# Neumann Laplacian in a perturbed domain

Diana Barseghyan1,2, Baruch Schneider1 and Ly Hong Hai1

(1 Department of Mathematics, University of Ostrava, 30.dubna 22, Ostrava
70103, Czech Republic
2 Nuclear Physics Institute, Czech Academy of Sciences, 250 68 Rez,
Czech Republic
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>)

###### Abstract

We consider a domain with a small compact set of zero Lebesgue measure of
removed. Our main result concerns the spectrum of the Neumann Laplacian
defined on such domain. We prove that the spectrum of the Laplacian converges
in the Hausdorff distance sense to the spectrum of the Laplacian defined on
the unperturbed domain.

Keywords. Neumann Laplacian, spectral convergence, domain perforation

Mathematics Subject Classification. Primary: 58J50; Secondary: 35P15, 47A10.

##  1 Introduction

What kind of spectral convergence can we expect for the Laplace operator under
perturbations of the domain such as removing small holes? It is a common
expectation that small perturbations of the physical situation lead only to a
small change of the spectrum. In the case of domain perturbations this is
largely true for Dirichlet boundary conditions while the Neumann case is more
delicate.

Such questions received already quite a lot of answers starting from the
seminal work of Rauch and Taylor concerning the spectrum of the Laplace
operator of the domains with holes [13].

In Neumann case even small perturbations may cause abrupt change of the
spectrum. For example, such an effect is observed when the hole has a ”split-
ring” geometry [14]. The split ring (even being very small) may produce
additional eigenvalues having nothing in common with the eigenvalues of the
Neumann Laplacian on the unperturbed domain.

In [11] Ozawa have studied the asymptotic behavior of the eigenvalues of
Laplace operator on the domain with small spherical obstacles imposing the
Neumann condition at their boundary and the Dirichlet condition at the rest
part of the boundary.

Maz’ya, Nazarov and Plamenewskii, see [10, Ch.9, vol.I], have considered the
Laplace operator on the domain with obstacles, imposing the Dirichlet
condition on their boundary and have proved the validity of a complete
asymptotic expansion for the eigenvalues.

In [2], [3], [4], [8] authors have considered the Dirichlet Laplacians on
Euclidean domains or manifolds with holes and studied the problems of the
resolvent convergence.

The problems with Neumann obstacles having more general geometry appeared in
[3]. The authors required the hole to satisfy the so called ”uniform extension
property”. This is requirement, which means that $\mathcal{H}^{1}$\- functions
on the domain with a hole can be extended to $\mathcal{H}^{1}$ function on the
unperturbed domain and the norm of this extension operator does not depend on
the hole diameter. In this case the authors established the spectral
convergence.

Another related paper, which makes use imposing Neumann boundary conditions on
the boundary of hole which has satisfies the suitable geometrical assumptions
to consider the upper and lower estimates for the ground state eigenvalue,
have been extensively studied by Hempel, see for instance [7].

There is one case, which was not considered in [3] (and other similar papers).
These are holes with zero Lebesgue measure (e.g. an interval or a piece of a
curve). Evidently, for the holes with Lebesgue measure zero such an extension
required in [3] is not possible.

We will be interested in a two-dimensional bounded domain with a single hole
$K_{\varepsilon}$ (for a fixed parameter $\varepsilon$) having zero Lebesgue
measure. The main purpose of this paper is to prove the spectral convergence
of the Neumann Laplacian on $\Omega_{K_{\varepsilon}}$ as $\varepsilon\to 0$
in terms of the Hausdorff distance under some additional assumptions on the
geometry of $K_{\varepsilon}$.

The outline of the paper is as follows. It consists of four sections besides
the introduction. In the second section we present the main tools of the
spectral convergence of operators on varying Hilbert spaces. In the third
section we present the main results, and consider a general theorem, namely
Theorem 3.1. We will use Theorem 3.1 in the proof of Theorem 3.2, and where
consider the spectral convergence for the Laplacians on $\Omega$ and
$\Omega_{K_{\varepsilon}}$. In the fourth and last section we prove our
results to which we already alluded.

##  2 Main tool of the spectral convergence of operators on varying Hilbert
spaces

We begin this section by reviewing some basic facts which assures a spectral
convergence for two operators having the different domains. For more
information we refer the reader to [12].

To a Hilbert space $H$ with inner product $(\cdot,\cdot)$ and norm $\|\cdot\|$
together with a non-negative, unbounded, operator $A$, we associate the scale
of Hilbert spaces

$H_{k}:=\mathrm{Dom}((A+I)^{k/2}),\quad\|u\|_{k}:=\|(A+I)^{k/2}u\|,\,\,k\geq
0,$

where $I$ is the identity operator.

We think of $(H^{\prime},A^{\prime})$ being some perturbation of $(H,A)$ and
want to lessen the assumption such that the spectral properties are not the
same but still are close.

######  Definition 1.

(see [12]) Suppose we have linear operators

$\displaystyle J:H\longrightarrow H^{\prime},\quad\quad
J_{1}:H_{1}\longrightarrow H_{1}^{\prime}$ $\displaystyle
J^{\prime}:H^{\prime}\longrightarrow H,\quad\quad
J_{1}^{\prime}:H_{1}^{\prime}\longrightarrow H_{1}.$

Let $\delta>0$ and $k\geq 1$. We say that $(H,A)$ and
$(H^{\prime},A^{\prime})$ are $\delta$-close of order $k$ iff the following
conditions are fulfilled:

$\displaystyle\|Jf-J_{1}f\|_{0}\leq\delta\|f\|_{1},$ (2.1)
$\displaystyle|(Jf,u)-(f,J^{\prime}u)|\leq\delta\|f\|_{0}\|u\|_{0},$ (2.2)
$\displaystyle\|u-JJ^{\prime}u\|_{0}\leq\delta\|u\|_{1},$ (2.3)
$\displaystyle\quad\|Jf\|_{0}\leq 2\|f\|_{0},\,\,\,\|J^{\prime}u\|_{0}\leq
2\|u\|_{0},$ (2.4) $\displaystyle\|(f-J^{\prime}Jf)\|_{0}\leq\delta\|f\|_{1},$
(2.5) $\displaystyle\|J^{\prime}u-J_{1}^{\prime}u\|_{0}\leq\delta\|u\|_{1}$
(2.6)
$\displaystyle|a(f,J_{1}^{\prime}u)-a^{\prime}(J_{1}f,u)|\leq\delta\|f\|_{k}\|u\|_{1},$
(2.7)

for all $f,u$ in the appropriate spaces. Here, $a$ and $a^{\prime}$ denote the
sesquilinear forms associated to $A$ and $A^{\prime}$.

We denote by $d_{\mathrm{Haussdorff}}(A,B)$ the Hausdorff distance for subsets
$A,B\subset\mathbb{R}$

$d_{\mathrm{Haussdorff}}(A,B):=\mathrm{max}\left\\{\underset{a\in
A}{\mathrm{sup}}\,d(a,B),\,\underset{b\in B}{\mathrm{sup}}\,d(b,A)\right\\},$
(2.8)

where $d(a,B):=\mathrm{inf}_{b\in B}|a-b|$. We set

$\overline{d}(A,B):=d_{\mathrm{Hausdorff}}\left((A+I)^{-1},(B+I)^{-1}\right)$
(2.9)

for closed subsets of $[0,\infty)$. For the next result, which originates with
the work of O. Post [12] we have the following spectral convergence theorem in
terms of the distance $\overline{d}$.

######  Theorem 2.1.

[12] There exists $\eta(\delta)>0$ with $\eta(\delta)\to 0$ as $\delta\to 0$
such that

$\overline{d}(\sigma_{\bullet}(A),\,\sigma_{\bullet}(A^{\prime}))\leq\eta(\delta)$
(2.10)

for all pairs of non-negative operators and Hilbert spaces $(H,A)$ and
$(H^{\prime},A^{\prime})$ which are $\delta$-close. Here,
$\sigma_{\bullet}(A)$ denotes either the entire spectrum, the essential or the
discrete spectrum of $A$. Furthermore, the multiplicity of the discrete
spectrum, $\sigma_{\mathrm{disc}}$, is preserved, i.e. if
$\lambda\in\sigma_{\mathrm{disc}}(A)$ has multiplicity $\mu>0$, then there
exist $\mu$ eigenvalues (not necessarily all distinct) of operator
$A^{\prime}$ belonging to interval
$(\lambda-\eta(\delta),\lambda+\eta(\delta))$.

##  3 Main results

The starting point is to consider a bounded domain
$\Omega\subset\mathbb{R}^{2}$ and a compact set $K\subset\Omega$ with zero
Lebesgue measure (e.g. an interval or a piece of a curve). We denote
$\Omega_{K}:=\Omega\setminus K$. The Neumann Laplacian
$-\Delta_{N}^{\Omega_{K}}$ is defined on the Sobolev space
$\mathcal{H}^{1}(\Omega_{K})$ via the quadratic form

$\int_{\Omega}|\nabla u|^{2}\,dx\,dy,\quad u\in\mathcal{H}^{1}(\Omega_{K}).$

In case if $K$ is empty set then unperturbed Neumann Laplacian denoted by
$-\Delta_{N}^{\Omega}$ which is defined via the same form
$\int_{\Omega}|\nabla u|^{2}\,dx\,dy$ on $\mathcal{H}^{1}(\Omega)$.

During our paper we suppose the additional property∗: for any
$(x_{0},y_{0})\in\Omega_{K}$ at least one of the following conditions takes
place

  * •

The line $l(x_{0})=\\{x=x_{0}\\}\subset\Omega_{K}$ has no intersection with
$K.$

  * •

The line $h(y_{0})=\\{y=y_{0}\\}\subset\Omega_{K}$ has no intersection with
$K.$

The main result of this section is the following theorem.

######  Theorem 3.1.

Let $\Omega$ be an open bounded domain in $\mathbb{R}^{2}$ and let
$p\in\Omega$ be some fixed point. Suppose that
${\mathbb{B}}_{\varepsilon}\subset\Omega$ is a ball with center at $p$ and
radius $\varepsilon>0$. Let $K=K(\varepsilon)\subset\mathbb{B}_{\varepsilon}$
be a compact set with zero Lebesgue measure (e.g. an interval or a piece of a
curve). Moreover, suppose that $\Omega_{K}$ satisfies property∗. Let
$-\Delta_{\Omega}^{N}$ and $-\Delta_{\Omega_{K_{\varepsilon}}}^{N}$ be the
Neumann Laplacians defined on $\Omega$ and $\Omega_{K_{\varepsilon}}$,
respectively. Then for small enough $\varepsilon$ the Neumann Laplacians
$-\Delta_{\Omega}^{N}$ and $-\Delta_{\Omega_{K_{\varepsilon}}}^{N}$ are
$\mathcal{O}\left(\varepsilon^{1/6}\right)$ close of order $2$.

As a consequence we formulate the following theorem:

######  Theorem 3.2.

Using the above assumptions, let $-\Delta_{\Omega}^{N}$ and
$-\Delta_{\Omega_{K_{\varepsilon}}}^{N}$ be the Neumann Laplacians defined on
$\Omega$ and $\Omega_{K_{\varepsilon}}$, respectively. Then there exists
$\eta(\varepsilon)>0$ with $\eta(\varepsilon)\to 0$ as $\varepsilon\to 0$ such
that the following spectral convergence takes place

$\overline{d}\left(\sigma_{\bullet}\left(-\Delta_{\Omega_{K_{\varepsilon}}}^{N}\right),\,\sigma_{\bullet}\left(-\Delta_{\Omega}^{N}\right)\right)\leq\eta(\varepsilon),$

where $\overline{d}$ is defined in (2.9) and $\sigma_{\bullet}(\cdot)$ denotes
either the entire spectrum, the essential or the discrete spectrum. Moreover,
the multiplicity of the discrete spectrum is preserved.

Proof of Theorem 3.2. The proof follows directly from the inequality (2.10)
and Theorem 3.1. ∎

As consequences of Theorem 3.2:

######  Corollary 3.3.

Suppose that $-\Delta_{\Omega}^{N}$ has purely discrete spectrum denoted by
$\lambda_{k}(\Omega)$ (repeated according to multiplicity). Then the infimum
of the essential spectrum of $-\Delta_{\Omega_{K_{\varepsilon}}}^{N}$ tends to
infinity and there exists $\eta_{k}(\varepsilon)>0$ with
$\eta_{k}(\varepsilon)\to 0$ as $\varepsilon\to 0$ such that

$|\lambda_{k}(\Omega)-\lambda_{k}(\Omega_{K_{\varepsilon}})|\leq\eta_{k}(\varepsilon)$

for small enough $\varepsilon$. Here, $\lambda_{k}(\Omega_{K_{\varepsilon}})$
denotes the discrete spectrum of $-\Delta^{N}_{\Omega_{K_{\varepsilon}}}$
(below the essential spectrum) repeated according to multiplicity.

######  Corollary 3.4.

The Hausdorff distance between the spectra of
$-\Delta_{\Omega_{K_{\varepsilon}}}^{N}$ and $-\Delta_{\Omega}^{N}$ converges
to zero on any compact interval $[0,\Lambda]$.

We now turn to the proof of Theorem 3.1.

##  4 Proof of Theorem 3.1

We proceed in a number of steps.

STEP 1: Construction of the mappings $J,J^{\prime},J_{1},J_{1}^{\prime}$.

It is easy to notice that $H=H^{\prime}=L^{2}(\Omega),\,A=A^{\prime}=-\Delta$;
$H_{1},\,H_{1}^{\prime}$ correspond to Sobolev spaces
$\mathcal{H}^{1}(\Omega)$ and $\mathcal{H}^{1}(\Omega_{K_{\varepsilon}})$ and
$H_{2}=\mathrm{Dom}(-\Delta_{N}^{\Omega})$. The norm $\|\cdot\|_{0}$
corresponds with the $L^{2}$ norm and

$\|u\|_{1}=(\|u\|^{2}_{0}+\|\nabla u\|_{0}^{2})^{1/2},\quad\|f\|_{2}=\|-\Delta
f+f\|_{0}.$

Since $H=H^{\prime}$ and $H_{1}\subset H_{1}^{\prime}$ we choose
$J=J^{\prime}=I$, where $I$ is the identity operator and $J_{1}$ is the
restriction operator: $J_{1}u=u\restriction_{\Omega_{K_{\varepsilon}}}$ for
$u\in H_{1}$.

Let us now construct the mapping $J_{1}^{\prime}:H_{1}^{\prime}\to H_{1}$.
Without loss of generality, assume that the ball $\mathbb{B}_{\varepsilon}$
mentioned in Theorem 3.1 is centered at the origin. Let
$\epsilon\in(\varepsilon,2\varepsilon)$ be a number to be chosen later and let
$\mathbb{B}_{\epsilon}\supset\mathbb{B}_{\varepsilon}$ be the ball with center
again at the origin and radius $\epsilon$,
$\Omega_{\epsilon}:=\Omega\backslash\mathbb{B}_{\epsilon}$.

We are going to construct mapping $J_{1}^{\prime}$ first for smooth functions.
For any $v\in C^{\infty}(\Omega_{K_{\varepsilon}})$ we define

$J_{1}^{\prime}v:=\begin{cases}v,\quad\text{on}\quad\Omega_{\epsilon},\\\
\frac{r}{\epsilon}\tilde{v}(\epsilon,\varphi),\quad\text{on}\quad\mathbb{B}_{\epsilon},\end{cases}$

where $\tilde{v}(r,\varphi)=v(r\cos\varphi,r\sin\varphi)$.

Now let us construct the mapping $J_{1}^{\prime}u$ for any $u\in
H_{1}^{\prime}$. Employing the approximation method described in [5, Thm.2,
5.3.2], for the fixed sequence $\\{\eta_{k}\\}_{k=1}^{\infty}$ converging to
zero we construct the sequence $v_{\eta_{k}}\in
C^{\infty}(\Omega_{K_{\varepsilon}})$ which satisfies

$\|u-v_{\eta_{k}}\|_{1}=\|u-v_{\eta_{k}}\|_{\mathcal{H}^{1}(\Omega_{K_{\varepsilon}})}<\eta_{k}\,\|u\|_{1}.$
(4.1)

Let us mention that in view of the inequalities (4.9) and (4.16) which will be
proved later it follows for any smooth function $v$

$\|J_{1}^{\prime}v\|_{1}^{2}=\int_{\Omega}|\nabla
J_{1}^{\prime}v|^{2}\,dx\,dy+\int_{\Omega}|J_{1}^{\prime}v|^{2}\,dx\,dy\leq\overline{C}(\varepsilon)\|v\|_{1}^{2},$

where $\overline{C}(\varepsilon)$ is some constant.

Therefore using the completeness of space $H_{1}$ we are able to define

$J_{1}^{\prime}u=\underset{k\to\infty}{\mathrm{lim}}\,J_{1}^{\prime}v_{\eta_{k}}.$
(4.2)

STEP 2: The conditions (2.1)-(2.7) hold for the mappings
$J,J^{\prime},J_{1},J_{1}^{\prime}$.

Indeed, we have that the estimates (2.1)-(2.5) are satisfied with $\delta=0$.

We now prove (2.6), i.e. under the assumptions stated in Theorem 3.1
inequality (2.6) is satisfied with $\delta=\mathcal{O}(\sqrt{\varepsilon})$
for small enough $\varepsilon$.

In view of our construction we have

$\|J^{\prime}u-J_{1}^{\prime}u\|_{0}^{2}=\int_{\Omega}|u-J_{1}^{\prime}u|^{2}\,dx\,dy.$
(4.3)

Let us estimate the right-hand side of (4.3). By virtue of (4.1) and (4.2) it
is enough to prove (4.3) for $u\in C^{\infty}(\Omega_{K_{\varepsilon}})$.

Taking into account the relation $J_{1}^{\prime}u=u$ on $\Omega_{\epsilon}$
and using (4.2) the above bound implies

$\displaystyle\int_{\Omega}|u-J_{1}^{\prime}u|^{2}\,dx\,dy=\int_{\mathbb{B}_{\varepsilon}}|u-J_{1}^{\prime}u|^{2}\,dx\,dy$
$\displaystyle\leq
2\int_{\mathbb{B}_{\epsilon}}|u|^{2}\,dx\,dy+2\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy.$
(4.4)

To estimate the first integral of the right-hand side of (4.4) we need the
following lemma (the proof is given in the Appendix):

######  Lemma 4.1.

Let $\omega\subset\mathbb{R}^{2}$ be an open domain satisfying property∗. Then
for any $u\in\mathcal{H}^{1}(\omega)$ and for almost any
$(x_{0},y_{0})\in\omega$ the following statement takes place:

$\displaystyle|u(x_{0},y_{0})|\leq C\left(\int_{l(x_{0})}\left|\frac{\partial
u}{\partial
y}(x_{0},z)\right|^{2}\,dz+\int_{l(x_{0})}|u(x_{0},z)|^{2}\,dz\right)^{1/2},$
(4.5) $\displaystyle\text{if}\quad(x_{0},y_{0})\quad\text{satisfies first
condition of {property${}^{*}$}}$ $\displaystyle|u(x_{0},y_{0})|\leq
C\left(\int_{h(y_{0})}\left|\frac{\partial u}{\partial
z}(z,y_{0})\right|^{2}+\int_{h(y_{0})}|u(z,y_{0})|^{2}\,dz\right)^{1/2},$
(4.6) $\displaystyle\text{if}\quad(x_{0},y_{0})\quad\text{satisfies second
condition of {property${}^{*}$}},$

where $C>0$ is a constant depending on the diameter of $\omega$ and the
distance between point $(x_{0},y_{0})$ and the boundary of $\omega$.

In view of Lemma 4.1 one guarantees the existence of constant $C^{\prime}$
depending on the distance between the boundary of $\mathbb{B}_{\epsilon}$ and
the boundary of $\Omega$ such that for any $(x,y)\in\mathbb{B}_{\epsilon}$

$\displaystyle|u(x,y)|^{2}\leq C^{\prime}\left(\int_{l(x)}\left|\frac{\partial
u}{\partial y}(x,z)\right|^{2}\,dz+\int_{l(x)}|u(x,z)|^{2}\,dz\right)$
$\displaystyle+C^{\prime}\left(\int_{h(y)}\left|\frac{\partial u}{\partial
z}(z,y)\right|^{2}\,dz+\int_{h(y)}|u(z,y)|^{2}\,dz\right).$

Therefore we obtain

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|u(x,y)|^{2}\,dx\,dy=C^{\prime}\int_{\mathbb{B}_{\epsilon}}\int_{l(x)}\left(\left|\frac{\partial
u}{\partial z}(x,z)\right|^{2}+|u(x,z)|^{2}\right)\,dz\,dx\,dy$
$\displaystyle+C^{\prime}\int_{\mathbb{B}_{\epsilon}}\int_{h(y)}\left(\left|\frac{\partial
u}{\partial z}(z,y)\right|^{2}+|u(z,y)|^{2}\right)\,dz\,dx\,dy$
$\displaystyle\leq
C^{\prime}\int_{-\epsilon}^{\epsilon}\int_{-\sqrt{\epsilon^{2}-x^{2}}}^{\sqrt{\epsilon^{2}-x^{2}}}\int_{l(x)}\left(\left|\frac{\partial
u}{\partial z}(x,z)\right|^{2}+|u(x,z)|^{2}\right)\,dz\,dx\,dy$
$\displaystyle+C^{\prime}\int_{-\epsilon}^{\epsilon}\int_{-\sqrt{\epsilon^{2}-y^{2}}}^{\sqrt{\epsilon^{2}-y^{2}}}\int_{h(y)}\left(\left|\frac{\partial
u}{\partial z}(z,y)\right|^{2}+|u(z,y)|^{2}\right)\,dz\,dy\,dx$
$\displaystyle\leq C^{\prime}\epsilon\int_{\Omega}(|\nabla
u|^{2}+2|u|^{2})\,dx\,dy\leq 2C^{\prime}\epsilon\|u\|_{1}^{2}.$

In view of the fact that $\epsilon\leq 2\varepsilon$ the above inequality
means

$\int_{\mathbb{B}_{\epsilon}}|u|^{2}\,dx\,dy\leq
4C^{\prime}\varepsilon\|u\|_{1}^{2},$ (4.7)

which estimates the first term in the right-hand side of (4.4).

Let us now investigate the second term in the right-hand side of (4.4). We are
going to establish the upper bound for the larger integral
$\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy$.

Passing to polar coordinates one gets

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy=\int_{0}^{\epsilon}\int_{0}^{2\pi}r\left|\frac{r}{\epsilon}\tilde{u}(\epsilon,\varphi)\right|^{2}\,dr\,d\varphi$
$\displaystyle\leq\int_{0}^{\epsilon}\int_{0}^{2\pi}r|\tilde{u}(\epsilon,\varphi)|^{2}\,dr\,d\varphi\leq\epsilon\int_{0}^{\epsilon}\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,dr\,d\varphi$
$\displaystyle=\epsilon^{2}\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi.$

Taking into account that
$\epsilon\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi$ coincides
with the curvilinear integral
$\int_{\partial\mathbb{B}_{\epsilon}}|u|_{\partial\mathbb{B}_{\epsilon}}^{2}\,d\mu$,
where $|u|_{\partial\mathbb{B}_{\epsilon}}$ is the trace of $|u|$ on the
circle $\partial\mathbb{B}_{\epsilon}$, the above bound performs to

$\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy\leq\epsilon\int_{\partial\mathbb{B}_{\epsilon}}|u|_{\partial\mathbb{B}_{\epsilon}}^{2}\,d\mu.$
(4.8)

The suitable bound for the above estimate is guaranteed by the following trace
inequality:

######  Lemma 4.2.

Let $v\in\mathcal{H}^{1}(\Omega\setminus\mathbb{B})$, where $\mathbb{B}$ is a
ball with center in zero and radius $\tau>0$. Then there exists a constant
$C^{\prime\prime}>0$ depending on the diameter of $\Omega$ and the distance
between the boundary of $\mathbb{B}$ and the boundary of $\Omega$ such that

$\int_{\partial\mathbb{B}}|v|_{\partial\mathbb{B}}^{2}\,d\mu\leq
C^{\prime\prime}\int_{\Omega}(|\nabla v|^{2}+|v|^{2})\,dx\,dy.$

Hence, having the fact that $\epsilon\leq 2\varepsilon$ inequality (4.8) means

$\int_{\mathbb{B}_{\epsilon}}|J_{1}^{\prime}u|^{2}\,dx\,dy\leq
2C^{\prime\prime}\varepsilon\|u\|_{1}^{2}.$ (4.9)

In view of the above bound and (4.3), (4.4), (4.7) it follows from that

$\|J^{\prime}u-J_{1}^{\prime}u\|_{0}^{2}\leq
8C^{\prime}\varepsilon\|u\|_{1}^{2}+4C^{\prime\prime}\varepsilon\|u\|_{1}^{2}.$

which proves (2.6) with $\delta=\mathcal{O}(\sqrt{\varepsilon})$.

We now give the proof of the estimate (2.7), i.e. under the assumptions stated
in Theorem 3.1 inequality (2.7) takes place with $k=2$ and
$\delta=\mathcal{O}(\varepsilon^{1/6})$ for small enough $\varepsilon$.

In view of our construction

$\displaystyle|a(f,J_{1}^{\prime}u)-a^{\prime}(J_{1}f,u)|=\left|\int_{\Omega}\nabla
f\,\overline{\nabla J_{1}^{\prime}u}\,dx\,dy-\int_{\Omega}\nabla
f\overline{\nabla u}\,dx\,dy\right|$ $\displaystyle=\left|\int_{\Omega}\nabla
f\,\overline{(\nabla J_{1}^{\prime}u-\nabla u)}\,dx\,dy\right|.$ (4.10)

We are going to give the upper estimate of the right-hand side of (4.10). As
before without loss of generality suppose that $u\in
C^{\infty}(\Omega_{K_{\varepsilon}})$.

Using the fact that $J_{1}^{\prime}u=u$ on $\Omega_{\epsilon}$ writing

$\left|\int_{\Omega}\nabla f\,\overline{(\nabla J_{1}^{\prime}u-\nabla
u)}\,dx\,dy\right|=\left|\int_{\mathbb{B}_{\epsilon}}\nabla
f\,\overline{(\nabla J_{1}^{\prime}u-\nabla u)}\,dx\,dy\right|.$ (4.11)

In view of the Cauchy inequality, the right-hand side of (4.11) can be
estimated as follows

$\displaystyle\left|\int_{\mathbb{B}_{\epsilon}}\nabla f\,\overline{(\nabla
J_{1}^{\prime}u-\nabla u)}\,dx\,dy\right|$
$\displaystyle\leq\left(\int_{\mathbb{B}_{\epsilon}}|\nabla
f|^{2}\,dx\,dy\right)^{1/2}\,\left(\int_{\mathbb{B}_{\epsilon}}|\nabla
J_{1}^{\prime}u-\nabla u|^{2}\,dx\,dy\right)^{1/2}$
$\displaystyle\leq\sqrt{2}\left(\int_{\mathbb{B}_{\epsilon}}|\nabla
f|^{2}\,dx\,dy\right)^{1/2}\,\left(\int_{\mathbb{B}_{\epsilon}}|\nabla
J_{1}^{\prime}u|^{2}\,dx\,dy+\int_{\mathbb{B}_{\epsilon}}|\nabla
u|^{2}\,dx\,dy\right)^{1/2}.$ (4.12)

The suitable upper bound for $\int_{\mathbb{B}_{\epsilon}}|\nabla
f|^{2}\,dx\,dy$ is guaranteed by the following lemma (the proof is given in
the Appendix):

######  Lemma 4.3.

For any function $g\in\mathrm{Dom}(-\Delta^{\Omega}_{N})$ the following
estimate takes place

$\int_{\mathbb{B}_{\epsilon}}|\nabla
g|^{2}\,dx\,dy\leq\tilde{C}\epsilon^{4/3}\,\|g\|^{2}_{2},$

with the constant $\tilde{C}$ depends on the distance of
$\mathbb{B}_{\varepsilon}$ from the boundary of $\Omega$.

To proceed further we need to estimate the integral
$\int_{\mathbb{B}_{\epsilon}}|\nabla J_{1}^{\prime}u|^{2}\,dx\,dy$. Passing to
polar coordinates one gets

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|\nabla
J_{1}^{\prime}u|^{2}\,dx\,dy=\int_{\mathbb{B}_{\epsilon}}\left(\left|\frac{\partial(J_{1}^{\prime}u)}{\partial
x}\right|^{2}+\left|\frac{\partial(J_{1}^{\prime}u)}{\partial
y}\right|^{2}\right)\,dx\,dy$
$\displaystyle=\int_{0}^{\epsilon}\int_{0}^{2\pi}r\left|\frac{1}{\epsilon}\tilde{u}(\epsilon,\varphi)\,\cos\varphi-\frac{1}{\epsilon}\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\,\sin\varphi\right|^{2}\,dr\,d\varphi$
$\displaystyle+\int_{0}^{\varepsilon}\int_{0}^{2\pi}r\left|\frac{1}{\epsilon}\tilde{u}(\epsilon,\varphi)\,\sin\varphi+\frac{1}{\epsilon}\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\,\cos\varphi\right|^{2}\,dr\,d\varphi$
$\displaystyle\leq\frac{4}{\epsilon}\int_{0}^{\epsilon}\int_{0}^{2\pi}\left(|\tilde{u}(\epsilon,\varphi)|^{2}+\left|\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\right|^{2}\right)\,dr\,d\varphi$
$\displaystyle=4\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi+4\int_{0}^{2\pi}\left|\frac{\partial\tilde{u}}{\partial\varphi}(\epsilon,\varphi)\right|^{2}\,d\varphi.$
(4.13)

As in proof of (2.6) we again note that

$4\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi=\frac{4}{\epsilon}\,\int_{\partial\mathbb{B}_{\epsilon}}|u|_{\partial\mathbb{B}_{\epsilon}}^{2}\,d\mu,$

which can be estimated using Lemma 4.2:

$4\int_{0}^{2\pi}|\tilde{u}(\epsilon,\varphi)|^{2}\,d\varphi\leq\frac{4C^{\prime\prime}}{\epsilon}\|u\|_{1}^{2}.$
(4.14)

The upper bound for the second term of (4.13) can be found by the following
auxiliary lemma (the proof is given in the Appendix).

######  Lemma 4.4.

Let $\mathbb{B}_{2\varepsilon}$ and $\mathbb{B}_{\varepsilon}$ be the balls
with center in origin and radii $\varepsilon$ and $2\varepsilon$. Let
$g\in\mathcal{H}^{1}(\mathbb{B}_{2\varepsilon}\setminus\mathbb{B}_{\varepsilon})$.
Then there exists $\tau\in(\varepsilon,2\varepsilon)$ such that

$\int_{0}^{2\pi}\left|\frac{\partial\tilde{g}}{\partial\varphi}(\tau,\varphi)\right|^{2}\,d\varphi\leq
4\|g\|_{1}^{2},$ (4.15)

where $\tilde{g}(r,\varphi):=g(r\cos\varphi,r\sin\varphi)$.

Now we are able to choose the number $\epsilon$. Let us apply Lemma 4.4 for
$g=u$. In case if mentioned in above lemma $\tau$ belongs to interval
$(\varepsilon,3\varepsilon/2]$ then we choose $\epsilon$ as the supremum of
all such numbers in $(\varepsilon,3\varepsilon/2]$. In the opposite case if
$\tau\in(3\varepsilon/2,2\varepsilon)$ then let $\epsilon$ be the infimum of
such numbers. Since $u$ is smooth function then inequality (4.15) is satisfied
with $\tau=\epsilon$. Combining (4.14) together with Lemma 4.4 inequality
(4.13) implies

$\int_{\mathbb{B}_{\epsilon}}|\nabla
J_{1}^{\prime}u|^{2}\,dx\,dy\leq\frac{4C^{\prime\prime}}{\epsilon}\,\|u\|_{1}^{2}+16\|u\|_{1}^{2}.$
(4.16)

Finally, employing (4.16) and Lemma 4.3 in inequality (4.12) and having that
$\int_{\mathbb{B}_{\epsilon}}|\nabla u|^{2}\,dx\,dy\leq\|u\|_{1}^{2}$ it
follows that

$\displaystyle\left|\int_{\mathbb{B}_{\epsilon}}\nabla f\,\overline{(\nabla
J_{1}^{\prime}u-\nabla
u)}\,dx\,dy\right|\leq(2\tilde{C})^{1/2}{\epsilon}^{2/3}\left(\frac{4C^{\prime\prime}}{\epsilon}+17\right)^{1/2}\|f\|_{2}\,\|u\|_{1}$
$\displaystyle=(2\tilde{C}(4C^{\prime\prime}+17{\epsilon}))^{1/2}{\epsilon}^{1/6}\,\|f\|_{2}\,\|u\|_{1}.$

Using the above estimate and the fact that ${\epsilon}\leq 2\varepsilon$
inequality (4.11) implies

$\left|\int_{\Omega}\nabla f\,\overline{(\nabla J_{1}^{\prime}u-\nabla
u)}\,dx\,dy\right|\leq
2^{2/3}(\tilde{C}(4C^{\prime\prime}+34\varepsilon))^{1/2}\varepsilon^{1/6}\|f\|_{2}\,\|u\|_{1}.$
(4.17)

Returning to (4.10) and applying the bound (4.17) we have

$|a(f,J_{1}^{\prime}u)-a^{\prime}(J_{1}f,u)|\leq
2^{2/3}(\tilde{C}(4C^{\prime\prime}+34\varepsilon))^{1/2}\varepsilon^{1/6}\|f\|_{2}\,\|u\|_{1}.$
(4.18)

It is easy to notice that the right-hand side of inequality (4.18) for small
enough $\varepsilon$ satisfies

$\mathrm{r.h.s.}(\ref{as.})=\mathcal{O}\left(\varepsilon^{1/6}\right)\,\|f\|_{2}\,\|u\|_{1},$

which ends the proof.∎

##  5 Appendix

In this section we give the proofs of Lemmata 4.1, 4.2, 4.3 and 4.4.

###  5.1 Proof of Lemma 4.1

Let $(x_{0},y_{0})\in\Omega_{K_{\varepsilon}}$. Assume the validity of the
first condition of property∗. Let $y_{1}(x_{0})$ be a point of intersection of
$l(x_{0})$ and the boundary of $\Omega$. Without less of generation suppose
that $y_{1}(x_{0})<y_{0}$. One can easily check that for almost any
$(x_{0},y_{0})\in\omega$ there exists $y_{2}(x_{0})\in(y_{1}(x_{0}),y_{0})$
such that

$|u(x_{0},y_{0})|\leq\frac{1}{\sqrt{y_{0}-y_{1}(x_{0})}}\sqrt{\int_{y_{1}(x_{0})}^{y_{0}}|u(x_{0},z)|^{2}\,dz}.$

Therefore

$\displaystyle|u(x_{0},y_{0})|^{2}=\left|u(x_{0},y_{2}(x_{0}))+\int_{y_{2}(x_{0})}^{y_{0}}\frac{\partial
u}{\partial z}(x_{0},z)\,dz\right|^{2}$ $\displaystyle\leq
2|u(x_{0},y_{2}(x_{0}))|^{2}+2(y_{0}-y_{2}(x_{0}))\int_{y_{2}(x)}^{y}\left|\frac{\partial
u}{\partial z}(x,z)\right|^{2}\,dy$
$\displaystyle\leq\frac{2}{y_{0}-y_{1}(x_{0})}\int_{y_{1}(x_{0})}^{y_{0}}|u(x_{0},z)|^{2}\,dz+2(y_{0}-y_{2}(x_{0}))\int_{y_{2}(x_{0})}^{y_{0}}\left|\frac{\partial
u}{\partial z}(x_{0},z)\right|^{2}\,dz$
$\displaystyle\leq\frac{2}{\mathrm{dist}((x_{0},y_{0}),\partial\Omega)}\int_{y_{1}(x_{0})}^{y_{0}}|u(x_{0},z)|^{2}\,dz+2\mathrm{diam}(\Omega)\int_{y_{1}(x_{0})}^{y_{0}}\left|\frac{\partial
u}{\partial z}(x_{0},z)\right|^{2}\,dz$
$\displaystyle\leq\frac{2}{\mathrm{dist}((x_{0},y_{0}),\partial\Omega)}\int_{l(x_{0})}|u(x_{0},z)|^{2}\,dz+2\mathrm{diam}(\Omega)\int_{l(x_{0})}\left|\frac{\partial
u}{\partial z}(x_{0},z)\right|^{2}\,dz,$

where $\mathrm{diam}(\Omega)$ is the diameter of $\Omega$ and
$\mathrm{dist}((x_{0},y_{0}),\partial\Omega)$ is the distance between
$(x_{0},y_{0})$ and the boundary of $\Omega$. This proves (4.5).

The case when $(x_{0},y_{0})$ satisfies second condition of property∗ can be
studied similarly. Repeating the same ideas one gets the validity of (4.6). ∎

###  5.2 Proof of Lemma 4.2

Let us estimate separately the curvilinear integrals
$\int_{\gamma^{i}}|v|_{\partial\gamma^{i}}^{2}\,d\mu,\,i=1,2,3,4$, where
$\gamma^{i}$ are given in polar coordinates as follows

$\displaystyle\gamma^{1}=\left\\{r=\tau,\frac{\pi}{4}\leq\varphi\leq\frac{3\pi}{4}\right\\},\,\,\gamma^{2}=\left\\{r=\tau,\frac{3\pi}{4}\leq\varphi\leq\frac{5\pi}{4}\right\\},$
$\displaystyle\gamma^{3}=\left\\{r=\tau,\frac{5\pi}{4}\leq\varphi\leq\frac{7\pi}{4}\right\\},\,\,\gamma^{4}=\left\\{r=\tau,-\frac{\pi}{4}\leq\varphi\leq\frac{\pi}{4}\right\\}$

and $|v|_{\gamma^{i}}$ is the trace of $v$ on $\gamma^{i}$.

Let us start from $\int_{\gamma^{1}}|v|_{\gamma^{1}}^{2}\,d\mu$. We have

$\displaystyle\int_{\gamma^{1}}|v|_{\gamma^{1}}^{2}\,d\mu=\tau\int_{-\tau/\sqrt{2}}^{\tau/\sqrt{2}}\frac{1}{\sqrt{\tau^{2}-x^{2}}}|v(x,\sqrt{\tau^{2}-x^{2}})|^{2}\,dx$
$\displaystyle\leq\sqrt{2}\int_{-\tau/\sqrt{2}}^{\tau/\sqrt{2}}|v(x,\sqrt{\tau^{2}-x^{2}})|^{2}\,dx.$

Since $(x,\sqrt{\tau^{2}-x^{2}}))$ for each $x,|x|\leq\tau/\sqrt{2}$,
obviously satisfies first condition of property∗ for domain
$\Omega_{K_{\varepsilon}}$ then using estimate 4.5 one gets

$\displaystyle\int_{\gamma^{1}}|v|_{\gamma^{1}}^{2}\,d\mu\leq\sqrt{2}C^{2}\int_{-\tau/\sqrt{2}}^{\tau/\sqrt{2}}\left(\int_{l(x)}\left|\frac{\partial
v}{\partial z}(x,z)\right|^{2}\,dz+\int_{l(x)}|v(x,z)|^{2}\,dz\right)\,dx\,dy$
$\displaystyle\leq\sqrt{2}C^{2}\int_{\Omega\cap\\{|x|\leq\tau/\sqrt{2}\\}}|v(x,y)|^{2}\,dx\,dy$
$\displaystyle+\sqrt{2}C^{2}\int_{\Omega\cap\\{|x|\leq\tau/\sqrt{2}\\}}\left|\frac{\partial
v}{\partial y}(x,y)\right|^{2}\,dx\,dy$
$\displaystyle\leq\sqrt{2}C^{2}\int_{\Omega}|v(x,y)|^{2}\,dx\,dy+\sqrt{2}C^{2}\int_{\Omega}\left|\frac{\partial
v}{\partial y}(x,y)\right|^{2}\,dx\,dy.$ (5.1)

In the same way we are able to estimate the curvilinear integral

$\displaystyle\int_{\gamma^{3}}|v|_{\gamma^{3}}^{2}\,d\mu\leq\sqrt{2}C^{2}\int_{\Omega}|v(x,y)|^{2}\,dx\,dy$
(5.2) $\displaystyle+\sqrt{2}C^{2}\int_{\Omega}\left|\frac{\partial
v}{\partial y}(x,y)\right|^{2}\,dx\,dy.$

In a similar way one deals with the integrals over curves
$\gamma^{2},\gamma^{4}$. We establish

$\displaystyle\int_{\gamma^{2}}|v|_{\gamma^{2}}^{2}\,d\mu\leq\sqrt{2}C^{2}\int_{\Omega}|v(x,y)|^{2}\,dx\,dy$
$\displaystyle+\sqrt{2}C^{2}\int_{\Omega}\left|\frac{\partial v}{\partial
x}(x,y)\right|^{2}\,dx\,dy,$
$\displaystyle\int_{\gamma^{4}}|v|_{\gamma^{4}}^{2}\,d\mu\leq\sqrt{2}C^{2}\int_{\Omega}|v(x,y)|^{2}\,dx\,dy$
$\displaystyle+\sqrt{2}C^{2}\int_{\Omega}\left|\frac{\partial v}{\partial
x}(x,y)\right|^{2}\,dx\,dy.$

Combining the above bounds together with (5.1) and (5.2) we finish the proof
of lemma. ∎

###  5.3 Proof of Lemma 4.3

To proceed with a proof we need the following auxiliary material [9, Lemma
4.9]:

######  Lemma 5.1.

Let $\Pi\subset\mathbb{R}^{n}$ be a convex set and let $G$ and $Q$ be
arbitrary measurable sets in $\Pi$ with $\mu\,(G)\neq 0$. Then, for all
$v\in\mathcal{H}^{1}(\Pi)$, the following inequality holds:

$\displaystyle\int_{Q}|v|^{2}\,dx\,dy$ (5.1)
$\displaystyle\leq\frac{2\mu\,(Q)}{\mu\,(G)}\int_{G}|v|^{2}\,dx\,dy+\frac{C(n)(d(\Pi))^{n+1}(\mu\,(Q))^{1/n}}{\mu\,(G)}\int_{\Pi}|\nabla
v|^{2}\,dx\,dy,$

where $d(\Pi)$ is the parameter of $\Pi$, $\mu$ is the Lebesque measure on
$\mathbb{R}^{n}$, and the constant $C(n)$ depends only on the dimension of
$\mathbb{R}^{n}$.

Let $G_{\epsilon}$ be a convex subset of $\Omega$ to be chosen later which
contains $\mathbb{B}_{\epsilon}$. Since
$g\in\mathcal{H}^{2}_{\mathrm{loc}}(\Omega)$ in view of the interior
regularity theory then applying (5.1) for $Q=\mathbb{B}_{\epsilon}$ and
$G=\Pi=G_{\epsilon}$ one infers that

$\displaystyle\int_{\mathbb{B}_{\epsilon}}\left|\frac{\partial g}{\partial
x}\right|^{2}\,dx\,dy\leq\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\frac{\partial
g}{\partial x}\right|^{2}\,dx\,dy$
$\displaystyle+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\nabla\left(\frac{\partial
g}{\partial x}\right)\right|^{2}\,dx\,dy$
$\displaystyle=\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\frac{\partial
g}{\partial
x}\right|^{2}\,dx\,dy+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left(\left|\frac{\partial^{2}g}{\partial
x^{2}}\right|^{2}+\left|\frac{\partial^{2}g}{\partial x\partial
y}\right|^{2}\right)\,dx\,dy$

and

$\displaystyle\int_{\mathbb{B}_{\epsilon}}\left|\frac{\partial g}{\partial
y}\right|^{2}\,dx\,dy\leq\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\frac{\partial
g}{\partial y}\right|^{2}\,dx\,dy$
$\displaystyle+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\nabla\left(\frac{\partial
g}{\partial y}\right)\right|^{2}\,dx\,dy$
$\displaystyle=\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left|\frac{\partial
g}{\partial
y}\right|^{2}\,dx\,dy+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left(\left|\frac{\partial^{2}g}{\partial
y\partial x}\right|^{2}+\left|\frac{\partial^{2}g}{\partial
y^{2}}\right|^{2}\right)\,dx\,dy.$

Combining the above inequalities we arrive

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|\nabla
g|^{2}\,dx\,dy\leq\frac{2\mu\,(\mathbb{B}_{\epsilon})}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}|\nabla
g|^{2}\,dx\,dy$
$\displaystyle+\frac{C(2)(d\,(G_{\epsilon}))^{3}(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\int_{G_{\epsilon}}\left(\left|\frac{\partial^{2}g}{\partial
x^{2}}\right|^{2}+2\left|\frac{\partial^{2}g}{\partial x\partial
y}\right|^{2}+\left|\frac{\partial^{2}f}{\partial
y^{2}}\right|^{2}\right)\,dx\,dy$
$\displaystyle\leq\frac{(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}}{\mu\,(G_{\epsilon})}\left(2(\mu\,(\mathbb{B}_{\epsilon}))^{1/2}+C(2)(d\,(G_{\epsilon}))^{3}\right)\|g\|_{\mathcal{H}^{2}(G_{\epsilon})}^{2}$
$\displaystyle=\frac{\sqrt{\pi}\epsilon}{\mu\,(G_{\epsilon})}\left(2\sqrt{\pi}\epsilon+C(2)(d\,(G_{\epsilon}))^{3}\right)\|g\|_{\mathcal{H}^{2}(G_{\epsilon})}^{2}.$
(5.2)

Let us choose $G_{\epsilon}=\\{r:\,0\leq r\leq\epsilon^{1-\alpha}\\}$ with
some $\alpha\in\left(\frac{1}{2},1\right)$ to be chosen later. One can easily
notice that $\mathbb{B}_{\epsilon}\subset G_{\epsilon}$. The inequality (5.2)
performs to

$\int_{\mathbb{B}_{\epsilon}}|\nabla g|^{2}\,dx\,dy\leq
2\epsilon\left(\epsilon^{2\alpha-1}+\frac{4C(2)}{\sqrt{\pi}}\epsilon^{1-\alpha}\right)\,\|g\|^{2}_{\mathcal{H}^{2}(G_{\epsilon})}.$
(5.3)

To make the further estimates we need the interior regularity theorem [1] and
the following lemma:

######  Theorem 5.2.

(Interior Regularity Theorem.) Suppose that $h\in\mathcal{H}^{1}(\Omega)$ is a
weak solution to $-\Delta h=w$. Then
$h\in\mathcal{H}^{2}_{\mathrm{loc}}(\Omega)$ and for each
$\Omega_{0}\subset\Omega$ there is a constant $c=c(\Omega_{0})$ independent of
$h$ and $w$ such that:

$\|h\|_{\mathcal{H}^{2}(\Omega_{0})}\leq
c\left(\|h\|_{L^{2}(\Omega)}+\|w\|_{L^{2}(\Omega)}\right).$ (5.4)

######  Lemma 5.3.

[The proof will be given in the next subsection] For any
$z\in\mathrm{Dom}(-\Delta_{N}^{\Omega})$ the following estimate is valid

$\int_{\Omega}|-\Delta z+z|^{2}\,dx\,dy\geq\frac{1}{16}\int_{\Omega}(|\Delta
z|^{2}+|z|^{2})\,dx\,dy.$

Let us go back to inequality (5.3). Using (5.4) and Lemma 5.3 one establishes
that

$\displaystyle\int_{\mathbb{B}_{\epsilon}}|\nabla g|^{2}\,dx\,dy\leq
4\,c^{2}\epsilon\left(\epsilon^{2\alpha-1}+\frac{4C(2)}{\sqrt{\pi}}\epsilon^{1-\alpha}\right)\,(\|\Delta
g\|_{L^{2}(\Omega)}^{2}+\|g\|_{L^{2}(\Omega)}^{2})$ $\displaystyle\leq
64\,c^{2}\epsilon\left(\epsilon^{2\alpha-1}+\frac{4C(2)}{\sqrt{\pi}}\epsilon^{1-\alpha}\right)\,\|-\Delta
g+g\|_{L^{2}(\Omega)}^{2}$
$\displaystyle=64\,c^{2}\epsilon\left(\epsilon^{2\alpha-1}+\frac{4C(2)}{\sqrt{\pi}}\epsilon^{1-\alpha}\right)\,\|g\|_{2}^{2}\leq\tilde{C}\left(\epsilon^{2\alpha}+\epsilon^{2-\alpha}\right)\,\|g\|_{2}^{2},$
(5.5)

where
$\tilde{C}:=64c^{2}\mathrm{max}\left\\{1,\frac{4C(2)}{\sqrt{\pi}}\right\\}$
and $c$ depends on the distance of $\mathbb{B}_{\varepsilon}$ from the
boundary of $\Omega$.

Let us investigate the function
$F(\alpha):=\epsilon^{2\alpha}+\epsilon^{2-\alpha}$ in (5.3) on interval
$\left(\frac{1}{2},1\right)$. It attains its minimum at
$\alpha_{0}=\frac{2}{3}-\frac{1}{3}\frac{\ln 2}{\ln\epsilon}$ and takes the
value $F(\alpha_{0})=\epsilon^{4/3}\left(\frac{1}{4^{1/3}}+2^{1/3}\right)$.

Let us now choose $\alpha=\alpha_{0}$ in inequality (5.5). Then

$\int_{\mathbb{B}_{\epsilon}}|\nabla
g|^{2}\,dx\,dy\leq\epsilon^{4/3}\left(\frac{1}{4^{1/3}}+2^{1/3}\right)\tilde{C}\,\|g\|_{2}^{2},$

which concludes the proof of Lemma 4.3 with
$C^{\prime}=\left(\frac{1}{4^{1/3}}+2^{1/3}\right)\tilde{C}$.∎

###  5.4 Proof of Lemma 4.4

We first prove the following auxiliary statement: there exists
$\tau\in(\varepsilon,2\varepsilon)$ such that

$\int_{0}^{2\pi}\left|\nabla
g(\tau\cos\varphi,\tau\sin\varphi)\right|^{2}\,d\varphi\leq\frac{1}{\varepsilon^{2}}\|g\|_{1}^{2}.$
(5.6)

Let us assume the opposite: for any $r\in(\varepsilon,2\varepsilon)$ we have

$\int_{0}^{2\pi}\left|\nabla
g(r\cos\varphi,r\sin\varphi)\right|^{2}\,d\varphi>\frac{1}{\varepsilon^{2}}\|g\|_{1}^{2}.$

Passing to polar coordinates in integral
$\int_{\mathbb{B}_{2\varepsilon}\backslash\mathbb{B}_{\varepsilon}}|\nabla
g|^{2}\,dx\,dy$ and using the above bound we get

$\int_{\mathbb{B}_{2\varepsilon}\backslash\mathbb{B}_{\varepsilon}}|\nabla
g|^{2}\,dx\,dy=\int_{\varepsilon}^{2\varepsilon}\,\int_{0}^{2\pi}r\left|\nabla
g(r\cos\varphi,r\sin\varphi)\right|^{2}\,d\varphi\,dr>\|g\|_{1}^{2}.$

This contradicts with the fact that the left-hand side of the above inequality
does not exceed $\|g\|_{1}^{2}$. Hence there exists at least one number
$\tau\in(\varepsilon,2\varepsilon)$ such that

$\int_{0}^{2\pi}|\nabla
g(\tau\cos\varphi,\tau\sin\varphi)|^{2}\,d\varphi\leq\frac{1}{\varepsilon^{2}}\|g\|_{1}^{2}.$
(5.7)

In view of the representation of the derivative of function
$\tilde{g}(r,\varphi)=g(r\cos\varphi,r\sin\varphi)$

$\frac{\partial\widetilde{g}}{\partial\varphi}(\tau,\varphi)=-\tau\frac{\partial
g}{\partial x}(\tau\cos\varphi,\tau\sin\varphi)\sin\varphi+\tau\frac{\partial
g}{\partial y}(\tau\cos\varphi,\tau\sin\varphi)\cos\varphi$

and the Cauchy inequality we conclude that

$\left|\frac{\partial\widetilde{g}}{\partial\varphi}(\tau,\varphi)\right|\leq
2\varepsilon\left|\nabla g(\tau\cos\varphi,\tau\sin\varphi)\right|.$

Hence, employing (5.7) we establish that

$\int_{0}^{2\pi}\left|\frac{\partial\tilde{g}}{\partial\varphi}(\tau,\varphi)\right|^{2}\,d\varphi\leq
4\varepsilon^{2}\int_{0}^{2\pi}\left|\nabla
g(\tau\cos\varphi,\tau\sin\varphi)\right|^{2}\,d\varphi\leq 4\|g\|_{1}^{2},$

which concludes the proof.∎

###  5.5 Proof of Lemma 5.3

It is straightforward to check that

$\displaystyle\int_{\Omega}|-\Delta u+u|^{2}\,dx\,dy=\int_{\Omega}|\Delta
u|^{2}\,dx\,dy+2\int_{\Omega}|\nabla
u|^{2}\,dx\,dy+\int_{\Omega}|u|^{2}\,dx\,dy$
$\displaystyle\geq\int_{\Omega}|u|^{2}\,dx\,dy.$ (5.8)

Let us consider two cases:

$\displaystyle\int_{\Omega}|\Delta u|^{2}\,dx\,dy\geq
4\int_{\Omega}|u|^{2}\,dx\,dy,$ (5.9) $\displaystyle\int_{\Omega}|\Delta
u|^{2}\,dx\,dy<4\int_{\Omega}|u|^{2}\,dx\,dy.$ (5.10)

Starting from the first one and employing (5.8) we have

$\displaystyle\sqrt{\int_{\Omega}|-\Delta
u+u|^{2}\,dx\,dy}\geq\sqrt{\int_{\Omega}|\Delta
u|^{2}\,dx\,dy}-\sqrt{\int_{\Omega}|u|^{2}\,dx\,dy}$
$\displaystyle\geq\frac{1}{2}\sqrt{\int_{\Omega}|\Delta
u|^{2}\,dx\,dy}\geq\frac{1}{4}\sqrt{\int_{\Omega}|\Delta
u|^{2}\,dx\,dy}+\frac{1}{2}\sqrt{\int_{\Omega}|u|^{2}\,dx\,dy}$
$\displaystyle\geq\frac{1}{4}\left(\sqrt{\int_{\Omega}|\Delta
u|^{2}\,dx\,dy}+\sqrt{\int_{\Omega}|u|^{2}\,dx\,dy}\right).$

Hence we arrive at the bound

$\int_{\Omega}|-\Delta
u+u|^{2}\,dx\,dy\geq\frac{1}{16}\left(\int_{\Omega}|\Delta
u|^{2}\,dx\,dy+\int_{\Omega}|u|^{2}\,dx\,dy\right).$ (5.11)

Now let us consider the case (5.10). In view of inequality (5.8) we conclude

$\displaystyle\int_{\Omega}|-\Delta
u+u|^{2}\,dx\,dy\geq\int_{\Omega}|u|^{2}\,dx\,dy\geq\frac{1}{2}\int_{\Omega}|u|^{2}\,dx\,dy+\frac{1}{8}\int_{\Omega}|\Delta
u|^{2}\,dx\,dy$ $\displaystyle\geq\frac{1}{8}\left(\int_{\Omega}|\Delta
u|^{2}\,dx\,dy+\int_{\Omega}|u|^{2}\,dx\,dy\right).$

Combining the above estimate together with (5.11) we complete the proof of the
lemma. ∎

### Acknowledgements

D.B. acknowledges support from the Czech Science Foundation (GACR) within the
project 21-07129S.

Conflict of Interests
The authors declare that they have no conflict of interest regarding the
publication of this paper.

Authors’ contributions
All authors contributed equally to the manuscript and typed, read, and
approved the final form of the manuscript, which is the result of an intensive
collaboration.

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