Sigma)}\leq C(1+\|H_{\Sigma}\|_{H^{k}(\Sigma)}).$

Finally let $m\geq 3$ be an integer and assume that $\Sigma$ satisfies in
addition the condition (Hm) for $m$. Then the above estimate holds for all
half-integers $k\leq m$.

###### Proof.

The first claim follows from standard Calderon-Zygmund estimate [25] and we
omit it. Let us proof the second claim for $k=\frac{1}{2}$. We recall the
geometric fact

$\Delta_{\Sigma}x_{i}=-H_{\Sigma}\nu_{i},$

where $x_{i}=x\cdot e_{i}$ and $\nu_{i}=\nu_{\Sigma}\cdot e_{i}$. Then we have
by Proposition 2.11 and (2.17)

$\begin{split}\|B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}&\leq
C\sum_{i=1}^{3}(1+\|\nabla^{2}_{\Sigma}x_{i}\|_{H^{\frac{1}{2}}(\Sigma)})\leq\sum_{i=1}^{3}C(1+\|\Delta_{\Sigma}x_{i}\|_{H^{\frac{1}{2}}(\Sigma)})\\\
&=\sum_{i=1}^{3}C(1+\|H_{\Sigma}\nu_{i}\|_{H^{\frac{1}{2}}(\Sigma)})\\\ &\leq
C(1+\|H_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}+\|H_{\Sigma}\|_{L^{4}(\Sigma)}\|\nu_{\Sigma}\|_{W^{1,4}(\Sigma)})\leq
C(1+\|H_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}).\end{split}$

The argument for $k=1$ is similar.

In the case $k=2$ we use the Simon’s identity (2.11) to deduce

$\|\Delta_{\Sigma}B\|_{L^{2}(\Sigma)}^{2}\leq\|\bar{\nabla}^{2}H\|_{L^{2}(\Sigma)}^{2}+C\|B\|_{L^{6}(\Sigma)}^{6}.$

Proposition 2.11 yields $\|B\|_{H^{2}(\Sigma)}\leq
2\|\Delta_{\Sigma}B\|_{L^{2}(\Sigma)}+C$. The claim then follows from
interpolation inequality (Proposition 2.8)

$\|B\|_{L^{6}(\Sigma)}^{6}\leq\|B\|_{H^{2}(\Sigma)}^{\frac{2}{3}}\|B\|_{L^{4}(\Sigma)}^{\frac{16}{3}}\leq\varepsilon\|B\|_{H^{2}(\Sigma)}+C_{\varepsilon}.$

Let us then fix $m\geq 3$, assume that $\Sigma$ satisfies the condition (Hm)
for $m$ and let $k\leq m$. We use the Simon’s identity (2.11) and Proposition
2.10 to deduce

$\begin{split}\|\Delta_{\Sigma}B\|_{H^{k-2}(\Sigma)}&\leq\|H\|_{H^{k}(\Sigma)}+C\|B\star
B\star B\|_{H^{k-2}(\Sigma)}\\\
&\leq\|H\|_{H^{k}(\Sigma)}+C\|B\|_{L^{\infty}(\Sigma)}^{2}\|B\|_{H^{k-2}(\Sigma)}\\\
&\leq\|H\|_{H^{k}(\Sigma)}+\varepsilon\|B\|_{H^{k}(\Sigma)}+C_{\varepsilon},\end{split}$

where the last inequality follows from $\|B\|_{L^{\infty}}\leq C$ and from
interpolation. The claim then follows from Proposition 2.11. ∎

Note that by the definition of the space $\|\cdot\|_{H^{k}(\Sigma)}$ in
Definition 2.6 it is not yet clear if it holds

$\|\nabla_{\tau}u\|_{H^{k-1}(\Sigma)}\leq C\|u\|_{H^{k}(\Sigma)}$

when $k$ is not an integer. We conclude this by section by proving this in the
following technical lemma.

######  Lemma 2.13.

Let $m$ be an integer with $m\geq 3$ and assume that $\Sigma$ satisfies the
condition (Hm). Then it holds

$\|\nabla_{\tau}u\|_{H^{m-\frac{3}{2}}(\Sigma)}\leq
C\|u\|_{H^{m-\frac{1}{2}}(\Sigma)}.$

###### Proof.

Let us denote $\tilde{u}=u\circ\Psi:\Gamma\to\mathbb{R}$ and in order to
simplify the notation denote the extension given by the projection in (2.7)
$(\tilde{u}\circ\pi_{\Gamma})$ simply by $\tilde{u}$. We observe that there is
a matrix field $A(x)=A(x,h,\bar{\nabla}h)$ such that

$(\nabla_{\tau}u\circ\Psi)(x)=A(x)\nabla\tilde{u}(x)\qquad\text{for
}\,x\in\Gamma.$

Therefore we have by Definition 2.6 and by Proposition 2.10

$\begin{split}\|\nabla_{\tau}u\|_{H^{m-\frac{3}{2}}(\Sigma)}&=\|\nabla_{\tau}u\circ\Psi\|_{H^{m-\frac{3}{2}}(\Gamma)}=\|A\,\nabla\tilde{u}\|_{H^{m-\frac{3}{2}}(\Gamma)}\\\
&\leq
C\|A\|_{L^{\infty}}\,\|\nabla\tilde{u}\|_{H^{m-\frac{3}{2}}(\Gamma)}+C\|A\|_{H^{m-\frac{3}{2}}(\Gamma)}\,\|\nabla\tilde{u}\|_{L^{\infty}}.\end{split}$

The assumption $\|B\|_{L^{\infty}(\Sigma)},\|B\|_{H^{m-2}(\Sigma)}\leq C$
implies for the height function $\|h\|_{C^{2}(\Gamma)}\leq C$ and
$\|h\|_{H^{m}(\Gamma)}\leq C$ and therefore
$\|A\|_{L^{\infty}(\Gamma)},\|A\|_{H^{m-1}(\Gamma)}\leq C$. Moreover, since
$m\geq 3$ the Sobolev embedding yields
$\|\nabla\tilde{u}\|_{L^{\infty}(\Gamma)}\leq
C\|\nabla\tilde{u}\|_{H^{m-\frac{3}{2}}(\Gamma)}$. Therefore we have

$\|\nabla_{\tau}u\|_{H^{m-\frac{3}{2}}(\Sigma)}\leq
C\|\nabla\tilde{u}\|_{H^{m-\frac{3}{2}}(\Gamma)}\leq
C\|\tilde{u}\|_{H^{m-\frac{1}{2}}(\Gamma)}=C\|u\|_{H^{m-\frac{1}{2}}(\Sigma)}.$

∎

##  3\. Elliptic estimates for vector fields and functions

In this section we recall some known and provide some new div-curl type
estimates for vector fields in the domain, i.e., $F:\Omega\to\mathbb{R}^{3}$.
We will need estimates where we control the norm $\|F\|_{H^{k}(\Omega)}$ by
the $\operatorname{div}F$, $\operatorname{curl}F$ in $\Omega$ and with $F_{n}$
on the boundary $\Sigma$. The main result of the section is Theorem 3.6 where
we prove this estimate for $k=1$ and require the boundary merely to satisfy
$\|B_{\Sigma}\|_{L^{4}}\leq C$. We do not expect the $L^{4}$-integrability to
be the optimal condition. However, related to this we note that we may
construct a cone $\Omega\subset\mathbb{R}^{3}$ and a harmonic function
$u:\Omega\to\mathbb{R}$ with zero Neumann boundary data $\partial_{\nu}u=0$
arguing as in [21, Section 3], such that $u$ can be written in spherical
coordinates as $u(\rho,\theta)=\sqrt{\rho}f(\theta)$ for a smooth functions
$f$. In particular, $u\notin H^{2}(\Omega\cap B_{R})$ and therefore we may
deduce that a necessary condition for the curvature is at least
$\|B_{\Sigma}\|_{L^{2}}\leq C$ for Lemma 3.5 and Theorem 3.6 to hold.

We will also prove boundary regularity estimates for harmonic functions in
Theorem 3.9, which quantifies the boundary regularity of the harmonic
functions with respect to the regularity of the boundary. We note that in
Theorem 3.9 it is crucial to assume that the boundary is uniformly
$C^{1,\alpha}(\Gamma)$-regular. Indeed, the statement does not hold for
Lipschitz domains.

###  3.1. Regularity estimates for vector fields

We begin this section by recalling the following result which is essentially
from [10] (see also [51]). Recall that we define

$\operatorname{curl}F=\nabla F-(\nabla F)^{T}.$

Throughout the section we assume that $\Omega$ is connected, but its boundary
$\Sigma=\partial\Omega$ may have many components.

######  Theorem 3.1.

Let $l\geq 2$ be an integer and let $\Omega$ be a smooth domain with
$\Sigma=\partial\Omega$, such that
$\|B_{\Sigma}\|_{H^{\frac{3}{2}l-1}(\Sigma)}\leq C$. Then for all smooth
vector fields $F:\Omega\to\mathbb{R}^{3}$ and every half-integers $1\leq
k\leq\frac{3}{2}l$ it holds

$\|F\|_{H^{k}(\Omega)}\leq
C(\|F_{n}\|_{H^{k-\frac{1}{2}}(\Sigma)}+\|F\|_{L^{2}(\Omega)}+\|\operatorname{div}F\|_{H^{k-1}(\Omega)}+\|\operatorname{curl}F\|_{H^{k-1}}).$

Moreover, for $k=\lfloor\frac{3}{2}(l+1)\rfloor$ it holds

$\|F\|_{H^{k}(\Omega)}\leq
C(\|\nabla_{\tau}F_{n}\|_{H^{k-\frac{3}{2}}(\Sigma)}+(1+\|B_{\Sigma}\|_{H^{\frac{3}{2}l}})\|F\|_{L^{\infty}}+\|\operatorname{div}F\|_{H^{k-1}(\Omega)}+\|\operatorname{curl}F\|_{H^{k-1}}).$

###### Proof.

We first note that the assumption $\|B\|_{H^{\frac{3}{2}l-1}(\Sigma)}\leq C$
implies that $\Sigma$ satisfies the condition (Hm) for
$m=\lfloor\frac{3}{2}l+1\rfloor\geq 4$. We use [10, Theorem 1.3] to deduce

$\|F\|_{H^{k}(\Omega)}\leq
C(\|\nabla_{\tau}F\cdot\nu\|_{H^{k-\frac{3}{2}}(\Sigma)}+\|F\|_{L^{2}(\Omega)}+\|\operatorname{div}F\|_{H^{k-1}(\Omega)}+\|\operatorname{curl}F\|_{H^{k-1}(\Omega)})$

for all $k\leq\lfloor\frac{3}{2}(l+1)\rfloor$. We write
$\nabla_{\tau}F\cdot\nu=\nabla_{\tau}F_{n}+F\star B$ and use Proposition 2.10
to obtain

$\|\nabla_{\tau}F\cdot\nu\|_{H^{k-\frac{3}{2}}(\Sigma)}\leq\|\nabla_{\tau}F_{n}\|_{H^{k-\frac{3}{2}}(\Sigma)}+C\|F\|_{L^{\infty}}\|B\|_{H^{k-\frac{3}{2}}(\Sigma)}+C\|B\|_{L^{\infty}}\|F\|_{H^{k-\frac{3}{2}}(\Sigma)}.$

Interpolation inequality yields

$\|F\|_{H^{k-\frac{3}{2}}(\Sigma)}\leq\|F\|_{H^{k-1}(\Omega)}\leq\varepsilon\|F\|_{H^{k}(\Omega)}+C_{\varepsilon}\|F\|_{L^{\infty}(\Omega)}.$

Thus we have the second inequality. The first one follows from the fact that
for $k\leq\frac{3}{2}l$ it holds $\|B\|_{H^{k-\frac{3}{2}}(\Sigma)}\leq C$ by
the assumption. ∎

We combine Proposition 2.11 and Theorem 3.1 and obtain the following
inequality which is suitable to our purpose.

######  Proposition 3.2.

Let $l$ and $\Omega$ be as in Theorem 3.1. Then for all smooth vector fields
$F:\Omega\to\mathbb{R}^{3}$ and every half-integer $\frac{5}{2}\leq
k\leq\frac{3}{2}l$ it holds

$\|F\|_{H^{k}(\Omega)}\leq
C(\|\Delta_{\Sigma}F_{n}\|_{H^{k-\frac{5}{2}}(\Sigma)}+\|F\|_{L^{2}(\Omega)}+\|\operatorname{div}F\|_{H^{k-1}(\Omega)}+\|\operatorname{curl}F\|_{H^{k-1}(\Omega)}).$

Moreover, for $k=\lfloor\frac{3}{2}(l+1)\rfloor$ it holds

$\|F\|_{H^{k}(\Omega)}\leq
C(\|\Delta_{\Sigma}F_{n}\|_{H^{k-\frac{5}{2}}(\Sigma)}+(1+\|B\|_{H^{\frac{3}{2}l}})\|F\|_{L^{\infty}}+\|\operatorname{div}F\|_{H^{k-1}(\Omega)}+\|\operatorname{curl}F\|_{H^{k-1}(\Omega)}).$

###### Proof.

Recall that $\|B\|_{H^{\frac{3}{2}l-1}(\Sigma)}\leq C$ implies that $\Sigma$
satisfies the condition (Hm) for $m=\lfloor\frac{3}{2}l+1\rfloor\geq 4$. The
first inequality then follows from Theorem 3.1 and Proposition 2.11.

Let us then prove the last inequality. We have by Proposition 2.11 that

$\|\nabla_{\tau}F_{n}\|_{H^{k-\frac{3}{2}}(\Sigma)}\leq
C(\|\Delta_{\Sigma}\nabla_{\tau}F_{n}\|_{H^{k-\frac{7}{2}}(\Sigma)}+\|F_{n}\|_{L^{2}(\Sigma)}).$

We use the commutation formula (2.9) for the tangential gradient of
$u:\Sigma\to\mathbb{R}$ and obtain

$\Delta_{\Sigma}(\nabla_{\tau}u)=\nabla_{\tau}(\Delta_{\Sigma}u)+(B\star
B)\star\nabla_{\tau}u.$

Therefore we have by Lemma 2.13

$\begin{split}\|\Delta_{\Sigma}\nabla_{\tau}F_{n}\|_{H^{k-\frac{7}{2}}(\Sigma)}&\leq
C\left(\|\nabla_{\tau}(\Delta_{\Sigma}F_{n})\|_{H^{k-\frac{7}{2}}(\Sigma)}+\|(B\star
B)\star\nabla_{\tau}F_{n}\|_{H^{k-\frac{7}{2}}(\Sigma)}\right)\\\ &\leq
C(\|\Delta_{\Sigma}F_{n}\|_{H^{k-\frac{5}{2}}(\Sigma)}+\|(B\star
B)\star\nabla_{\tau}F_{n}\|_{H^{k-\frac{7}{2}}(\Sigma)}).\end{split}$

Recall that $k=\lfloor\frac{3}{2}(l+1)\rfloor$. We have by Proposition 2.10,
by Lemma 2.13 and by the assumption $\|B\|_{H^{\frac{3}{2}l-1}(\Sigma)}\leq C$
that

$\begin{split}\|(B\star
B)\star\nabla_{\tau}F_{n}\|_{H^{k-\frac{7}{2}}(\Sigma)}&\leq
C\left(\|B\|_{L^{\infty}}^{2}\|\nabla_{\tau}F_{n}\|_{H^{k-\frac{7}{2}}}+\|B\|_{L^{\infty}}\|B\|_{H^{k-\frac{7}{2}}}\|\nabla_{\tau}F_{n}\|_{L^{\infty}}\right)\\\
&\leq
C(\|F_{n}\|_{H^{k-\frac{5}{2}}(\Sigma)}+\|\nabla_{\tau}F_{n}\|_{L^{\infty}(\Sigma)}).\end{split}$

Finally we have by the Sobolev embedding and by Corollary 2.9

$\|\nabla_{\tau}F_{n}\|_{L^{\infty}(\Sigma)}+\|F_{n}\|_{H^{k-\frac{5}{2}}(\Sigma)}\leq\varepsilon\|\nabla_{\tau}F_{n}\|_{H^{k-\frac{3}{2}}(\Sigma)}+\varepsilon\|F\|_{H^{k}(\Omega)}+C_{\varepsilon}\|F\|_{L^{\infty}}.$

The second inequality then follows from Theorem 3.1 and combining the above
inequalities. ∎

Proposition 3.2 provides the inequality we need when we have the bound
$\|B_{\Sigma}\|_{H^{\frac{3}{2}l-1}(\Sigma)}\leq C$ for $l\geq 2$. When $l=1$
the above bound reduces to $\|B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}$, which
is the bound that we are able to prove in Section 6, but is not enough to
apply the results from [10, 51]. Note that by the Sobolev embedding this
implies $\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq C$. We need to work more in order
to prove the first inequality in Theorem 3.1 under the assumption
$\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq C$.

We begin by recalling the following Reilly’s type identity for vector fields.
First, if $\psi:\Omega\to\mathbb{R}^{3}$ is a smooth divergence free vector
field such that $\psi\cdot\nu=0$ on $\Sigma$ then it holds

(3.1)
$\|\nabla\psi\|_{L^{2}(\Omega)}^{2}=\frac{1}{2}\|\operatorname{curl}\psi\|_{L^{2}(\Omega)}^{2}-\int_{\Sigma}\langle
B_{\Sigma}\,\psi,\psi\rangle\,d\mathcal{H}^{2}.$

Second, if $u:\Omega\to\mathbb{R}$ is a smooth function then it holds

(3.2) $\begin{split}\|\nabla^{2}u\|_{L^{2}(\Omega)}^{2}=&\|\Delta
u\|_{L^{2}(\Omega)}^{2}-2\int_{\Sigma}\Delta_{\Sigma}u\,\partial_{\nu}u\,d\mathcal{H}^{2}\\\
&-\int_{\Sigma}\langle
B_{\Sigma}\bar{\nabla}u,\bar{\nabla}u\rangle\,d\mathcal{H}^{2}-\int_{\Sigma}H_{\Sigma}(\partial_{\nu}u)^{2}\,d\mathcal{H}^{2}.\end{split}$

We give the calculations for (3.1) and (3.2) for the reader’s convenience.
First, for a generic smooth vector field $F:\Omega\to\mathbb{R}^{3}$ it holds

$\begin{split}\int_{\Omega}|\operatorname{div}F|^{2}+\frac{1}{2}|\text{curl}\,F|^{2}\,dx&=\sum_{i,j=1}^{3}\int_{\Omega}(\partial_{i}F_{j})^{2}\,dx+\sum_{i,j=1}^{3}\int_{\Omega}(\partial_{i}F_{i}\partial_{j}F_{j}-\partial_{i}F_{j}\partial_{j}F_{i})\,dx\\\
&=\|\nabla
F\|_{L^{2}(\Omega)}^{2}+\sum_{i,j=1}^{3}\int_{\Omega}(\partial_{i}F_{i}\partial_{j}F_{j}-\partial_{i}F_{j}\partial_{j}F_{i})\,dx.\end{split}$

By using divergence theorem twice we obtain

$\begin{split}\int_{\Omega}\partial_{i}F_{i}\partial_{j}F_{j}\,dx&=-\int_{\Omega}F_{i}\partial_{i}\partial_{j}F_{j}\,dx+\int_{\Sigma}\partial_{j}F_{j}\,F_{i}\nu_{i},\
d\mathcal{H}^{2}\\\
&=\int_{\Omega}\partial_{j}F_{i}\partial_{i}F_{j}\,dx+\int_{\Sigma}\partial_{j}F_{j}\,F_{i}\nu_{i},\
d\mathcal{H}^{2}-\int_{\Sigma}F_{i}\partial_{i}F_{j}\nu_{j}\
d\mathcal{H}^{2}.\end{split}$

Combining the two above equalities yield

(3.3) $\begin{split}\|\nabla
F\|_{L^{2}(\Omega)}^{2}&=\|\operatorname{div}F\|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\|\text{curl}\,F\|_{L^{2}(\Omega)}^{2}\\\
&+\int_{\Sigma}(\nabla F\,F)\cdot\nu\
d\mathcal{H}^{2}-\int_{\Sigma}\operatorname{div}F\,F_{n}\,d\mathcal{H}^{2}.\end{split}$

Assume now that $\psi$ is a divergence free vector field such that
$\psi\cdot\nu=0$ on $\Sigma$. Since $\psi$ is a tangent field on $\Sigma$ we
have $\nabla_{\psi}(\psi\cdot\nu)=0$ and thus
$\nabla\psi\,\psi\cdot\nu=-\langle\bar{\nabla}_{\psi}\nu,\psi\rangle=-\langle
B_{\Sigma}\psi,\psi\rangle$. Therefore the equality (3.1) follows from (3.3)
and from $F_{n}=\psi\cdot\nu=0$.

To obtain (3.2) we apply (3.3) for $F=\nabla u$ and deduce

$\|\nabla^{2}u\|_{L^{2}(\Omega)}^{2}=\|\Delta
u\|_{L^{2}(\Omega)}^{2}+\int_{\Sigma}(\nabla^{2}u\,\nabla u\cdot\nu-\Delta
u\,\partial_{\nu}u)\,d\mathcal{H}^{2}.$

We write $\nabla u=\nabla_{\tau}u+\partial_{\nu}u\,\nu$ and observe

$\begin{split}\nabla^{2}u\,\nabla
u\cdot\nu&=\nabla^{2}u\,\nabla_{\tau}u\cdot\nu+(\nabla^{2}u\,\nu\cdot\nu)\,\partial_{\nu}u\\\
&=\nabla_{\tau}(\partial_{\nu}u)\cdot\nabla_{\tau}u-\langle
B_{\Sigma}\,\bar{\nabla}u,\bar{\nabla}u\rangle+(\nabla^{2}u\,\nu\cdot\nu)\,\partial_{\nu}u.\end{split}$

Again by divergence theorem

$\int_{\Sigma}\nabla_{\tau}(\partial_{\nu}u)\cdot\nabla_{\tau}u\,d\mathcal{H}^{2}=-\int_{\Sigma}\Delta_{\Sigma}u\,\partial_{\nu}u\,d\mathcal{H}^{2}.$

The equality (3.2) then follows from

(3.4) $\Delta_{\Sigma}u=\Delta
u-(\nabla^{2}u\,\nu\cdot\nu)-H_{\Sigma}\,\partial_{\nu}u.$

We remark that it is crucial in Theorem 3.1 that the boundary term on the RHS
has only the normal component of the vector field. The next lemma is a
generalization of [34] and it essentially states that we may control the
vector field on the boundary by its normal or its tangential component.

######  Lemma 3.3.

Let $\Omega\subset\mathbb{R}^{3}$ with $\Sigma=\partial\Omega$ be
$C^{1}(\Gamma)$-regular. Then for all vector fields
$F\in\dot{H}^{1}(\Omega;\mathbb{R}^{3})$ it holds

$\|F\|_{L^{2}(\Sigma)}^{2}\leq
C\left(\|F_{n}\|_{L^{2}(\Sigma)}^{2}+\|F\|_{L^{2}(\Omega)}^{2}+\|\operatorname{div}F\|_{L^{2}(\Omega)}^{2}+\|\operatorname{curl}F\|_{L^{2}(\Omega)}^{2}\right)$

and

$\|F\|_{L^{2}(\Sigma)}^{2}\leq
C\left(\|F_{\tau}\|_{L^{2}(\Sigma)}^{2}+\|F\|_{L^{2}(\Omega)}^{2}+\|\operatorname{div}F\|_{L^{2}(\Omega)}^{2}+\|\operatorname{curl}F\|_{L^{2}(\Omega)}^{2},\right)$

where $F_{n}=F\cdot\nu$ and $F_{\tau}=F-F_{n}\,\nu$. Here
$F\in\dot{H}^{1}(\Omega;\mathbb{R}^{3})$ means that
$\|F\|_{\dot{H}^{1}(\Omega)}=\|\nabla
F\|_{L^{2}(\Omega)}+\|F\|_{L^{6}(\Omega)}<\infty$. Note that $\Omega$ may be
unbounded, but its boundary is compact.

###### Proof.

We only consider the case when $\Omega$ is bounded. Let us first assume that
$\Omega$ is uniformly star shaped with respect to the origin, i.e., we have
that there exists a constant $c_{0}>0$ such that $x\cdot\nu_{\Omega}\geq
c_{0}$ for all $x\in\Sigma$. We claim that the following identity holds

(3.5) $\operatorname{div}\left(|F|^{2}x-2(F\cdot
x)F\right)=|F|^{2}-2\operatorname{curl}F(F\cdot
x)-2\operatorname{div}F\,(F\cdot x).$

Indeed, this follows from the following straightforward computation, where we
denote the Dirac delta by $\delta_{i}^{j}$,

$\begin{split}\operatorname{div}(|F|^{2}x-&2(F\cdot
x)F)=\sum_{i,j=1}^{3}\partial_{i}\left(F_{j}^{2}x_{i}-2x_{j}F_{j}F_{i}\right)\\\
&=3|F|^{2}+\sum_{i,j=1}^{3}2x_{i}\partial_{i}F_{j}F_{j}-2\delta_{i}^{j}F_{j}F_{i}-2x_{j}\partial_{i}F_{j}F_{i}-2x_{j}F_{j}\partial_{i}F_{i}\\\
&=|F|^{2}-2(F\cdot
x)\operatorname{div}F-2\sum_{i,j=1}^{3}(\partial_{j}F_{i}-\partial_{i}F_{j})F_{j}x_{i}.\end{split}$

Thus we integrate (3.5) to find

$\int_{\Sigma}\big{(}(x\cdot\nu)|F|^{2}-2F_{n}(F\cdot
x)\big{)}\,d\mathcal{H}^{2}=\int_{\Omega}|F|^{2}-2\operatorname{curl}F(F\cdot
x)-2\operatorname{div}F(F\cdot x)\,dx.$

Note that $|F|^{2}=|F_{\tau}|^{2}+F_{n}^{2}$ and $(F\cdot
x)=(x\cdot\nu)F_{n}+(F_{\tau}\cdot x)$. Therefore we have the equality

$\begin{split}\int_{\Sigma}\big{(}-(x\cdot\nu)F_{n}^{2}&+(x\cdot\nu)|F_{\tau}|^{2}-2F_{n}(F_{\tau}\cdot
x)\big{)}\,d\mathcal{H}^{2}\\\
&=\int_{\Omega}|F|^{2}+2\operatorname{curl}F(F\cdot
x)-2\operatorname{div}F(F\cdot x)\,dx.\end{split}$

We use the fact that $x\cdot\nu\geq c_{0}$ on $\Sigma$ and obtain the first
claim by re-organizing the terms in above and estimating $|F_{n}(F_{\tau}\cdot
x)|\leq\varepsilon|F_{\tau}|^{2}+C_{e}F_{n}^{2}$

$\begin{split}c_{0}\int_{\Sigma}|F_{\tau}|^{2}\,d\mathcal{H}^{2}\leq&\int_{\Sigma}(\varepsilon|F_{\tau}|^{2}+C_{\varepsilon}|F_{n}|^{2})\,d\mathcal{H}^{2}\\\
&+C(\|F\|_{L^{2}(\Omega)}^{2}+\|\operatorname{curl}F\|_{L^{2}(\Omega)}^{2}+\|\operatorname{div}F\|_{L^{2}(\Omega)}^{2}).\end{split}$

This yields the first inequality. The second follows from similar argument.

To prove the general case, i.e. when $\Omega$ is not starshaped, we use a
localization argument which is similar to [1]. ∎

######  Remark 3.4.

We observe that the proof gives us a slightly stronger estimate. Indeed, we
may improve the second inequality in Lemma 3.3 as

$\|F\|_{L^{2}(\Sigma)}^{2}\leq
C\left(\|F_{\tau}\|_{L^{2}(\Sigma)}^{2}+\|F\|_{L^{2}(\Omega)}^{2}+\|F\operatorname{div}F\|_{L^{1}(\Omega)}+\|F\operatorname{curl}F\|_{L^{1}(\Omega)})\right).$

In order to estimate $\|\nabla F\|_{H^{1}(\Omega)}$ we first consider the case
when $F$ is curl-free, i.e., $F=\nabla u$. Since we define the norm
$\|\partial_{\nu}u\|_{H^{\frac{1}{2}}(\Sigma)}$ via the harmonic extension, we
prove the next lemma using standard results from harmonic analysis instead of
localizing and flattening the boundary.

######  Lemma 3.5.

Assume that $\Omega$, with $\Sigma=\partial\Omega$, is $C^{1}(\Gamma)$-regular
and $\|B_{\Sigma}\|_{L^{4}}\leq C$ and $u:\Omega\to\mathbb{R}$ is a smooth
function. Then it holds

$\|u\|_{H^{2}(\Omega)}\leq
C\left(\|\partial_{\nu}u\|_{H^{\frac{1}{2}}(\Sigma)}+\|u\|_{L^{2}(\Omega)}+\|\Delta
u\|_{L^{2}(\Omega)}\right).$

The reverse also holds $\|\partial_{\nu}u\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C\|u\|_{H^{2}(\Omega)}$.

###### Proof.

We have by (3.2) and (3.4) that

$\|\nabla^{2}u\|_{L^{2}(\Omega)}^{2}\leq\|\Delta
u\|_{L^{2}(\Omega)}^{2}+2\int_{\Sigma}(\nabla^{2}u\,\nu\cdot\nu-\Delta
u)\,\partial_{\nu}u\,d\mathcal{H}^{2}+C\int_{\Sigma}|B_{\Sigma}|\,|\nabla
u|^{2}\,d\mathcal{H}^{2}.$

To estimate the last terms, we use the interpolation inequality (Proposition
2.8), Lemma 3.3 and the assumption $\|B\|_{L^{4}}\leq C$ and have for some
$\theta\in(0,1)$

(3.6) $\begin{split}\int_{\Sigma}|B_{\Sigma}|\,|\nabla
u|^{2}\,d\mathcal{H}^{2}&\leq C\|B\|_{L^{4}(\Sigma)}\|\nabla
u\|_{L^{\frac{8}{3}}(\Sigma)}^{2}\leq\|\nabla
u\|_{H^{\frac{1}{2}}(\Sigma)}^{2\theta}\|\nabla
u\|_{L^{2}(\Sigma)}^{2(1-\theta)}\\\
&\leq\varepsilon\|u\|_{H^{2}(\Omega)}^{2}+C_{\varepsilon}\|\nabla
u\|_{L^{2}(\Sigma)}^{2}\\\
&\leq\varepsilon\|u\|_{H^{2}(\Omega)}^{2}+C_{\varepsilon}(\|\partial_{\nu}u\|_{L^{2}(\Sigma)}^{2}+\|\nabla
u\|_{L^{2}(\Omega)}^{2}+\|\Delta u\|_{L^{2}(\Omega)}^{2})\\\
&\leq\varepsilon\|u\|_{H^{2}(\Omega)}^{2}+C_{\varepsilon}(\|\partial_{\nu}u\|_{L^{2}(\Sigma)}^{2}+\|u\|_{L^{2}(\Omega)}^{2}+\|\Delta
u\|_{L^{2}(\Omega)}^{2}).\end{split}$

Let us then show that

(3.7) $\int_{\Sigma}(\nabla^{2}u\,\nu\cdot\nu-\Delta
u)\,\partial_{\nu}u\,d\mathcal{H}^{2}\leq\varepsilon\|u\|_{H^{2}(\Omega)}^{2}+C_{\varepsilon}(\|\partial_{\nu}u\|_{H^{\frac{1}{2}}(\Sigma)}^{2}+\|u\|_{L^{2}(\Omega)}^{2}+\|\Delta
u\|_{L^{2}(\Omega)}^{2}).$

To this aim we denote the harmonic extension of $\nu$ by $\tilde{\nu}$ and
denote $f=\Delta u$. Then we have

$\begin{split}\int_{\Sigma}(\nabla^{2}u\,\nu\cdot\nu)\,\partial_{\nu}u\,d\mathcal{H}^{2}&=\int_{\Sigma}\big{(}\partial_{\nu}(\nabla
u\cdot\tilde{\nu})-(\partial_{\nu}\tilde{\nu}\cdot\nabla u)\big{)}\,(\nabla
u\cdot\tilde{\nu})\,d\mathcal{H}^{2}\\\
&\leq\int_{\Sigma}\partial_{\nu}(\nabla u\cdot\tilde{\nu})\,(\nabla
u\cdot\tilde{\nu})\,d\mathcal{H}^{2}+C\|\partial_{\nu}\tilde{\nu}\|_{L^{4}(\Sigma)}\|\nabla
u\|_{L^{\frac{8}{3}}(\Sigma)}^{2}.\end{split}$

We argue as in (3.6) and obtain

$\|\nabla
u\|_{L^{\frac{8}{3}}(\Sigma)}^{2}\leq\varepsilon\|u\|_{H^{2}(\Omega)}^{2}+C_{\varepsilon}(\|\partial_{\nu}u\|_{L^{2}(\Sigma)}^{2}+\|u\|_{L^{2}(\Omega)}^{2}+\|f\|_{L^{2}(\Omega)}^{2}).$

Next we use the result from [19] for harmonic functions
$\varphi:\Omega\to\mathbb{R}$ in $C^{1,\alpha}$-domains which states that

$\|\partial_{\nu}\varphi\|_{L^{p}(\Sigma)}\leq
C_{p}\|\nabla_{\tau}\varphi\|_{L^{p}(\Sigma)}\qquad\text{for
}\,p\in(1,\infty).$

We use this for $\tilde{\nu}$ component-wise, use the fact that on $\Sigma$ it
holds $\tilde{\nu}=\nu$ and obtain

(3.8) $\|\nabla\tilde{\nu}\|_{L^{4}(\Sigma)}\leq
C\|\nabla_{\tau}\tilde{\nu}\|_{L^{4}(\Sigma)}\leq\|B\|_{L^{4}(\Sigma)}\leq C.$

Therefore we have

(3.9)
$\begin{split}\int_{\Sigma}(\nabla^{2}u\,\nu\cdot\nu)\,\partial_{\nu}u\,d\mathcal{H}^{2}\leq&\int_{\Sigma}\partial_{\nu}(\nabla
u\cdot\tilde{\nu})\,(\nabla u\cdot\tilde{\nu})\,d\mathcal{H}^{2}\\\
&+\varepsilon\|u\|_{H^{2}(\Omega)}^{2}+C_{\varepsilon}(\|\partial_{\nu}u\|_{L^{2}(\Sigma)}^{2}+\|u\|_{L^{2}(\Omega)}^{2}+\|f\|_{L^{2}(\Omega)}^{2}).\end{split}$

Let us denote $\tilde{u}=(\nabla u\cdot\tilde{\nu})$ for short and let $v$ be
the harmonic extension of $\tilde{u}$ to $\Omega$. Let us show that $v$ is
close to $\tilde{u}$, i.e., we show that

(3.10)
$\|\nabla(\tilde{u}-v)\|_{L^{2}(\Omega)}\leq\varepsilon\|\nabla^{2}u\|_{L^{2}(\Omega)}+C_{\varepsilon}(\|f\|_{L^{2}(\Omega)}+\|\tilde{u}-v\|_{L^{2}(\Omega)}).$

To this aim we calculate (recall that $f=\Delta u$)

(3.11) $\Delta\tilde{u}=\nabla
f\cdot\tilde{\nu}+2\nabla^{2}u:\nabla\tilde{\nu}.$

This implies by integration by parts

$\begin{split}\|\nabla&(\tilde{u}-v)\|_{L^{2}(\Omega)}^{2}=-\int_{\Omega}\Delta\tilde{u}(\tilde{u}-v)\,dx=\int_{\Omega}(\nabla
f\cdot\tilde{\nu}+2\nabla^{2}u:\nabla\tilde{\nu})(\tilde{u}-v)\,dx\\\
&=\int_{\Omega}f\operatorname{div}\,((\tilde{u}-v)\,\tilde{\nu})-2(\nabla^{2}u:\nabla\tilde{\nu})(\tilde{u}-v)\,dx\\\
&\leq
C\|\nabla(\tilde{u}-v)\|_{L^{2}(\Omega)}\|f\|_{L^{2}(\Omega)}+C(\|\nabla^{2}u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)})\|\nabla\tilde{\nu}\|_{L^{4}(\Omega)}\|(\tilde{u}-v)\|_{L^{4}(\Omega)}.\end{split}$

By standard estimates from harmonic analysis [16] and by (3.8) it holds

(3.12) $\|\nabla\tilde{\nu}\|_{L^{4}(\Omega)}\leq
C\|\nabla\tilde{\nu}\|_{L^{4}(\Sigma)}\leq C.$

On the other hand we have by Hölder’s inequality and by Sobolev embedding
(recall that $\tilde{u}-v=0$ on $\Sigma$)

$\|(\tilde{u}-v)\|_{L^{4}(\Omega)}\leq\|(\tilde{u}-v)\|_{L^{6}(\Omega)}^{\frac{1}{2}}\|\tilde{u}-v\|_{L^{2}(\Omega)}^{\frac{1}{2}}\leq
C\|\nabla(\tilde{u}-v)\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|\tilde{u}-v\|_{L^{2}(\Omega)}^{\frac{1}{2}}.$

Therefore by combining the previous inequalities we obtain (3.10) by Young’s
inequality.

We proceed by using (3.11) and by integrating by parts

$\begin{split}\int_{\Sigma}&\partial_{\nu}\tilde{u}\,\tilde{u}\,d\mathcal{H}^{2}=\int_{\Sigma}\partial_{\nu}\tilde{u}\,v\,d\mathcal{H}^{2}=\int_{\Omega}(\nabla\tilde{u}\cdot\nabla
v+\Delta\tilde{u}\,v)\,dx\\\ &\leq 2\|\nabla
v\|_{L^{2}(\Omega)}^{2}+2\|\nabla(\tilde{u}-v)\|_{L^{2}(\Omega)}^{2}+\int_{\Omega}(\nabla
f\cdot\tilde{\nu}+2\nabla^{2}u:\nabla\tilde{\nu})\,v\,dx\\\ &=2\|\nabla
v\|_{L^{2}(\Omega)}^{2}+2\|\nabla(\tilde{u}-v)\|_{L^{2}(\Omega)}^{2}+\int_{\Sigma}f\,v\,(\tilde{\nu}\cdot\nu)\,d\mathcal{H}^{2}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\int_{\Omega}-f\operatorname{div}\,(v\,\tilde{\nu})+2(\nabla^{2}u:\nabla\tilde{\nu})v\,dx.\end{split}$

Recall that $\tilde{\nu}=\nu$ and $v=\tilde{u}$ on $\Sigma$. Therefore we
obtain by the above inequality, by $\|v\|_{L^{4}(\Omega)}\leq
C\|v\|_{H^{1}(\Omega)}$, (3.10) and (3.12) that

$\begin{split}\int_{\Sigma}\partial_{\nu}\tilde{u}\,\tilde{u}-f\,\tilde{u}\,d\mathcal{H}^{2}&\leq\varepsilon\|\nabla^{2}u\|_{L^{2}(\Omega)}\\\
&\,\,\,\,\,\,\,\,\,\,\,+C_{\varepsilon}(\|\nabla\nu\|_{L^{4}(\Omega)}\|v\|_{L^{4}(\Omega)}+\|v\|_{H^{1}(\Omega)}^{2}+\|f\|_{L^{2}(\Omega)}^{2}+\|\tilde{u}-v\|_{L^{2}(\Omega)}^{2})\\\
&\leq\varepsilon\|\nabla^{2}u\|_{L^{2}(\Omega)}+C_{\varepsilon}(\|v\|_{H^{1}(\Omega)}^{2}+\|f\|_{L^{2}(\Omega)}^{2}+\|\tilde{u}-v\|_{L^{2}(\Omega)}^{2}).\end{split}$

The inequality (3.7) then follows from the above and (3.9) together with

$\|v\|_{H^{1}(\Omega)}=\|\partial_{\nu}u\|_{H^{\frac{1}{2}}(\Sigma)},\qquad\|\tilde{u}-v\|_{L^{2}(\Omega)}^{2}\leq\varepsilon\|\nabla^{2}u\|_{L^{2}(\Omega)}+C_{\varepsilon}\|u\|_{L^{2}(\Omega)}^{2}+\|v\|_{L^{2}(\Omega)}^{2},$

and by recalling that $\tilde{u}=(\nabla u\cdot\tilde{\nu})=\partial_{\nu}u$
on $\Sigma$ and $f=\Delta u$. This yields the first claim. The second
inequality follows from reversing the previous calculations. ∎

We state our lower order version of Theorem 3.1.

######  Theorem 3.6.

Assume that $\Omega$, with $\Sigma=\partial\Omega$, is $C^{1}(\Gamma)$-regular
and $\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq C$. Then for all vector fields $F\in
H^{1}(\Omega;\mathbb{R}^{3})$ it holds

$\|F\|_{H^{1}(\Omega)}\leq
C(\|F_{n}\|_{H^{\frac{1}{2}}(\Sigma)}+\|F\|_{L^{2}(\Omega)}+\|\operatorname{div}F\|_{L^{2}(\Omega)}+\|\operatorname{curl}F\|_{L^{2}(\Omega)})$

###### Proof.

By approximation argument we may assume that $F$ and $\Omega$ are smooth. We
use the Helmholtz-Hodge decomposition and write $F=\nabla\phi+\psi$ where
$\phi$ is the unique solution of the Neuman problem

$\begin{cases}\Delta\phi=\operatorname{div}F\qquad&x\in\Omega\\\
\partial_{\nu}\phi=F_{n}&x\in\Sigma\end{cases}$

with zero average and $\psi$ solves

$\begin{cases}\operatorname{curl}\psi=\operatorname{curl}F\qquad&x\in\Omega\\\
\operatorname{div}\psi=0&x\in\Omega\\\ \psi\cdot\nu=0&x\in\Sigma.\end{cases}$

We also note that $\nabla\phi$ and $\psi$ are orthogonal in $L^{2}(\Omega)$
and thus

$\int_{\Omega}|\nabla\phi|^{2}+|\psi|^{2}dx=\int_{\Omega}|F|^{2}dx.$

For $\phi$ we have by Lemma 3.5 that

(3.13) $\begin{split}\|\phi\|_{H^{2}(\Omega)}&\leq
C(\|\partial_{\nu}\phi\|_{H^{\frac{1}{2}}(\Sigma)}+\|\nabla\phi\|_{L^{2}(\Omega)}+\|\Delta\phi\|_{L^{2}(\Omega)})\\\
&\leq
C(\|F_{n}\|_{H^{\frac{1}{2}}(\Sigma)}+\|F\|_{L^{2}(\Omega)}+\|\operatorname{div}F\|_{L^{2}(\Omega)}).\end{split}$

For $\psi$ we have by (3.1)

$\|\nabla\psi\|_{L^{2}(\Omega)}^{2}\,dx=\frac{1}{2}\|\text{curl}\,\psi\|_{L^{2}(\Omega)}^{2}-\int_{\Sigma}\langle
B_{\Sigma}\,\psi,\psi\rangle\,d\mathcal{H}^{2}.$

We use the assumption $\|B_{\Sigma}\|_{L^{4}}\leq C$, Hölder’s inequality and
interpolation inequality to deduce

$\begin{split}-\int_{\Sigma}\langle
B_{\Sigma}\,\psi,\psi\rangle\,d\mathcal{H}^{2}&\leq\|B_{\Sigma}\|_{L^{4}(\Sigma)}\|\psi\|_{L^{\frac{8}{3}}(\Sigma)}^{2}\\\
&\leq\varepsilon\|\psi\|_{H^{\frac{1}{2}}(\Sigma)}^{2}+C_{\varepsilon}\|\psi\|_{L^{2}(\Omega)}^{2}\leq\varepsilon\|\nabla\psi\|_{L^{2}(\Omega)}^{2}+C_{\varepsilon}\|F\|_{L^{2}(\Omega)}^{2}.\end{split}$

Thus we have

$\|\nabla\psi\|_{L^{2}(\Omega)}\leq
C(\|\operatorname{curl}F\|_{L^{2}(\Omega)}+\|F\|_{L^{2}(\Omega)})$

This together with (3.13) yields the claim. ∎

We proceed by using Theorem 3.6 to control the higher order norms
$\|F\|_{H^{2}(\Omega)}$ and $\|F\|_{H^{3}(\Omega)}$. We do not need the sharp
dependence on the curvature for these estimates and we state the result in a
form that is suitable for us. We also treat the case
$\|F\|_{H^{\frac{3}{2}}(\Omega)}$, but only for curl-free vector fields. In
this case we need the ’sharp’ curvature dependence but this time we have non-
optimal dependence on the divergence.

######  Lemma 3.7.

Assume that $\Omega$, with $\Sigma=\partial\Omega$, is $C^{1}(\Gamma)$-regular
and $\|B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}\leq C$. Then for all vector
fields $F\in H^{3}(\Omega;\mathbb{R}^{3})$ it holds

$\|F\|_{H^{3}(\Omega)}\leq
C\big{(}\|\Delta_{\Sigma}F_{n}\|_{H^{\frac{1}{2}}(\Sigma)}+(1+\|H_{\Sigma}\|_{H^{2}(\Sigma)})\|F\|_{L^{\infty}(\Omega)}+\|\operatorname{div}F\|_{H^{2}(\Omega)}+\|\operatorname{curl}F\|_{H^{2}(\Omega)}\big{)}$

and

$\|F\|_{H^{2}(\Omega)}\leq
C\big{(}\|\Delta_{\Sigma}F_{n}\|_{H^{\frac{1}{2}}(\Sigma)}+\|F\|_{L^{\infty}(\Omega)}+\|\operatorname{div}F\|_{H^{1}(\Omega)}+\|\operatorname{curl}F\|_{H^{1}(\Omega)}\big{)}.$

Moreover, if $F=\nabla u$ then it holds

$\|\nabla u\|_{H^{\frac{3}{2}}(\Omega)}\leq
C(\|\partial_{\nu}u\|_{H^{1}(\Sigma)}+\|u\|_{L^{2}(\Omega)}+\|\Delta
u\|_{H^{1}(\Omega)})$

and for $k=\frac{1}{2},1$ it holds

$\|u\|_{H^{k}(\Sigma)}\leq C\|u\|_{H^{k+\frac{1}{2}}(\Omega)}.$

###### Proof.

By approximation argument we may assume that $F$ and $\Omega$ are smooth. Note
also that by Sobolev embedding
$\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq\|B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C$.

Let $\tilde{\nu}$ be the harmonic extension of the normal field $\nu$ to
$\Omega$. Let us define the vector fields $\tau_{i}=e_{i}-(\tilde{\nu}\cdot
e_{i})\tilde{\nu}$ for $i=1,2,3$, where $\\{e_{i}\\}_{i}$ is a coordinate
basis of $\mathbb{R}^{3}$. For $i,j$ we define a vector field
$F_{ij}:\Omega\to\mathbb{R}^{3}$ as
$(F_{ij})_{k}=\nabla^{2}F_{k}\tau_{i}\cdot\tau_{j}$. We apply Theorem 3.6 for
$F_{ij}$ and obtain

$\|\nabla F_{ij}\|_{L^{2}(\Omega)}\leq
C(\|F_{ij}\cdot\nu\|_{H^{\frac{1}{2}}(\Sigma)}+\|F_{ij}\|_{L^{2}(\Omega)}+\|\operatorname{div}F_{ij}\|_{L^{2}(\Omega)}+\|\operatorname{curl}F_{ij}\|_{L^{2}(\Omega)}).$

Recall that (3.12) implies $\|\nabla\tilde{\nu}\|_{L^{4}(\Omega)}\leq C$.
Moreover by maximum principle it holds
$\|\tilde{\nu}\|_{L^{\infty}(\Omega)}\leq C$. Therefore

$\begin{split}\|\operatorname{div}F_{ij}\|_{L^{2}(\Omega)}&\leq
C\|\operatorname{div}F\|_{H^{2}(\Omega)}+C\|\nabla^{2}F\|_{L^{4}(\Omega)}\|\nabla\nu\|_{L^{4}(\Omega)}\\\
&\leq
C\|\operatorname{div}F\|_{H^{2}(\Omega)}+C\|\nabla^{2}F\|_{L^{4}(\Omega)}.\end{split}$

By interpolation we have
$\|\nabla^{2}F\|_{L^{4}(\Omega)}\leq\varepsilon\|F\|_{H^{3}(\Omega)}+C_{\varepsilon}\|F\|_{L^{2}(\Omega)}$
and thus

$\|F_{ij}\|_{L^{2}(\Omega)}+\|\operatorname{div}F_{ij}\|_{L^{2}(\Omega)}\leq\varepsilon\|F\|_{H^{3}(\Omega)}+C_{\varepsilon}(\|\operatorname{div}F\|_{H^{2}(\Omega)}+\|F\|_{L^{2}(\Omega)}).$

By a similar argument

$\|\operatorname{curl}F_{ij}\|_{L^{2}(\Omega)}\leq\varepsilon\|F\|_{H^{3}(\Omega)}+C_{\varepsilon}(\|\operatorname{curl}F\|_{H^{2}(\Omega)}+\|F\|_{L^{2}(\Omega)})$

and

$\|\nabla
F_{ij}\|_{L^{2}(\Omega)}^{2}\geq\sum_{k,l=1}^{3}\|\nabla^{2}\nabla_{l}F_{k}\,\tau_{i}\cdot\tau_{j}\|_{L^{2}(\Omega)}^{2}-\varepsilon\|F\|_{H^{3}(\Omega)}^{2}-C_{\varepsilon}\|F\|_{L^{2}(\Omega)}^{2}.$

Let us fix a point $x\in\Omega$ and estimate the norm

$\sum_{i,j,k,l=1}^{3}|\nabla^{2}\nabla_{l}F_{k}(x)\,\tau_{i}\cdot\tau_{j}|^{2}.$

First we observe that the above quantity does not depend on the choice of the
coordinates in $\mathbb{R}^{3}$. Let us choose the coordinates such that
$\tilde{\nu}(x)\cdot e_{i}=0$ for $i=1,2$. Then we have

$\sum_{i,j,k,l=1}^{3}|\nabla^{2}\nabla_{l}F_{k}(x)\,\tau_{i}\cdot\tau_{j}|^{2}\geq\sum_{k,l=1}^{3}\sum_{i,j=1}^{2}|\nabla_{i}\nabla_{j}\nabla_{l}F_{k}(x)|^{2}.$

By a simple combinatorial argument we deduce

$\begin{split}&\sum_{i,j,k,l=1}^{3}|\nabla_{i}\nabla_{j}\nabla_{l}F_{k}(x)|^{2}\\\
&\leq
C\sum_{k,l=1}^{3}\sum_{i,j=1}^{2}|\nabla_{i}\nabla_{j}\nabla_{l}F_{k}(x)|^{2}+C|\nabla^{2}\operatorname{div}F(x)|^{2}+C|\nabla^{2}\operatorname{curl}F(x)|^{2}.\end{split}$

By applying the above argument for every $x$ we have

$\sum_{i,j,k,l=1}^{3}\|\nabla F_{ij}\|_{L^{2}(\Omega)}^{2}\geq
c_{0}\|\nabla^{3}F\|_{L^{2}(\Omega)}^{2}-C(\|\operatorname{div}F\|_{H^{2}(\Omega)}^{2}+\|\operatorname{curl}F\|_{H^{2}(\Omega)}^{2}+\|F\|_{L^{2}(\Omega)}^{2}).$

Combing all the previous estimates we obtain

$\|\nabla^{3}F\|_{L^{2}(\Omega)}\leq\sum_{i,j=1}^{3}C(\|F_{ij}\cdot\nu\|_{H^{\frac{1}{2}}(\Sigma)}+\|F\|_{L^{2}(\Omega)}+\|\operatorname{div}F\|_{H^{2}(\Omega)}+\|\operatorname{curl}F\|_{H^{2}(\Omega)}).$

The first inequality follows once we show

(3.14) $\sum_{i,j=1}^{3}\|F_{ij}\cdot\nu\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C\|\Delta_{\Sigma}F_{n}\|_{H^{\frac{1}{2}}(\Sigma)}+\varepsilon\|F\|_{H^{3}(\Omega)}+C_{\varepsilon}(1+\|H_{\Sigma}\|_{H^{2}})\|F\|_{L^{\infty}(\Sigma)}.$

To this aim we first note that on $\Sigma$ it holds $\tau_{i}=e_{i}-(\nu\cdot
e_{i})\nu$ and therefore $\tau_{i}$ is tangential on $\Sigma$. Thus we have

(3.15)
$(\nabla^{2}F_{n})\tau_{i}\cdot\tau_{j}=F_{ij}\cdot\nu+\nabla_{\tau}F\star
B_{\Sigma}+F\star\nabla_{\tau}B_{\Sigma}.$

We use Proposition 2.10 and get

$\|\nabla_{\tau}F\star B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}\leq C\|\nabla
F\|_{H^{\frac{1}{2}}(\Sigma)}\|B_{\Sigma}\|_{L^{\infty}(\Sigma)}+C\|\nabla_{\tau}F\|_{L^{\infty}(\Sigma)}\|B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}.$

Recall that $\|B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}\leq C$. By interpolation
we have

$\|\nabla_{\tau}F\|_{L^{\infty}(\Sigma)}\leq\varepsilon\|F\|_{H^{3}(\Omega)}+C_{\varepsilon}\|F\|_{L^{\infty}}.$

Moreover, by Sobolev embedding, Proposition 2.8 and by Proposition 2.12 we
have for $\theta<\frac{1}{2}$

$\|B_{\Sigma}\|_{L^{\infty}(\Sigma)}\leq
C\|B_{\Sigma}\|_{W^{1,\frac{7}{3}}(\Sigma)}\leq
C\|B_{\Sigma}\|_{H^{2}(\Sigma)}^{\theta}\|B_{\Sigma}\|_{L^{4}(\Sigma)}^{1-\theta}\leq
C(1+\|H_{\Sigma}\|_{H^{2}(\Sigma)}^{\theta}).$

On the other hand, Corollary 2.9 implies

$\|\nabla F\|_{H^{\frac{1}{2}}(\Sigma)}\leq C\|\nabla F\|_{H^{1}(\Omega)}\leq
C\|F\|_{H^{3}(\Omega)}^{\frac{1}{2}}\|\nabla
F\|_{L^{2}(\Omega)}^{\frac{1}{2}}.$

Therefore we have by the above estimates and by Young’s inequality

$\begin{split}\|\nabla_{\tau}F\star
B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}&\leq\varepsilon\|F\|_{H^{3}(\Omega)}+C_{\varepsilon}\|F\|_{L^{\infty}}+C_{\varepsilon}(1+\|H_{\Sigma}\|_{H^{2}(\Sigma)}^{\theta})\|\nabla
F\|_{L^{2}(\Omega)}\\\
&\leq\varepsilon\|F\|_{H^{3}(\Omega)}+C_{\varepsilon}(1+\|H_{\Sigma}\|_{H^{2}(\Sigma)})\|F\|_{L^{\infty}(\Omega)},\end{split}$

where the last inequality follows from interpolation.

Let us then bound the last term in (3.15). We have by Proposition 2.10

$\begin{split}\|F\star\nabla_{\tau}B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}&\leq\|F\star\nabla_{\tau}B_{\Sigma}\|_{H^{1}(\Sigma)}\\\
&\leq
C\|F\|_{L^{\infty}(\Sigma)}\|\bar{\nabla}B_{\Sigma}\|_{H^{1}(\Sigma)}+C\|F\|_{W^{1,3}(\Sigma)}\|\bar{\nabla}B_{\Sigma}\|_{L^{6}(\Sigma)}.\end{split}$

Proposition 2.12 yields $\|B_{\Sigma}\|_{H^{2}(\Sigma)}\leq
C(1+\|H_{\Sigma}\|_{H^{2}(\Sigma)})$. Interpolation implies

$\|\bar{\nabla}B_{\Sigma}\|_{L^{6}(\Sigma)}\leq
C\|B_{\Sigma}\|_{H^{2}(\Sigma)}^{\theta}\|B_{\Sigma}\|_{L^{2}(\Sigma)}^{1-\theta}$

and

$\|F\|_{W^{1,3}(\Sigma)}\leq\|F\|_{H^{3}(\Omega)}^{1-\theta}\|F\|_{L^{\infty}(\Omega)}^{\theta}$

for some $\theta\in(0,1)$. Therefore we have by (3.15)

$\|F_{ij}\cdot\nu\|_{H^{\frac{1}{2}}(\Sigma)}\leq\|\bar{\nabla}^{2}F_{n}\|_{H^{\frac{1}{2}}(\Sigma)}+\varepsilon\|F\|_{H^{3}(E)}+C_{\varepsilon}(1+\|H_{\Sigma}\|_{H^{2}(\Sigma)})\|F\|_{L^{\infty}(\Omega)}.$

The inequality (3.14) then follows from Proposition 2.11 as

$\|\bar{\nabla}^{2}F_{n}\|_{H^{\frac{1}{2}}(\Sigma)}\leq
2\|\Delta_{\Sigma}F_{n}\|_{H^{\frac{1}{2}}(\Sigma)}+C_{\varepsilon}\|F_{n}\|_{L^{2}(\Sigma)}.$

The second inequality follows from a similar argument.

Let us next prove the last part of the statement, i.e. the inequalities when
$F=\nabla u$. Let $u$ be a solution of the Neumann boundary problem

$\begin{cases}\Delta u=f\qquad&\text{in }\Omega\\\ \partial_{\nu}u=g&\text{on
}\Sigma,\end{cases}$

where $\int_{\Sigma}g=\int_{\Omega}f$ and $\int_{\Omega}u\,dx=0$. First,
clearly $\|u\|_{H^{\frac{1}{2}}(\Sigma)}\leq C\|u\|_{H^{1}(\Omega)}$. By the
equation and by divergence theorem

$\int_{\Omega}|\nabla
u|^{2}\,dx+\int_{\Omega}u\,f\,dx=\int_{\Sigma}u\,\partial_{\nu}u\,d\mathcal{H}^{2}\leq\|u\|_{H^{\frac{1}{2}}(\Sigma)}\|g\|_{H^{-\frac{1}{2}}(\Sigma)}\leq
C\|u\|_{H^{1}(\Omega)}\|g\|_{H^{-\frac{1}{2}}(\Sigma)}.$

Therefore by

$\left|\int_{\Omega}u\,f\,dx\right|\leq\|u\|_{L^{2}(\Omega)}\|f\|_{L^{2}(\Omega)}$

and by Poincaré inequality $\|u\|_{L^{2}(\Omega)}\leq C\|\nabla
u\|_{L^{2}(\Omega)}$ we have

$\|u\|_{H^{1}(\Omega)}\leq
C(\|g\|_{H^{-\frac{1}{2}}(\Sigma)}+\|f\|_{L^{2}(\Omega)}).$

On the other hand Lemma 3.5 implies

$\begin{split}\|u\|_{H^{2}(\Omega)}&\leq
C(\|g\|_{H^{\frac{1}{2}}(\Sigma)}+\|u\|_{L^{2}}+\|f\|_{L^{2}(\Omega)})\\\
&\leq C(\|g\|_{H^{\frac{1}{2}}(\Sigma)}+\|f\|_{L^{2}(\Omega)}).\end{split}$

We use the two above inequalities and standard interpolation argument to
deduce

(3.16) $\|u\|_{H^{3/2}(\Omega)}\leq
C(\|g\|_{L^{2}(\Sigma)}+\|f\|_{L^{2}(\Omega)}).$

We proceed by applying (3.16) for $u_{x_{i}}=\nabla u\cdot e_{i}-c_{i}$, for
$i=1,2,3$, $c_{i}=\fint_{E}u_{x_{i}}$, and obtain

(3.17) $\|\nabla u\|_{H^{3/2}(\Omega)}\leq C(\|\partial_{\nu}(\nabla
u)\|_{L^{2}(\Sigma)}+\|f\|_{H^{1}(\Omega)}).$

In order to treat the first term on the RHS we let $\tilde{\nu}$ be the
harmonic extension of $\nu$ to $\Omega$. We write $\nabla
u=\nabla_{\tau}u+(\partial_{\nu}u)\,\nu$ and have

$\partial_{\nu}(\nabla u)=\nabla_{\tau}(\partial_{\nu}u)+\partial_{\nu}(\nabla
u\cdot\tilde{\nu})\nu+\nabla\tilde{\nu}\star\nabla u.$

Recall that we have by maximum principle $\|\tilde{\nu}\|_{L^{\infty}}\leq C$
and by (3.12) $\|\nabla\tilde{\nu}\|_{L^{4}(\Omega)}\leq C$. We argue as in
(3.6) and obtain

$\|\nabla\tilde{\nu}\star\nabla
u\|_{L^{2}(\Sigma)}\leq\|\nabla\tilde{\nu}\|_{L^{4}(\Sigma)}\,\|\nabla
u\|_{L^{4}(\Sigma)}\leq C\|u\|_{H^{2}(\Omega)}.$

We use Remark 3.4 for $F=\nabla(\nabla u\cdot\tilde{\nu})$ and (3.11) and have

$\begin{split}\|\partial_{\nu}(\nabla
u\cdot\tilde{\nu})\|_{L^{2}(\Sigma)}^{2}&\leq C(\|\nabla_{\tau}(\nabla
u\cdot\tilde{\nu})\|_{L^{2}(\Sigma)}^{2}+\|\nabla(\nabla
u\cdot\tilde{\nu})\,\Delta(\nabla
u\cdot\tilde{\nu})\|_{L^{1}(\Omega)}+\|\nabla
u\cdot\tilde{\nu}\|_{H^{1}(\Omega)}^{2})\\\ &\leq
C(\|\partial_{\nu}u\|_{H^{1}(\Sigma)}^{2}+\|u\|_{H^{2}(\Omega)}^{2}+\|f\|_{H^{1}(\Omega)}^{2}+\|\nabla
u\cdot\tilde{\nu}\|_{H^{1}(\Omega)}^{2})\\\
&\,\,\,\,\,\,\,+C(\|\nabla^{2}u\star\nabla^{2}u\star\nabla\tilde{\nu}\|_{L^{1}(\Omega)}+\|\nabla^{2}u\star\nabla
u\star\nabla\tilde{\nu}\star\nabla\tilde{\nu}\|_{L^{1}(\Omega)}).\end{split}$

First, we obtain by using the previous estimates

$\|\nabla u\cdot\tilde{\nu}\|_{H^{1}(\Omega)}^{2}\leq
C\|u\|_{H^{2}(\Omega)}^{2}.$

We bound the second last term by Hölder’s inequality and by the Sobolev
embedding

$\|\nabla^{2}u\star\nabla^{2}u\star\nabla\tilde{\nu}\|_{L^{1}(\Omega)}\leq
C\|\nabla\tilde{\nu}\|_{L^{4}(\Omega)}\|\nabla^{2}u\|_{L^{\frac{8}{3}}(\Omega)}^{2}\leq\varepsilon\|\nabla
u\|_{H^{\frac{3}{2}}(\Omega)}^{2}+C_{\varepsilon}\|u\|_{H^{2}(\Omega)}^{2}.$

Similarly we estimate the last term

$\begin{split}\|\nabla^{2}u\star\nabla
u\star\nabla\tilde{\nu}\star\nabla\tilde{\nu}\|_{L^{1}(\Omega)}&\leq
C\|\nabla\tilde{\nu}\|_{L^{4}(\Omega)}^{2}\|\nabla^{2}u\|_{L^{\frac{8}{3}}(\Omega)}\|\nabla
u\|_{L^{8}(\Omega)}\\\ &\leq\varepsilon\|\nabla
u\|_{H^{\frac{3}{2}}(\Omega)}^{2}+C_{\varepsilon}\|u\|_{H^{2}(\Omega)}^{2}.\end{split}$

Therefore we have

$\|\partial_{\nu}(\nabla u)\|_{L^{2}(\Sigma)}\leq\varepsilon\|\nabla
u\|_{H^{\frac{3}{2}}(\Omega)}+C_{\varepsilon}(\|\partial_{\nu}u\|_{H^{1}(\Sigma)}+\|u\|_{H^{2}(\Omega)}+\|f\|_{H^{1}(\Omega)}).$

Recall that we have

$\|u\|_{H^{2}(\Omega)}\leq
C(\|g\|_{H^{\frac{1}{2}}(\Sigma)}+\|f\|_{L^{2}(\Omega)}).$

Therefore the third inequality follows from (3.17).

For the last inequality we recall that while the Trace operator is not bounded
$T:H^{\frac{1}{2}}(\mathbb{R}^{3})\to L^{2}(\mathbb{R}^{2})$ it is bounded as
$T:H^{\frac{3}{2}}(\mathbb{R}^{3})\to H^{1}(\mathbb{R}^{2})$. We prove the
statement by localization argument similar to the one in the proof of
Proposition 2.10 and we only give the sketch of the proof.

We cover $\Sigma$ with balls of radius $\delta$,
$B_{\delta}(x_{i}),i=1,\dots,N$ such that the set $\Sigma\cap
B_{2\delta}(x_{i})$ is contained in the graph of $\phi_{i}$ and $\Omega$ is
above the graph. We denote the partition of unity by $\eta_{i}$. Let us fix
$i$ and we may assume that $x_{i}=0$ and $\phi_{i}(0)=\nabla\phi_{i}(0)=0$. By
the regularity assumptions it holds
$\|\phi_{i}\|_{C^{1,\alpha}(\mathbb{R}^{2})},\|\phi_{i}\|_{W^{2,4}(\mathbb{R}^{2})}\leq
C$. We define $u_{i}(x)=\eta_{i}(x)u_{i}(x)$ and
$v_{i}(x^{\prime},x_{3})=u_{i}(x^{\prime},x_{3}+\phi_{i}(x^{\prime}))$ for
$x_{3}\geq 0$ and extend $v_{i}$ to $\mathbb{R}^{3}$ by the extension
operator. Then we have by the Trace Theorem

$\|u_{i}\|_{H^{1}(\Sigma\cap B_{\delta})}\leq
C\|v_{i}\|_{H^{1}(\mathbb{R}^{2})}\leq
C\|v_{i}\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}.$

Recall that the assumption $\|B_{\Sigma}\|_{L^{4}}\leq C$ guarantees that
$\Omega$ is an $H^{2}$-extension domain. Therefore it holds
$\|v_{i}\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}\leq
C\|u\|_{H^{\frac{3}{2}}(\Omega)}$ and the last inequality follows. ∎

###  3.2. Regularity estimates for functions

In this subsection we prove regularity estimates for functions
$u:\Omega\to\mathbb{R}$ defined as a solution of the Dirichlet problem

(3.18) $\begin{cases}\Delta u=f\qquad&x\in\Omega\\\
u=g\qquad&x\in\Sigma\end{cases}$

We first consider the case when $g=0$ and improve in this case the third
inequality in Lemma 3.7. Here we assume that the boundary has the regularity
$\|B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}\leq C$. Note that by the Sobolev
embedding this implies $\|B_{\Sigma}\|_{L^{4}(\Sigma)}\leq C$.

######  Proposition 3.8.

Assume $\Omega$, with $\Sigma=\partial\Omega$, is $C^{1}(\Gamma)$-regular and
$\|B_{\Sigma}\|_{H^{\frac{1}{2}}(\Sigma)}\leq C$. Then the solution of the
problem (3.18) with zero Dircihlet boundary datum, i.e., $g=0$, satisfies

$\|\partial_{\nu}u\|_{H^{1}(\Sigma)}+\|\nabla
u\|_{H^{\frac{3}{2}}(\Omega)}\leq C\|f\|_{H^{\frac{1}{2}}(\Omega)}.$

###### Proof.

First we note that since $u=0$ on $\Sigma$ then by (3.2) we have

$\|\nabla^{2}u\|_{L^{2}(\Omega)}^{2}=\|f\|_{L^{2}(\Omega)}^{2}-\int_{\Sigma}H_{\Sigma}|\partial_{\nu}u|^{2}\,d\mathcal{H}^{2}.$

By (3.6) it holds

$-\int_{\Sigma}H_{\Sigma}|\partial_{\nu}u|^{2}\,d\mathcal{H}^{2}\leq\varepsilon\|u\|_{H^{2}(\Omega)}^{2}+C_{\varepsilon}\|\nabla
u\|_{L^{2}(\Sigma)}^{2}.$

We apply Lemma 3.3 for $F=\nabla u$ and recall that $u=0$ on $\Sigma$ to
deduce

$\begin{split}\|\nabla u\|_{L^{2}(\Sigma)}&\leq
C(\|\nabla_{\tau}u\|_{L^{2}(\Sigma)}+\|u\|_{H^{1}(\Omega)}+\|f\|_{L^{2}(\Omega)})\\\
&\leq\varepsilon\|u\|_{H^{2}(\Omega)}+C_{\varepsilon}(\|u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}).\end{split}$

Therefore we have

$\|\nabla^{2}u\|_{L^{2}(\Omega)}^{2}\leq
C(\|u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)})$

We bound $\|u\|_{L^{2}(\Omega)}$ simply by multiplying the equation (3.18) by
$u$ and integrating by parts $\|\nabla
u\|_{L^{2}(\Omega)}^{2}\leq\|f\|_{L^{2}(\Omega)}\|u\|_{L^{2}(\Omega)}$.
Poincaré inequality then implies $\|u\|_{L^{2}(\Omega)}\leq
C\|f\|_{L^{2}(\Omega)}$ and we have

(3.19) $\|u\|_{H^{2}(\Omega)}\leq C\|f\|_{L^{2}(\Omega)}.$

Let $\tilde{\nu}$ be the harmonic extension of the normal field and let us
define $\tau_{i}=e_{i}-\langle e_{i},\tilde{\nu}\rangle\tilde{\nu}$ as in the
proof of Lemma 3.7. Define $u_{i}=\nabla u\cdot\tau_{i}$. Observe that
$u_{i}=0$ on $\Sigma$ and apply (3.19) to deduce

(3.20) $\|\nabla^{2}u_{i}\|_{L^{2}(\Omega)}\leq C\|\Delta
u_{i}\|_{L^{2}(\Omega)}.$

We have (recall $\Delta u=f$)

$\Delta u_{i}=\nabla f\star\tau_{i}+\nabla^{2}u\star\nabla\tilde{\nu}+\nabla
u\star\nabla\tilde{\nu}\star\nabla\tilde{\nu}.$

Arguing similarly as in the proof of Lemma 3.5 and using (3.19) yields

(3.21) $\|\Delta
u_{i}\|_{L^{2}(\Omega)}\leq\varepsilon\|u\|_{H^{3}(\Omega)}+C\|f\|_{H^{1}(\Omega)}$

Let us then treat the LHS of (3.20). We have (recall that
$\tau_{i}=e_{i}-\langle e_{i},\tilde{\nu}\rangle\tilde{\nu}$)

$\nabla_{j}\nabla_{k}u_{i}=\nabla(\nabla_{j}\nabla_{k}u)\cdot\tau_{i}+\nabla^{2}u\star\nabla\tilde{\nu}+\nabla
u\star\nabla\tilde{\nu}\star\nabla\tilde{\nu}+\nabla
u\star\nabla^{2}\tilde{\nu}.$

Therefore arguing as in the proof of Lemma 3.5, we obtain

(3.22)
$\begin{split}\|\nabla^{2}u_{i}\|_{H^{2}(\Omega)}\geq\sum_{i,j,k=1}^{3}&\|\nabla(\nabla_{j}\nabla_{k}u)\cdot\tau_{i}\|_{L^{2}(\Omega)}\\\
&-\varepsilon\|u\|_{H^{3}(\Omega)}-C_{\varepsilon}\|f\|_{H^{1}(\Omega)}-C\|\nabla
u\|_{L^{\infty}(\Omega)}\|\nabla^{2}\tilde{\nu}\|_{L^{2}(\Omega)}.\end{split}$

Let us fix a point $x\in\Omega$ and as in the proof of Lemma 3.7 we may assume
that $\tilde{\nu}(x)\cdot e_{i}=0$ for $i=1,2$. Then it is easy to see that

$\begin{split}\sum_{i,j,k=1}^{3}|\nabla(\nabla_{j}\nabla_{k}u(x))\cdot\tau_{i}|^{2}&\geq\sum_{i=1}^{2}\sum_{j,k=1}^{3}|\langle\nabla\nabla_{j}\nabla_{k}u(x),\tau_{i}\rangle|^{2}\\\
&\geq
c\sum_{i,j,k=1}^{3}|\nabla_{i}\nabla_{j}\nabla_{k}u(x)|^{2}-C|\nabla\Delta
u(x)|^{2}.\end{split}$

This together with (3.22) yields

(3.23) $\|u_{i}\|_{H^{2}(\Omega)}\geq
c\|\nabla^{3}u\|_{L^{2}(\Omega)}-C\|f\|_{H^{1}(\Omega)}-C\|\nabla
u\|_{L^{\infty}}\|\nabla^{2}\tilde{\nu}\|_{L^{2}(\Omega)}.$

We proceed by recalling that $\tilde{\nu}$ is the harmonic extension of $\nu$
and therefore $\tilde{\nu}\cdot\nu=1$ on $\Sigma$. Therefore we obtain by the
second inequality in Lemma 3.7 and $\|\tilde{\nu}\|_{L^{\infty}}\leq C$ that

(3.24) $\|\nabla^{2}\tilde{\nu}\|_{L^{2}(\Omega)}\leq C.$

We have by interpolation inequality in Corollary 2.9

$\|\nabla u\|_{L^{\infty}(\Omega)}\leq C\|\nabla^{2}u\|_{L^{4}(\Omega)}\leq
C\|\nabla^{3}u\|_{L^{2}(\Omega)}^{\frac{3}{4}}\|\nabla^{2}u\|_{L^{2}(\Omega)}^{\frac{1}{4}}.$

Therefore by Young’s inequality and by (3.19)

$\begin{split}\|\nabla
u\|_{L^{\infty}(\Omega)}\|\nabla^{2}\tilde{\nu}\|_{L^{2}(\Omega)}&\leq\varepsilon\|\nabla^{3}u\|_{L^{2}(\Omega)}+C_{\varepsilon}\|\nabla^{2}u\|_{L^{2}(\Omega)}\\\
&\leq\varepsilon\|\nabla^{3}u\|_{L^{2}(\Omega)}+C_{\varepsilon}\|f\|_{L^{2}(\Omega)}.\end{split}$

Hence, (3.20), (3.21) and (3.23) imply

(3.25) $\|u\|_{H^{3}(\Omega)}\leq C\|f\|_{H^{1}(\Omega)}.$

We set $\mathcal{F}$ to be the linear operator such that it associates $f$
with the unique solution $u$ of the problem (3.18). Then we have by (3.19) and
(3.25)

$\|\mathcal{F}\|_{\mathcal{L}(L^{2},H^{2})}\leq
C\qquad\text{and}\qquad\|\mathcal{F}\|_{\mathcal{L}(H^{1},H^{3})}\leq C.$

Then we have the inequality

(3.26) $\|\nabla u\|_{H^{\frac{3}{2}}(\Omega)}\leq
C\|f\|_{H^{\frac{1}{2}}(\Omega)}$

by standard interpolation theory.

We need yet to bound $\|\partial_{\nu}u\|_{H^{1}(\Sigma)}$. To this aim we
extend $\nabla u$ to $\mathbb{R}^{3}$ by $T$ such that

$\|T(\nabla u)\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}\leq C\|\nabla
u\|_{H^{\frac{3}{2}}(\Omega)}.$

Let us denote $U=T(\nabla u)$. Let $\tilde{\nu}$ be the Harmonic extension of
$\nu$ as before, which we may also extend to $\mathbb{R}^{3}$. We note that we
may assume that the extensions have support in $B_{R}$. We have by Lemma 3.7

$\|\nabla u\cdot\nu\|_{H^{1}(\Sigma)}\leq
C\|U\cdot\tilde{\nu}\|_{H^{\frac{3}{2}}(\Omega)}\leq
C\|U\cdot\tilde{\nu}\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}.$

The Kato-Ponce inequality (2.15) with $p_{2}=8,q_{2}=8/3$ yields

$\|U\cdot\tilde{\nu}\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}\leq
C\|U\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}\|\tilde{\nu}\|_{L^{\infty}(\mathbb{R}^{3})}+C\|U\|_{L^{8}(\mathbb{R}^{3})}\|\tilde{\nu}\|_{W^{\frac{3}{2},\frac{8}{3}}(\mathbb{R}^{3})}.$

We have $\|\tilde{\nu}\|_{L^{\infty}(\mathbb{R}^{3})}\leq C$ and by the
Sobolev embedding $\|U\|_{L^{8}(\mathbb{R}^{3})}\leq
C\|U\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}$. We use (2.13) to deduce that

$\|\nabla\tilde{\nu}\|_{W^{\frac{1}{2},\frac{8}{3}}(\mathbb{R}^{3})}\leq
C\|\nabla\tilde{\nu}\|_{H^{1}(\mathbb{R}^{3})}^{\frac{1}{2}}\|\nabla\tilde{\nu}\|_{L^{4}(\mathbb{R}^{3})}^{\frac{1}{2}}.$

By (3.12) we have

$\|\nabla\tilde{\nu}\|_{L^{4}(\mathbb{R}^{3})}\leq\|\tilde{\nabla}\nu\|_{L^{4}(\Omega)}\leq
C$

and by (3.24)

$\|\nabla\tilde{\nu}\|_{H^{1}(\mathbb{R}^{3})}\leq
C\|\nabla^{2}\tilde{\nu}\|_{L^{2}(\Omega)}\leq C.$

Therefore by combining the previous inequalities we have

$\|\nabla u\cdot\nu\|_{H^{1}(\Sigma)}\leq
C\|U\cdot\tilde{\nu}\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}\leq
C\|U\|_{H^{\frac{3}{2}}(\mathbb{R}^{3})}\leq C\|\nabla
u\|_{H^{\frac{3}{2}}(\Omega)}.$

The result then follows from (3.26). ∎

We conclude this section by proving the sharp boundary regularity estimate for
the Dirichlet problem. The proof follows the argument in [22, Theoreom 4.1],
with the difference that here we have Dirichlet boundary datum, instead of the
zero Neumann case.

######  Theorem 3.9.

Assume $\Omega$, with $\Sigma=\partial\Omega$, is
$C^{1,\alpha}(\Gamma)$-regular and satisfies $\eqref{eq:notationhp}$ for
$m\geq 2$. Let $u\in\dot{H}^{1}(\Omega^{c})$ be the solution of

(3.27) $\begin{cases}\Delta u=0\qquad&x\in\Omega^{c}\\\
u=g&x\in\Sigma.\end{cases}$

Then for all integers $0\leq k\leq m-1$ it holds

(3.28) $\|\nabla^{k}u\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C(1+\|B_{\Sigma}\|_{H^{k-1}(\Sigma)}+\|g\|_{H^{k+\frac{1}{2}}(\Sigma)}).$

Moreover, if $g$ is constant then the above holds for all $k\in\mathbb{N}$.

###### Proof.

Step 1: Flattening the boundary. Since $\Sigma$ is $C^{1,\alpha}(\Gamma)$, for
any $x\in\Sigma$ we find $\delta>0$ such that after rotating and translating
the coordinates

$\Omega^{c}\cap B_{\delta}=\\{(x^{\prime},x_{3}):\,x_{3}>\phi(x^{\prime})\\}$

with $\phi\in C^{1,\alpha}(B_{\delta})$, $\phi(0)=0$ and $\nabla\phi(0)=0$.
Consider the diffeomorfism $\Psi:\Omega^{c}\cap B_{\delta}\to B_{\delta}^{+}$
$\Psi(x^{\prime},x_{3})\to(x^{\prime},x_{3}-\phi(x^{\prime}))$ and let
$v:=u\circ\Psi^{-1}$ and $w:=g\circ\Psi^{-1}$. Let us extend $g$ by its
harmonic extension, denote it simply by $g$, and thus $w$ is defined in
$B_{\delta}^{+}$. By standard calculations we deduce that $v$ is the solution
of

(3.29) $\begin{cases}\operatorname{div}(A_{\phi}\nabla v)=0\qquad&x\in
B^{+}_{\delta}\\\ v=w&x_{3}=0,\end{cases}$

where $A_{\phi}$ is symmetric matrix which can be written as
$A_{\phi}=I+\tilde{A}(\nabla\phi)$ where $\tilde{A}(\nabla\phi(x))=0$ if
$\nabla\phi(x)=0$. In particular, by choosing $\delta$ small enough $A_{\phi}$
is postitive definite. In weak form (3.29) reads as

$\int_{B_{\delta}^{+}}A_{\phi}\nabla v\cdot\nabla\varphi\,dx=0$

for all $\varphi\in C_{0}^{\infty}(B_{\delta}^{+})$.

Let $k$ be an integer as in the statement. Let us differentiate the equation
(3.29) $k$ times in tangential directions. To this aim let us fix an index
vector $\gamma=(\gamma_{1},\gamma_{2},0)$ with $\gamma_{1}+\gamma_{2}=k$, and
denote $\bar{v}=\nabla^{\gamma}v$ and $\bar{w}=\nabla^{\gamma}w$. Then
$\bar{v}$ is the solution of

(3.30)
$\begin{cases}\operatorname{div}(A_{\phi}\nabla\bar{v})=-\sum_{\tilde{\alpha},\beta}\operatorname{div}(\nabla^{\tilde{\alpha}}A_{\phi}\nabla\nabla^{\beta}v)\qquad&x\in
B^{+}_{\delta}\\\ \bar{v}=\bar{w}&x_{3}=0.\end{cases}$

with $\tilde{\alpha}=(\tilde{\alpha}_{1},\tilde{\alpha}_{2},0)$,
$\beta=(\beta_{1},\beta_{2},0)$, $|\beta|\leq k-1$ and
$|\tilde{\alpha}|+|\beta|\leq k$. In the weak form this reads as

$\int_{B_{\delta}^{+}}A_{\phi}\nabla\bar{v}\cdot\nabla\varphi\,dx=-\sum_{\tilde{\alpha},\beta}\int_{B_{\delta}^{+}}(\nabla^{\tilde{\alpha}}A_{\phi}\nabla\nabla^{\beta}v)\cdot\nabla\varphi\,dx$

for all $\varphi\in C_{0}^{\infty}(B_{\delta}^{+})$.

Step 2: Choice of test function that has zero boundary value Let $\zeta\in
C_{0}^{\infty}(B_{\delta}^{+})$ be a smooth cut-off function such that
$\zeta(x)=1$ for $|x|\leq\frac{\delta}{2}$ and $0\leq\zeta\leq 1$. We choose a
test function $\varphi=(\bar{v}-\bar{w})\zeta^{2}$, which has zero boundary
value. With this choice we have

$\begin{split}&\int_{B_{\delta}^{+}}(A_{\phi}\nabla\bar{v}\cdot\nabla\bar{v})\,\eta^{2}dx=\int_{B_{\delta}^{+}}(A_{\phi}\nabla\bar{v}\cdot\nabla\bar{w})\zeta^{2}\,dx+\int_{B_{\delta}^{+}}(A_{\phi}\nabla\bar{v}\cdot\nabla\zeta)\,(\bar{w}-\bar{v})\eta
dx\\\
&-\sum_{\tilde{\alpha},\beta}\int_{B_{\delta}^{+}}(\nabla^{\tilde{\alpha}}A_{\phi}\nabla\nabla^{\beta}v\cdot\nabla(\bar{v}-\bar{w}))\zeta^{2}\,dx-2\sum_{\tilde{\alpha},\beta}\int_{B_{\delta}^{+}}(\nabla^{\tilde{\alpha}}A_{\phi}\nabla\nabla^{\beta}v\cdot\nabla\zeta)(\bar{v}-\bar{w})\zeta\,dx\\\
&=I_{1}+I_{2}+I_{3}+I_{4}.\end{split}$

By the assumption $\phi\in C^{1,\alpha}$ it holds
$\|A_{\phi}\|_{L^{\infty}}\leq C$. Thus we may bound the first two terms as

$I_{1}+I_{2}\leq
C\|\nabla\bar{v}\|_{L^{2}(B_{\delta}^{+})}(\|\nabla\bar{w}\|_{L^{2}(B_{\delta}^{+})}+\|\bar{v}-\bar{w}\|_{L^{2}(B^{+}_{\delta})}).$

The term $I_{3}$ is more difficult to treat. Note first since $A_{\phi}$ is of
the form $I+\tilde{A}(\nabla\phi)$ we have a point-wise bound by the Leibniz
rule

$\sum_{\tilde{\alpha},\beta}|\nabla^{\tilde{\alpha}}A_{\phi}|^{2}||\nabla\nabla^{\beta}v|^{2}\leq
C\sum_{\begin{subarray}{c}|\alpha|+|\beta|\leq k\\\ |\beta|\leq
k-1\end{subarray}}(1+|\nabla^{\alpha_{1}}\nabla\phi|^{2}\dots|\nabla^{\alpha_{k}}\nabla\phi|^{2})|\nabla\nabla^{\beta}v|^{2}.$

Hence, we obtain by Hölder’s inequality

$\begin{split}I_{3}\leq&C\|\nabla(\bar{v}-\bar{w})\|_{L^{2}(B^{+}_{\delta})}\\\
&\cdot\sum_{\begin{subarray}{c}|\alpha|+|\beta|\leq k\\\ |\beta|\leq
k-1\end{subarray}}\Big{(}1+\|\nabla^{\alpha_{1}}\nabla\phi\|_{L^{\frac{2k}{\alpha_{1}}}(B_{\delta}^{+})}\dots\|\nabla^{\alpha_{k}}\nabla\phi\|_{L^{\frac{2k}{\alpha_{k}}}(B^{+}_{\delta})}\Big{)}\|\nabla\nabla^{\beta}v\|_{L^{\frac{2k}{|\beta|}}(B^{+}_{\delta})}.\end{split}$

We use interpolation inequality to estimate

$\|\nabla^{\alpha_{i}}\nabla\phi\|_{L^{\frac{2k}{\alpha_{1}}}(B_{\delta}^{+})}\leq\|\nabla\phi\|_{H^{k}(B_{\delta}^{+})}^{\frac{\alpha_{i}}{k}}\|\nabla\phi\|^{1-\frac{\alpha_{i}}{k}}_{L^{\infty}(B_{\delta}^{+})}.$

Also by interpolation we have

$\|\nabla\nabla^{\beta}v\|_{L^{\frac{2k}{|\beta|}}(B_{\delta}^{+})}\leq\|v\|_{H^{k+1}(B_{\delta}^{+})}^{\frac{|\beta|}{k}}\|\nabla
v\|_{L^{\infty}(B_{\delta}^{+})}^{1-\frac{|\beta|}{k}}$

and $|\beta|\leq k-1$. Since $\Sigma$ is $C^{1,\alpha}$-regular, we have by
Schauder estimates [25] that $\nabla v\in C^{0,\alpha}(B_{\delta}^{+})$. Note
that $\sum_{i}\frac{\alpha_{i}}{k}\leq\frac{k-|\beta|}{k}$ and
$\frac{|\beta|}{k}<1$. Therefore by Young’s inequality we deduce

$\begin{split}|I_{3}|\leq&C\|\nabla(\bar{v}-\bar{w})\|_{L^{2}(B^{+}_{\delta})}(1+\|\nabla\phi\|_{H^{k}(B_{\delta}^{+})}^{\frac{k-|\beta|}{k}})\|v\|_{H^{k+1}(B_{\delta}^{+})}^{\frac{|\beta|}{k}}\\\
\leq&\varepsilon\|\nabla(\bar{v}-\bar{w})\|_{L^{2}(B^{+}_{\delta})}^{2}+\varepsilon\|v\|_{H^{k+1}(B_{\delta}^{+})}^{2}+C_{\varepsilon}(1+\|\nabla\phi\|_{H^{k}(B_{\delta}^{+})}^{2}).\end{split}$

We bound the last term $I_{4}$ similarly.

Finally we collect the previous estimates, use the positivity of the matrix
$A_{\phi}$ and the definition of $\bar{w}$ and obtain

$\|\nabla\bar{v}\|_{L^{2}(B^{+}_{\delta/2})}^{2}\leq
4\varepsilon\|v\|_{H^{k+1}(B_{\delta}^{+})}^{2}+C(1+\|\phi\|_{H^{k+1}(B_{\delta}^{+})}^{2}+\|w\|_{H^{k+1}(B_{\delta}^{+})}^{2}).$

Summing over all the multi index of the type $(\gamma_{1},\gamma_{2})$ we have
the control over the horizontal derivatives. To estimate the vertical
derivatives, we use the equation in the strong form as in [22], and obtain

(3.31) $\|v\|_{H^{k+1}(B^{+}_{\delta/2})}^{2}\leq
C\varepsilon\|v\|_{H^{k+1}(B_{\delta}^{+})}^{2}+C(1+\|\phi\|_{H^{k+1}(B_{\delta}^{+})}^{2}+\|w\|_{H^{k+1}(B_{\delta}^{+})}^{2}).$

Step 3: Going back to the original function. We need to go back to the
original function $u$. The argument is similar to [22] and we merely sketch
it. We note that arguing as in [22, Thorem 4.1] we may control

$\|\phi\|_{H^{k+1}(B_{\delta}^{+})}\leq C(1+\|B_{\Sigma}\|_{H^{k-1}(\Sigma)})$

for all $k\in\mathbb{N}$. Let $\tilde{g}$ be the harmonic extension of $g$.
Using the assumption that the curvature satisfies the condition (Hm) for $m$,
we may deduce, arguing as in the proof of Proposition 2.1, that for $k\leq
m-1$ it holds

$\|w\|_{H^{k+1}(B_{\delta}^{+})}\leq C\|\tilde{g}\|_{H^{k+1}(\Omega^{c}\cap
B_{\delta})}\leq C\|g\|_{H^{k+\frac{1}{2}}(\Sigma)}.$

Obviously if $g$ is constant the above inequality is trivial.

Fix $\sigma$ small such that $\cup_{x\in\Sigma}B_{\delta}(x)$ covers
$\mathcal{N}_{\delta}=\\{x\in\Omega^{c}:d(x,\Omega)\leq\delta\\}$ and
$\sigma_{1}<\sigma_{2}<\sigma$. By compactness we may choose a finite family
of balls covering $\mathcal{N}_{\delta}$. Choosing $\varepsilon$ small enough
we have by (3.31) and by the above inequalities

$\|u\|_{H^{k+1}(\mathcal{N}_{\sigma_{1}})}^{2}\leq
C(\|u\|_{H^{k+1}(\mathcal{N}_{\sigma_{2}}\setminus\mathcal{N}_{\sigma_{1}})}^{2}+1+\|B_{\Sigma}\|^{2}_{H^{k-1}(\Sigma)}+\|g\|^{2}_{H^{k+\frac{1}{2}}(\Sigma)}).$

Since $u$ is harmonic, the interior regularity yields

$\|u\|_{H^{k+1}(\mathcal{N}_{\sigma_{2}}\setminus\mathcal{N}_{\sigma_{1}})}\leq
C\|u\|_{L^{2}(\mathcal{N}_{\sigma_{2}})}.$

By standard estimates from harmonic analysis [16] it holds for $R$ large

$\|u\|_{L^{2}(N_{{\sigma}_{2}})}\leq\|u\|_{L^{2}(\Omega^{c}\cap B_{R})}\leq
C(\|u\|_{L^{2}(\Sigma)}+\|u\|_{L^{2}(\partial B_{R})})\leq
C(1+\|g\|_{L^{2}(\Sigma)}).$

Therefore we have

$\|u\|_{H^{k+1}(\mathcal{N}_{\sigma_{1}})}\leq
C(1+\|B_{\Sigma}\|_{H^{k-1}(\Sigma)}+\|g\|_{H^{k+\frac{1}{2}}(\Sigma)}).$

The claim follows from

$\|\nabla^{k}u\|_{H^{\frac{1}{2}}(\Sigma)}\leq
C\|\nabla^{k+1}u\|^{2}_{L^{2}(\mathcal{N}_{\sigma_{1}})}.$

∎

##  4\. Useful formulas

In this section we focus on the equations (1.3) and assume that the family of
sets $(\Omega_{t})_{t\in(0,T)}$ and velocities $v(\cdot,t)$ are solution of