A Simple Proof of Sklyanin’s Formula

A Simple Proof of Sklyanin’s Formula
for Canonical Spectral Coordinates
of the Rational Calogero–Moser System

Tamás F. GÖRBE

T.F. Görbe

Department of Theoretical Physics, University of Szeged,
Tisza Lajos krt 84-86, H-6720 Szeged, Hungary<EMAIL_ADDRESS>http://www.staff.u-szeged.hu/~tfgorbe/

Received January 19, 2016, in final form March 08, 2016; Published online
March 11, 2016

We use Hamiltonian reduction to simplify Falqui and Mencattini’s recent proof
of Sklyanin’s expression providing spectral Darboux coordinates of the
rational Calogero–Moser system. This viewpoint enables us to verify a
conjecture of Falqui and Mencattini, and to obtain Sklyanin’s formula as a
corollary.

integrable systems; Calogero–Moser type systems; spectral coordinates;
Hamiltonian reduction; action-angle duality

14H70; 37J15; 53D20

##  1 Introduction

Integrable many-body systems in one spatial dimension form an important class
of exactly solvable Hamiltonian systems with their diverse mathematical
structure and widespread applicability in physics [2, 6, 10]. Among these
many-body systems, one of the most widely known is the rational Calogero–Moser
model of equally massive interacting particles moving along a line with a pair
potential inversely proportional to the square of the distance. The model was
introduced and solved at the quantum level by Calogero [1]. The complete
integrability of its classical version was established by Moser [5], who
employed the Lax formalism to identify a complete set of commuting integrals
as coefficients of the characteristic polynomial of a certain Hermitian matrix
function, called the Lax matrix.

These developments might prompt one to consider the Poisson commuting
eigenvalues of the Lax matrix and be interested in searching for an expression
of conjugate variables. Such an expression was indeed formulated by Sklyanin
[8] in his work on bispectrality, and worked out in detail for the open Toda
chain [9]. Sklyanin’s formula for the rational Calogero–Moser model was
recently confirmed within the framework of bi-Hamiltonian geometry by Falqui
and Mencattini [3] in a somewhat circuitous way, although a short-cut was
pointed out in the form of a conjecture. The purpose of this paper is to prove
this conjecture and offer an alternative simple proof of Sklyanin’s formula
using results of Hamiltonian reduction.

Section 2 is a recap of complete integrability and action-angle duality for
the rational Calogero–Moser system in the context of Hamiltonian reduction. In
Section 3 we put these ideas into practice when we identify the canonical
variables of [3] in terms of the reduction picture, and prove the relation
conjectured in that paper. We attain Sklyanin’s formula as a corollary.
Section 4 contains our concluding remarks on possible generalizations.

##  2 The rational Calogero–Moser system via reduction

We begin by describing the rational Calogero–Moser system and recalling how it
originates from Hamiltonian reduction [4]. The content of this section is
standard and only included for the sake of self-consistency.

For $n$ particles, let the $n$-tuples $q=(q_{1},\dots,q_{n})$ and
$p=(p_{1},\dots,p_{n})$ collect their coordinates and momenta, respectively.
Then the Hamiltonian of the model reads

$\displaystyle
H(q,p)=\frac{1}{2}\sum_{j=1}^{n}p_{j}^{2}+g^{2}\sum_{\begin{subarray}{c}j,k=1\\\
(j<k)\end{subarray}}^{n}\frac{1}{(q_{j}-q_{k})^{2}},$ (1)

where $g$ is a real coupling constant tuning the strength of particle
interaction. The pair potential is singular at $q_{j}=q_{k}$ $(j\neq k)$,
hence any initial ordering of the particles remains unchanged during time-
evolution. The configuration space is chosen to be the domain
$\mathcal{C}=\\{q\in\mathbb{R}^{n}\,|\,q_{1}>\dots>q_{n}\\}$, and the phase
space is its cotangent bundle

$\displaystyle
T^{\ast}\mathcal{C}=\big{\\{}(q,p)\,|\,q\in\mathcal{C},\,p\in\mathbb{R}^{n}\big{\\}},$
(2)

endowed with the standard symplectic form

$\displaystyle\omega=\sum_{j=1}^{n}dq_{j}\wedge dp_{j}.$ (3)

The Hamiltonian system $(T^{\ast}\mathcal{C},\omega,H)$, called the rational
Calogero–Moser system, can be obtained as an appropriate Marsden–Weinstein
reduction of the free particle moving in the space of $n\times n$ Hermitian
matrices as follows.

Consider the manifold of pairs of $n\times n$ Hermitian matrices

$\displaystyle
M=\big{\\{}(X,P)\,|\,X,P\in\mathfrak{gl}(n,\mathbb{C}),\,X^{\dagger}=X,\,P^{\dagger}=P\big{\\}},$
(4)

equipped with the symplectic form

$\displaystyle\Omega=\operatorname{tr}(dX\wedge dP).$ (5)

The Hamiltonian of the analogue of a free particle reads

$\displaystyle\mathcal{H}(X,P)=\frac{1}{2}\operatorname{tr}\big{(}P^{2}\big{)}.$

The equations of motion can be solved explicitly for this Hamiltonian system
$(M,\Omega,\mathcal{H})$, and the general solution is given by
$X(t)=tP_{0}+X_{0}$, $P(t)=P_{0}$. Moreover, the functions
$\mathcal{H}_{k}(X,P)=\frac{1}{k}\operatorname{tr}\big{(}P^{k}\big{)}$,
$k=1,\dots,n$ form an independent set of commuting first integrals.

The group of $n\times n$ unitary matrices $U(n)$ acts on $M$ (4) by
conjugation

$\displaystyle(X,P)\to\big{(}UXU^{\dagger},UPU^{\dagger}\big{)},\qquad U\in
U(n),$

leaves both the symplectic form $\Omega$ (5) and the Hamiltonians
$\mathcal{H}_{k}$ invariant, and the matrix commutator $(X,P)\to[X,P]$ is a
momentum map for this $U(n)$-action. Consider the Hamiltonian reduction
performed by factorizing the momentum constraint surface

$\displaystyle[X,P]=\mathrm{i}g\big{(}vv^{\dagger}-\mathbf{1}_{n}\big{)}=:\mu,\qquad
v=(1\dots 1)^{\dagger}\in\mathbb{R}^{n},\qquad g\in\mathbb{R},$

with the stabilizer subgroup $G_{\mu}\subset U(n)$ of $\mu$, e.g., by
diagonalization of the $X$ component. This yields the gauge slice
$S=\\{(Q(q,p),L(q,p))\,|\,q\in\mathcal{C},\,p\in\mathbb{R}^{n}\\}$, where

$\displaystyle Q_{jk}=(UXU^{\dagger})_{jk}=q_{j}\delta_{jk},$ $\displaystyle
L_{jk}=(UPU^{\dagger})_{jk}=p_{j}\delta_{jk}+\mathrm{i}g\frac{1-\delta_{jk}}{q_{j}-q_{k}},\qquad
j,k=1,\dots,n.$ (6)

This $S$ is symplectomorphic to the reduced phase space and to
$T^{\ast}\mathcal{C}$ (2) since it inherits the reduced symplectic form
$\omega$ (3). The unreduced Hamiltonians project to a commuting set of
independent integrals $H_{k}=\frac{1}{k}\operatorname{tr}\big{(}L^{k}\big{)}$,
$k=1,\dots,n$, such that $H_{2}=H$ (1) and what’s more, the completeness of
Hamiltonian flows follows automatically from the reduction. Therefore the
rational Calogero–Moser system is completely integrable.

The similar role of matrices $X$ and $P$ in the derivation above can be
exploited to construct action-angle variables for the rational Calogero–Moser
system. This is done by switching to the gauge, where the $P$ component is
diagonalized by some matrix $\tilde{U}\in G_{\mu}$, and it boils down to the
gauge slice
$\tilde{S}=\big{\\{}\big{(}\tilde{Q}(\phi,\lambda),\tilde{L}(\phi,\lambda)\big{)}\,|\,\phi\in\mathbb{R}^{n},\,\lambda\in\mathcal{C}\big{\\}}$,
where

$\displaystyle\tilde{Q}_{jk}=\big{(}\tilde{U}X\tilde{U}^{\dagger}\big{)}_{jk}=\phi_{j}\delta_{jk}-\mathrm{i}g\frac{1-\delta_{jk}}{\lambda_{j}-\lambda_{k}},$
$\displaystyle\tilde{L}_{jk}=\big{(}\tilde{U}P\tilde{U}^{\dagger}\big{)}_{jk}=\lambda_{j}\delta_{jk},\qquad
j,k=1,\dots,n.$ (7)

By construction, $\tilde{S}$ with the symplectic form
$\tilde{\omega}=\sum\limits_{j=1}^{n}d\phi_{j}\wedge d\lambda_{j}$ is also
symplectomorphic to the reduced phase space, thus a canonical transformation
$(q,p)\to(\phi,\lambda)$ is obtained, where the reduced Hamiltonians depend
only on $\lambda$, viz.
$H_{k}=\frac{1}{k}\big{(}\lambda_{1}^{k}+\dots+\lambda_{n}^{k}\big{)}$,
$k=1,\dots,n$.

##  3 Sklyanin’s formula

Now, we turn to the question of variables conjugate to the Poisson commuting
eigenvalues $\lambda_{1},\dots,\lambda_{n}$ of $L$ (6), i.e., such functions
$\theta_{1},\dots,\theta_{n}$ in involution that

$\displaystyle\\{\theta_{j},\lambda_{k}\\}=\delta_{jk},\qquad j,k=1,\dots,n.$

At the end of Section 2 we saw that the variables $\phi_{1},\dots,\phi_{n}$
are such functions. These action-angle variables $\lambda$, $\phi$ were
already obtained by Moser [5] using scattering theory, and also appear in
Ruijsenaars’ proof of the self-duality of the rational Calogero–Moser system
[7].

Let us define the following functions over the phase space
$T^{\ast}\mathcal{C}$ (2) with dependence on an additional variable $z$:

$\displaystyle A(z)=\det(z\mathbf{1}_{n}-L),\qquad
C(z)=\operatorname{tr}\big{(}Q\operatorname{adj}(z\mathbf{1}_{n}-L)vv^{\dagger}\big{)},$
$\displaystyle
D(z)=\operatorname{tr}\big{(}Q\operatorname{adj}(z\mathbf{1}_{n}-L)\big{)},$
(8)

where $Q$ and $L$ are given by (6), $v=(1\dots 1)^{\dagger}\in\mathbb{R}^{n}$
and $\operatorname{adj}$ denotes the adjugate matrix, i.e., the transpose of
the cofactor matrix. Sklyanin’s formula [8] for $\theta_{1},\dots,\theta_{n}$
then reads

$\displaystyle\theta_{k}=\frac{C(\lambda_{k})}{A^{\prime}(\lambda_{k})},\qquad
k=1,\dots,n.$ (9)

In [3] Falqui and Mencattini have shown that

$\displaystyle\mu_{k}=\frac{D(\lambda_{k})}{A^{\prime}(\lambda_{k})},\qquad
k=1,\dots,n$ (10)

are conjugate variables to $\lambda_{1},\dots,\lambda_{n}$, and

$\displaystyle\theta_{k}=\mu_{k}+f_{k}(\lambda_{1},\dots,\lambda_{n}),\qquad
k=1,\dots,n,$ (11)

with such $\lambda$-dependent functions $f_{1},\dots,f_{n}$ that

$\displaystyle\frac{\partial f_{j}}{\partial\lambda_{k}}=\frac{\partial
f_{k}}{\partial\lambda_{j}},\qquad j,k=1,\dots,n$ (12)

thus $\theta_{1},\dots,\theta_{n}$ given by Sklyanin’s formula (9) are
conjugate to $\lambda_{1},\dots,\lambda_{n}$. This was done in a roundabout
way, although the explicit form of relation (11) was conjectured.

Here we take a different route by making use of the reduction viewpoint of
Section 2. From this perspective, the problem becomes transparent and can be
solved effortlessly. First, we show that $\mu_{1},\dots,\mu_{n}$ (10) are
nothing else than the angle variables $\phi_{1},\dots,\phi_{n}$.

The variables $\mu_{1},\dots,\mu_{n}$ defined in (10) are the angle variables
$\phi_{1},\dots,\phi_{n}$ of the rational Calogero–Moser system.

###### Proof.

Notice that, by definition, $\mu_{1},\dots,\mu_{n}$ are gauge invariant, thus
by working in the gauge, where the $P$ component is diagonal, that is with the
matrices $\tilde{Q}$, $\tilde{L}$ (7), we get

$\displaystyle\frac{D(z)}{A^{\prime}(z)}=\frac{\sum\limits_{j=1}^{n}\phi_{j}\prod\limits_{\begin{subarray}{c}\ell=1\\\
(\ell\neq
j)\end{subarray}}^{n}(z-\lambda_{\ell})}{\sum\limits_{j=1}^{n}\prod\limits_{\begin{subarray}{c}\ell=1\\\
(\ell\neq j)\end{subarray}}^{n}(z-\lambda_{\ell})}.$ (13)

Substituting $z=\lambda_{k}$ into (13) yields $\mu_{k}=\phi_{k}$, for each
$k=1,\dots,n$. ∎

Next, we prove the relation of functions $A$, $C$, $D$ (8), that was
conjectured in [3].

For any $n\in\mathbb{N}$, $(q,p)\in T^{\ast}\mathcal{C}$ (2), and
$z\in\mathbb{C}$ we have

$\displaystyle C(z)=D(z)+\frac{\mathrm{i}g}{2}A^{\prime\prime}(z).$

###### Proof.

Pick any point $(q,p)$ in the phase space $T^{\ast}\mathcal{C}$ and consider
the corresponding point $(\lambda,\phi)$ in the space of action-angle
variables. Since $A(z)=(z-\lambda_{1})\cdots(z-\lambda_{n})$ we have

$\displaystyle\frac{\mathrm{i}g}{2}A^{\prime\prime}(z)=\mathrm{i}g\sum_{\begin{subarray}{c}j,k=1\\\
(j<k)\end{subarray}}^{n}\prod_{\begin{subarray}{c}\ell=1\\\ (\ell\neq
j,k)\end{subarray}}^{n}(z-\lambda_{\ell}).$

The difference of functions $C$ and $D$ (8) reads

$\displaystyle
C(z)-D(z)=\operatorname{tr}\big{(}Q\operatorname{adj}(z\mathbf{1}_{n}-L)\big{(}vv^{\dagger}-\mathbf{1}_{n}\big{)}\big{)}.$
(14)

Due to gauge invariance, we are allowed to work with $\tilde{Q}$, $\tilde{L}$
(7) instead of $Q$, $L$ (6). Therefore (14) can be written as the sum of all
off-diagonal components of
$\tilde{Q}\operatorname{adj}(z\mathbf{1}_{n}-\tilde{L})$, that is

$\displaystyle C(z)-D(z)=\mathrm{i}g\sum_{\begin{subarray}{c}j,k=1\\\ (j\neq
k)\end{subarray}}^{n}\frac{-1}{\lambda_{j}-\lambda_{k}}\prod_{\begin{subarray}{c}\ell=1\\\
(\ell\neq
k)\end{subarray}}^{n}(z-\lambda_{\ell})=\mathrm{i}g\sum_{\begin{subarray}{c}j,k=1\\\
(j<k)\end{subarray}}^{n}\prod_{\begin{subarray}{c}\ell=1\\\ (\ell\neq
j,k)\end{subarray}}^{n}(z-\lambda_{\ell}).$

This concludes the proof. ∎

Our theorem confirms that indeed relation (11) is valid with

$\displaystyle
f_{k}(\lambda_{1},\dots,\lambda_{n})=\frac{\mathrm{i}g}{2}\frac{A^{\prime\prime}(\lambda_{k})}{A^{\prime}(\lambda_{k})}=\mathrm{i}g\sum_{\begin{subarray}{c}\ell=1\\\
(\ell\neq k)\end{subarray}}^{n}\frac{1}{\lambda_{k}-\lambda_{\ell}},\qquad
k=1,\dots,n,$

for which (12) clearly holds. An immediate consequence, as we indicated
before, is that $\theta_{1},\dots,\theta_{n}$ (9) are conjugate variables to
$\lambda_{1},\dots,\lambda_{n}$, thus Sklyanin’s formula is verified.

[Sklyanin’s formula] The variables $\theta_{1},\dots,\theta_{n}$ defined by

$\displaystyle\theta_{k}=\frac{C(\lambda_{k})}{A^{\prime}(\lambda_{k})},\qquad
k=1,\dots,n$

are conjugate to the eigenvalues $\lambda_{1},\dots,\lambda_{n}$ of the Lax
matrix $L$.

##  4 Discussion

There seem to be several ways for generalization. For example, one might
consider rational Calogero–Moser models associated to root systems other than
type $A_{n-1}$. The hyperbolic Calogero–Moser systems as well as, the
‘relativistic’ Calogero–Moser systems, also known as Ruijsenaars–Schneider
systems, are also of considerable interest.

### Acknowledgements

Thanks are due to László Fehér for drawing our attention to the duality
perspective. This work was supported in part by the Hungarian Scientific
Research Fund (OTKA) under the grant K-111697. The work was also partially
supported by COST (European Cooperation in Science and Technology) in COST
Action MP1405 QSPACE.

## References

  * [1] Calogero F., Solution of the one-dimensional $N$-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419–436, Erratum, J. Math. Phys. 37 (1996), 3646.
  * [2] Calogero F., Classical many-body problems amenable to exact treatments, Lecture Notes in Physics. New Series m: Monographs, Vol. 66, Springer-Verlag, Berlin, 2001.
  * [3] Falqui G., Mencattini I., Bi-Hamiltonian geometry and canonical spectral coordinates for the rational Calogero–Moser system, arXiv:1511.06339.
  * [4] Kazhdan D., Kostant B., Sternberg S., Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481–507.
  * [5] Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197–220.
  * [6] Perelomov A.M., Integrable systems of classical mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990.
  * [7] Ruijsenaars S.N.M., Action-angle maps and scattering theory for some finite-dimensional integrable systems. I. The pure soliton case, Comm. Math. Phys. 115 (1988), 127–165.
  * [8] Sklyanin E., Bispectrality and separation of variables in multiparticle hypergeometric systems, Talk given at the Workshop ‘Quantum Integrable Discrete Systems’, Cambridge, England, March 23–27, 2009.
  * [9] Sklyanin E., Bispectrality for the quantum open Toda chain, J. Phys. A: Math. Theor. 46 (2013), 382001, 8 pages, arXiv:1306.0454.
  * [10] Sutherland B., Beautiful models: 70 years of exactly solved quantum many-body problems, World Sci. Publ. Co., Inc., River Edge, NJ, 2004.


C^{\prime}_{1}\bigg{(}\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}+\mathcal{B}(|\nabla_{x}u|^{2},\Psi^{3})\bigg{)}+C^{\prime}_{2}\,\mathcal{B}(|\nabla_{\varphi}u|^{2},\Psi^{3})\\\
+C^{\prime}_{3}\bigg{(}\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}+\iint_{\Omega}|\nabla
u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}\bigg{)}.$

We take $\delta=C^{\prime}_{3}\kappa^{1/2}$ in Lemma 5.8 and we obtain that

$-\mathcal{B}(|\partial_{r}u|^{2},\Psi^{3})\leq
C^{\prime}_{3}\kappa^{1/2}\|\widetilde{N}(\nabla
u|\Psi)\|_{2}^{2}+C_{0}\kappa^{-1/2}K^{2}\|S(\overline{\nabla}u|\Psi^{3})\|_{2}^{2},$

for some $C_{0}$ depending on $\lambda$, $n$, that is

(5.21) $-\frac{1}{2}\|S(\overline{\nabla}u|\Psi^{3})\|_{2}^{2}\leq
C^{\prime}_{3}\kappa\|\widetilde{N}(\nabla
u|\Psi)\|_{2}^{2}+\frac{\kappa^{1/2}}{2C_{0}K^{2}}\mathcal{B}(|\partial_{r}u|^{2},\Psi^{3}),$

Add (5.21) to (5.20) implies

(5.22) $\frac{1}{2}\|S(\overline{\nabla}u|\Psi^{3})\|^{2}_{2}\leq
C^{\prime}_{1}\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}\\\
+\mathcal{B}(C^{\prime}_{1}|\nabla_{x}u|^{2}+C^{\prime}_{2}|\nabla_{\varphi}u|^{2}+(2C_{0}K^{2})^{-1}\kappa^{1/2}|\partial_{r}u|^{2},\Psi^{3})\\\
+2C^{\prime}_{3}\bigg{(}\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}+\iint_{\Omega}|\nabla
u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}\bigg{)}.$

The lemma follows by taking $c_{0}=(2C_{0})^{-1}$, $C_{1}=2C^{\prime}_{1}$,
$C_{2}=2C^{\prime}_{2}$, and $C_{3}=4C^{\prime}_{3}$. ∎

##  6\. Global Estimates for Energy Solutions

We define the weighted homogeneous Sobolev space $W$ as

$W:=\\{u\in L^{1}_{loc}(\Omega),\,\nabla u\in L^{2}(\Omega,|t|^{d+1-n}\\}$

which is equipped with the semi-norm

$\|u\|_{W}:=\left(\iint_{\Omega}|\nabla
u|^{2}\frac{dt\,dx}{|t|^{d+1-n}}\right)^{\frac{1}{2}}.$

The space $H:=\dot{W}^{1/2,2}(\mathbb{R}^{d})$ is the usual homogeneous space
of traces on the boundary $\partial\Omega=\mathbb{R}^{d}$, equipped with the
usual semi-norm

$\|g\|_{H}:=\left(\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{|f(x)-f(y)|^{2}}{|x-y|^{d+1}}\right)^{\frac{1}{2}}.$

The trace of a function $u\in W$ is defined for any $x\in\mathbb{R}^{d}$ as

$\operatorname{Tr}(u)(x):=\lim_{\epsilon\to
0}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{B(x,\epsilon)\cap\Omega}u\,dy\,dt$

if the limit exists and $+\infty$ otherwise. Equivalent definitions of trace
exist, and we choose the one constructed in [DFM21b] just because we shall
refer to this manuscript for basic results.

We know from [DFM21b]444when $d<n-1$, the case $d=n-1$ being general
knowledge. that $\operatorname{Tr}$ is a bounded linear operator from $W$ to
$H$. Moreover, $\|.\|_{W}$ is a norm for the subspace $W_{0}:=\\{u\in
W,\,\operatorname{Tr}(u)=0\\}$, and $(W_{0},\|.\|_{W})$ is complete. We can
apply Lax-Milgram’s theorem to obtain weak solution to $Lu=0$ with prescribed
data $g\in H$.

######  Lemma 6.1 (Lemma 9.1 and Lemma 9.4 (v) in [DFM21b]).

Let $L=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}\nabla]$ be an operator
satisfying (1.1). For each $g\in H$, there exists a unique function $u_{g}\in
W$ such that

$\iint_{\Omega}\mathcal{A}\nabla u_{g}\cdot\nabla
v\,\frac{dt\,dx}{|t|^{n-d-1}}=0\qquad\text{ for any }v\in W_{0}$

and $\operatorname{Tr}(u_{g})=g$. Moreover, we have the bound

$\|u_{g}\|_{W}\leq C\|g\|_{H}$

with a constant $C$ that depends only on the elliptic constant $\lambda$ in
(1.1).

When $g\in H$ is continuous and compactly supported, the solution given by
this lemma and the solution given by (1.3) are the same.

In this section, we assume that $u\in W$, and we observe to which extend we
can take non-compact cut-off functions in the local estimates given in Lemma
5.5, Lemma 5.9, and Lemma 4.6, and thus obtain global estimates.

######  Lemma 6.2.

Let $u\in W$. For any $\epsilon>0$, we have

$\|\widetilde{N}(u|\Psi_{\epsilon})\|_{2}^{2}\leq C\epsilon^{-1}\|u\|_{W}^{2}$

where $C$ depends only on $d$ and $n$.

###### Proof.

Take $x\in\mathbb{R}^{d}$ and $(z,r)\in\Gamma(x)\cap\\{r\geq\epsilon/2\\}$.
Then

$|(\nabla
u)_{W}(z,r)|^{2}:=\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(z,r)}|\nabla
u|^{2}\,dt\,dy\lesssim\epsilon^{-1}\iint_{W(z,r)}|\nabla
u|^{2}\,\frac{dt\,dy}{|t|^{n-1}}\leq\epsilon^{-1}\iint_{\widehat{\Gamma}^{*}(x)}|\nabla
u|^{2}\,\frac{dt\,dy}{|t|^{n-1}}$

where $\widehat{\Gamma}^{*}(x):=\\{(y,t)\in\Omega,\,|y-x|\leq C^{*}|t|\\}$ and
$C^{*}$ is a large constant that depends only on $d$ and $n$ and is such that
$W(z,r)\in\widehat{\Gamma}^{*}(x)$ for all $(z,r)\in\Gamma(x)$. We deduce that

$\widetilde{N}(u|\Psi_{\epsilon})(x)\lesssim\epsilon^{-1}\iint_{\widehat{\Gamma}^{*}(x)}|\nabla
u|^{2}\,\frac{dt\,dy}{|t|^{n-1}}$

and then

$\|\widetilde{N}(u|\Psi_{\epsilon})\|_{2}^{2}\lesssim\epsilon^{-1}\iint_{\Omega}|\nabla
u|^{2}\,\frac{dt\,dy}{|t|^{n-d-1}}=\epsilon^{-1}\|u\|_{W}^{2}$

by a simple variant of (3.1). ∎

When $u\in W$, we can see that $\|\widetilde{N}(u)\|_{2}^{2}$ will explode
only when we get close to the boundary. We can use the monotone convergence of
$\Psi_{B,l,\epsilon}\uparrow\Psi_{\epsilon}$ as $B\uparrow\mathbb{R}^{d}$ and
$l\to\infty$, and then take $\Psi=\Psi_{\epsilon}$ in (5.14), which shows that
$\|S(\overline{\nabla}u|\Psi_{\epsilon})\|_{2}^{2}$ is also finite and bounded
by $C\epsilon^{-1}\|u\|_{W}^{2}$.

Moreover, we can also take $\Psi=\Psi_{\epsilon}$ in (5.19). Indeed, any term
containing $\Psi_{\epsilon}\nabla\Psi_{B}$ can be bounded by
$C\epsilon^{-1}\|\nabla u\|^{2}_{L^{2}(\mathbb{R}^{d}\setminus
B,|t|^{d+1-n}dtdx)}$ and any term containing $\nabla\Psi_{l}$ can be bounded
by $Cl^{-1}\|u\|_{W}^{2}$, and both those terms converges to 0 as
$B\uparrow\mathbb{R}^{d}$ and $l\to\infty$. It means that the terms in (5.19)
that contain either $\nabla\Psi_{B}$ or $\nabla\Psi_{l}$ with eventually
disappear when we take the limit $\Psi_{B,l,\epsilon}\uparrow\Psi_{\epsilon}$.
Let us give a bit of details. In the left-hand side of 5.19, we first have

$\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi_{B,l,\epsilon}^{3}\frac{dtdx}{|t|^{n-d-1}}\\\
\leq\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi^{3}_{\epsilon}\frac{dtdx}{|t|^{n-d-1}}+\iint_{\Omega}|\nabla_{x}u|^{2}\Psi_{\epsilon}|\partial_{r}\Psi_{B}|\frac{dtdx}{|t|^{n-d-1}}+\iint_{\Omega}|\nabla_{x}u|^{2}|\partial_{r}\Psi_{l}|\frac{dtdx}{|t|^{n-d-1}}\\\
\leq\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi^{3}_{\epsilon}\frac{dtdx}{|t|^{n-d-1}}+C\epsilon^{-1}\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{d}\setminus
B,|t|^{d+1-n}dtdx)}+Cl^{-1}\|u\|_{W}^{2}\\\
\longrightarrow\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi^{3}_{\epsilon}\frac{dtdx}{|t|^{n-d-1}}\quad\text{
as $B\uparrow\mathbb{R}^{d}$ and $l\to\infty$.}$

The terms $\kappa\|\widetilde{N}(\nabla u|\Psi_{B,l,\epsilon}^{3})\|^{2}_{2}$
is easily bounded by $\kappa\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{3})\|^{2}_{2}$ and

$\iint_{\Omega}|\nabla
u|^{2}|\nabla_{x}\Psi_{B,l,\epsilon}|^{2}\Psi_{B,l,\epsilon}\frac{dtdx}{|t|^{n-d-2}}\leq\frac{Cl}{r_{B}^{2}}\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{d}\setminus B,|t|^{d+1-n}dtdx)}\longrightarrow 0$

as long as we always take the radius $r_{B}$ of $B$ bigger than $l$. The last
term in the left-hand side of 5.19 is $\mathcal{B}(|\nabla
u|^{2}_{\kappa},\Psi_{B,l,\epsilon}^{3})$. We have

$|\mathcal{B}(|\nabla
u|^{2}_{\kappa},\Psi_{B,l,\epsilon}^{3})-\mathcal{B}(|\nabla
u|^{2}_{\kappa},\Psi_{\epsilon}^{3})|\\\ \lesssim\iint_{\Omega}\big{(}|\nabla
u|^{2}+t|\nabla\overline{\nabla}u||\nabla
u|\big{)}\,\big{(}\Psi_{\epsilon}^{3}|\partial_{r}\Psi_{B,l}^{3}|+|1-\Psi_{B,l}^{3}||\partial_{r}\Psi_{\epsilon}^{3}|\big{)}\,\frac{dtdx}{|t|^{n-d-2}}\\\
\lesssim\iint_{\Omega}\big{(}|\nabla
u|^{2}+t^{2}|\nabla\overline{\nabla}u|^{2}\big{)}\,\big{(}\Psi_{\epsilon}^{3}|\partial_{r}\Psi_{B,l}^{3}|+|1-\Psi_{B,l}^{3}||\partial_{r}\Psi_{\epsilon}^{3}|\big{)}\,\frac{dtdx}{|t|^{n-d-2}}\\\
\longrightarrow 0\quad\text{ as $B\uparrow\mathbb{R}^{d}$ and $l\to\infty$}$

since
$\|u\|_{W}^{2}+\|S(\overline{\nabla}u|\Psi_{\epsilon})\|_{2}^{2}<+\infty$. A
similar argument gives that the term

$\bigg{\|}\bigg{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{l}(.)}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{l\leq|s|\leq 2l}|\nabla
u|^{2}\Psi_{B}^{3}ds\,dy\bigg{)}^{1/2}\bigg{\|}^{2}_{2}\approx\int_{\mathbb{R}^{d}}\fint_{l\leq|s|\leq
2l}|\nabla u|^{2}\Psi_{B}^{3}\,ds\,dy$

that appears in Lemma 4.6 is bounded by $Cl^{-1}\|u\|_{W}$ and also converges
to 0 as $l$ goes to infinity.

From those observations, Lemmas 5.9 and 4.6 combined with the fact that $u\in
W$ entail the following estimates.

######  Lemma 6.3.

Let $\kappa\in(0,1)$ and let $L$ be an elliptic operator satisfying
($\mathcal{H}$)λ,κ. There exist four constants $c_{0},C_{1},C_{2},C_{3}>0$
depending on $\lambda$ and $n$ such that, if

$|\nabla
u|^{2}_{\kappa}:=C_{1}|\nabla_{x}u|^{2}+C_{2}|\nabla_{\varphi}u|^{2}+c_{0}\kappa^{1/2}|\partial_{r}u|^{2},$

then for any weak solution $u\in W$ to $Lu=0$ and for any $\epsilon>0$, we
have that

(6.1) $\|S(\overline{\nabla}u|\Psi_{\epsilon}^{3})\|^{2}_{2}\leq
C_{1}\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi_{\epsilon}^{3}\frac{dtdx}{|t|^{n-d-1}}+\mathcal{B}(|\nabla
u|^{2}_{\kappa},\Psi_{\epsilon}^{3})\\\ +C_{3}\kappa\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{3})\|^{2}_{2}$

and

(6.2) $\|\widetilde{N}(\nabla u|\Psi_{\epsilon}^{3})\|_{2}^{2}\leq
C_{3}\|S(\overline{\nabla}u|\Psi_{\epsilon})\|_{2}^{2}+C_{3}\kappa\|\widetilde{N}(\nabla
u|\Psi_{\epsilon})\|_{2}^{2}.$

Moreover, all the quantities that appear in (6.1) and (6.2) are finite and
bounded by $C\epsilon^{-1}\|u\|_{W}^{2}$.

Note that the we only assume that $\kappa\in(0,1)$ in Lemma 6.3 for a
technical reason (that comes from the fact that $\delta\in(0,1)$ in Lemma 5.8)
and that condition can probably be removed. But all this does not really
matter because the proof of our next result (and thus the proof of the main
result of the article) requires $\kappa$ to be small anyway.

######  Theorem 6.4.

Take $\lambda>0$. There exists $\kappa\in(0,1)$ small enough (depending only
on $\lambda$, $d$ and $n$) such that if
$L:=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ is an elliptic operator
satisfying ($\mathcal{H}$)λ,M,κ, then for any weak solution $u\in W$ to $Lu=0$
we have that

(6.3) $\displaystyle\|\widetilde{N}(\nabla u)\|^{2}_{2}\leq
C\limsup_{\epsilon\to
0}\int_{\mathbb{R}^{d}}\fint_{\epsilon/2\leq|s|\leq\epsilon}|\nabla_{x}u|^{2}\,ds\,dy,$

where $C>0$ depends only on $\lambda$, $d$, and $n$.

Neither the right-hand side nor the left-hand side of (6.3) are guaranteed to
be finite, but the left-hand side is finite as long as the right-hand side is.
More precisely, there exists a sequence $\epsilon_{k}\to 0$ such that

(6.4) $\displaystyle\|\widetilde{N}(\nabla
u|\Psi_{\epsilon_{k}}^{9})\|^{2}_{2}\leq
C\int_{\mathbb{R}^{d}}\fint_{\epsilon_{k}/2\leq|s|\leq\epsilon_{k}}|\nabla_{x}u|^{2}\,ds\,dy.$

###### Proof.

The bound (6.3) is an immediate consequence of (6.4), so we just need to
establish the later.

Remember that $\Psi_{\epsilon}$ is constructed from a smooth function $\phi$.
All the constant depend on the fixed $\phi$, but we have a bit of freedom (as
long as we do not take $\phi$ so that $\Psi_{\epsilon}$ satisfies
($\mathcal{COF}$)with a controlled constant $K$. Therefore, we can replace
$\Psi_{\epsilon}$ by $\Psi_{\epsilon}^{3}$ in Lemma 6.3 and thus (6.2) gives
that

$\|\widetilde{N}(\nabla u|\Psi_{\epsilon}^{9})\|_{2}^{2}\leq
C_{3}\|S(\overline{\nabla}u|\Psi_{\epsilon}^{3})\|_{2}^{2}+C_{3}\kappa\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{3})\|_{2}^{2}.$

Observe also that $\partial_{r}\Psi_{\epsilon}^{3}$ is non-negative, supported
in $\\{\epsilon/2\leq|s|\leq s\\}$, and is bounded by $C\epsilon^{-1}$, so

$0\leq\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi_{\epsilon}^{3}\frac{dtdx}{|t|^{n-d-1}}\lesssim\int_{\mathbb{R}^{d}}\fint_{\epsilon/2\leq|s|\leq\epsilon}|\nabla_{x}u|^{2}\,ds\,dy.$

The two last bounds combined with (6.1) entail that

(6.5) $\|\widetilde{N}(\nabla u|\Psi_{\epsilon}^{9})\|_{2}^{2}\leq
C_{4}\bigg{(}\int_{\mathbb{R}^{d}}\fint_{\epsilon/2\leq|s|\leq\epsilon}|\nabla_{x}u|^{2}\,ds\,dy+\mathcal{B}_{+}(|\nabla
u|^{2}_{\kappa},\Psi_{\epsilon}^{3})+\kappa\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{3})\|^{2}_{2}\bigg{)},$

where $\mathcal{B}_{+}(v,\Phi)$ is the positive part of $\mathcal{B}(v,\Phi)$
and $C_{4}$ depends only $\lambda$, $n$. The proof consists to say that if
$\kappa$ is small enough, there exists $\epsilon>0$ as close as $0$ as we want
such that

(6.6) $\mathcal{B}_{+}(|\nabla
u|^{2}_{\kappa},\Psi_{\epsilon}^{3})=0\quad\text{ and }\quad
2C_{4}\kappa\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{3})\|^{2}_{2}\leq\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}^{2}.$

For such values of $\epsilon$, the bound (6.5) easily self-improves to (6.4),
which is exactly our objective.

The rough strategy of the proof consists in studying the quantity

(6.7) $\displaystyle\omega(\epsilon):=\iint_{\Omega}|\nabla
u|^{2}_{\kappa}\partial_{r}\Psi^{3}_{\epsilon}\frac{dtdx}{|s|^{n-d-2}},$

which is non-negative since $\Psi_{\epsilon}$ is increasing in $r$. Since
$\partial_{r}\Psi_{\epsilon}$ is supported in the strip
$\\{\epsilon/2\leq|s|\leq\epsilon\\}$ and since
$\partial_{r}\Psi_{\epsilon}\lesssim|s|^{-1}$, we deduce that

$\omega(\epsilon)\lesssim\int_{\mathbb{R}^{d}}\int_{\epsilon/2\leq|s|\leq\epsilon}|\nabla
u|^{2}_{\kappa}\,\frac{ds\,dy}{|s|^{n-d-1}}\lesssim\int_{\mathbb{R}^{d}}\int_{\epsilon/2\leq|s|\leq\epsilon}|\nabla
u|^{2}\,\frac{ds\,dy}{|s|^{n-d-1}}$

with constants that depends only on $\lambda$, $d$ and $n$. But the right-hand
side above converges to 0 as $\epsilon$ goes to 0 (because it is the tail of
$\|u\|_{W}^{2}<+\infty$). So we necessarily have that

(6.8) $\displaystyle\lim_{\epsilon\to 0}\omega(\epsilon)=0.$

We shall prove that if (6.6) fails for every $\epsilon$ in a small
neighborhood $(0,\epsilon_{0}]$ of zero, then (6.8) does not hold. So by
contraposition, (6.8) implies the existence of $\epsilon$ arbitrary close to
zero such that (6.6) holds.

Step 1: For this step, we look at the implications of the fact that

(6.9) $\mathcal{B}_{+}(|\nabla u|^{2}_{\kappa},\Psi_{\epsilon}^{3})>0.$

We write $I_{\mathcal{B}}$ for the values $\epsilon\in(0,\infty)$ for which
(6.9) holds. Due to the fact that $\nabla u\in L^{2}_{loc}(\Omega)$ and
$\partial_{r}(\Psi_{\epsilon}^{3})$ is smooth (and compactly supported in
$\Omega$), we have that the domain $I_{\mathcal{B}}$ is open.

Recall that $\Psi_{\epsilon}:=\phi(\frac{\epsilon}{|t|})$, where
$\phi:\mathbb{R^{+}}\rightarrow[0,1]$ such that $\phi\equiv 1$ on $[0,1]$ and
$\phi\equiv 0$ on $[2,+\infty)$. We compute

$\partial_{r}\Psi_{\epsilon}=-\frac{\epsilon}{|t|^{2}}\phi^{\prime},\qquad\partial_{r}\partial_{r}\Psi_{\epsilon}=\frac{2\epsilon}{|t|^{3}}\phi^{\prime}+\frac{\epsilon^{2}}{|t|^{4}}\phi^{\prime\prime},$

and we notice that

$\frac{\partial}{\partial_{\epsilon}}(\partial_{r}\Psi_{\epsilon})=-\frac{1}{|t|^{2}}\phi^{\prime}-\frac{\epsilon}{|t|^{3}}\phi^{\prime\prime}=-\frac{|t|}{\epsilon}\partial_{r}^{2}\Psi_{\epsilon}-\frac{1}{\epsilon}\partial_{r}\Psi_{\epsilon}=-\frac{1}{\epsilon}\partial_{r}(|t|\partial_{r}\Psi_{\epsilon}).$

The same argument shows that
$\frac{\partial}{\partial_{\epsilon}}(\partial_{r}\Psi_{\epsilon}^{3})=-\epsilon^{-1}\partial_{r}(|t|\partial_{r}\Psi_{\epsilon}^{3})$.
So we deduce

$\begin{split}\omega^{\prime}(\epsilon)&=-\frac{1}{\epsilon}\iint_{\Omega}|\nabla
u|_{\kappa}^{2}\partial_{r}\Big{(}|t|\partial_{r}\Psi^{3}_{\epsilon}\Big{)}\frac{dt\,dy}{|t|^{n-d-2}}=\frac{1}{\epsilon}\iint_{\Omega}\partial_{r}\Big{(}|t|\nabla
u|_{\kappa}^{2}\Big{)}\partial_{r}\Psi^{3}_{\epsilon}\frac{dt\,dy}{|t|^{n-d-2}}\\\
&=-\frac{2}{\epsilon}\mathcal{B}(\nabla
u|_{\kappa}^{2},\Psi_{\epsilon}^{3}).\end{split}$

By the integration by parts in $r$ (see Proposition 2.2), the function
$\epsilon\mapsto\omega(\epsilon)$ is decreasing on $I_{\mathcal{B}}$, that is

(6.10) $\omega(a)>\omega(b)\text{ whenever }(a,b)\subset I_{\mathcal{B}}.$

Step 2: Now, we look at the implications of the fact that
$\|\widetilde{N}(\nabla u|\Psi_{\epsilon}^{3})\|_{2}\gg\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}$.

We write $\alpha$ for the universal constant $5/4$. The exact value of
$\alpha$ does not matter, as long as we have
$\alpha^{-1}<1<\alpha<\alpha^{3}<2$. We have a bit of freedom on the function
$\phi$ that is used to construct $\Psi_{\epsilon}$ (see Definition 4.3). It is
always possible to choose $\phi$ such that $\phi(x)=2-x$ when

$1<\alpha\leq x\leq\max\\{2\alpha^{-1},\alpha^{3}\\}<2.$

For the same values of $x$, we have $\phi^{\prime}=1$. If
$\epsilon^{\prime}\geq\alpha\epsilon$, we have $\Psi_{\epsilon}\geq
2-2\alpha^{-1})>0$ on $\operatorname{supp}\Psi_{\epsilon^{\prime}}$, hence
$\epsilon^{\prime}\partial_{r}\Psi_{\epsilon^{\prime}}^{3}\lesssim\Psi_{\epsilon}^{9}$
and

$(\epsilon^{\prime})^{-1}\omega(\epsilon^{\prime})\lesssim\epsilon^{\prime}\iint_{\Omega}|\nabla
u|^{2}\partial_{r}\Psi_{\epsilon^{\prime}}^{3}\,\frac{ds\,dy}{|s|^{n-d}}\lesssim\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}^{2}.$

So there exists $C^{\prime}>0$ depending only on $d$, $n$ (and $\alpha$) such
that

(6.11)
$\sup_{\alpha\epsilon\leq\epsilon^{\prime}\leq\alpha^{2}\epsilon}\omega(\epsilon^{\prime})\leq
C^{\prime}\epsilon\|\widetilde{N}(\nabla u|\Psi_{\epsilon}^{9})\|_{2}^{2}.$

By the triangle inequality, we have

$\|\widetilde{N}(\nabla u|\Psi_{\epsilon}^{3})\|_{2}-\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}\leq\|\widetilde{N}(\nabla
u|(1-\Psi_{\epsilon}^{6})\Psi^{3}_{\epsilon})\|_{2}$

But since $(1-\Psi_{\epsilon}^{6})\Psi^{3}_{\epsilon}$ is supported in the
strip $\\{\epsilon/2\leq|s|\leq\epsilon\\}$, it is fairly easy to see that

$\|\widetilde{N}(\nabla
u|(1-\Psi_{\epsilon}^{6})\Psi^{3}_{\epsilon})\|_{2}^{2}\lesssim\int_{\mathbb{R}^{d}}\fint_{\epsilon/2\leq|s|\leq\epsilon}|\nabla
u|^{2}\,ds\,dy$

Take now $\epsilon^{\prime}\in[\alpha^{-2}\epsilon,\alpha^{-1}\epsilon]$. With
our choice of $\phi$ and $\alpha$, we have that
$\Psi_{\epsilon}+\epsilon^{\prime}\partial_{r}\Psi_{\epsilon^{\prime}}$ is
bounded from below by a uniform constant (i.e. depend only on $\alpha$) on
$[\epsilon/2,\epsilon]$. It implies that

(6.12) $\int_{\mathbb{R}^{d}}\fint_{\epsilon/2\leq|s|\leq\epsilon}|\nabla
u|^{2}\,ds\,dy\lesssim\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}^{2}+(\epsilon^{\prime})^{-1}\iint_{\Omega}|\nabla
u|^{2}\partial_{r}\Psi^{3}_{\epsilon^{\prime}}\frac{ds\,dy}{|s|^{n-d-2}}\\\
\lesssim\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}^{2}+\kappa^{-1/2}(\epsilon^{\prime})^{-1}\omega(\epsilon^{\prime}),$

where, in the last line, the coefficient $\kappa^{-1/2}$ appears because
$\omega$ is defined using the $|\nabla u|_{\kappa}^{2}$. By combining the last
three computations, we obtain the existence of a constant $C^{\prime\prime}>0$
depending on $n$ and $\lambda$ (and $\alpha$) such that

(6.13) $\epsilon\|\widetilde{N}(\nabla u|\Psi_{\epsilon}^{3})\|_{2}^{2}\leq
C^{\prime\prime}\epsilon\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}^{2}+C^{\prime\prime}\kappa^{-1/2}\inf_{\epsilon/\alpha^{2}\leq\epsilon^{\prime}\leq\epsilon/\alpha}\omega(\epsilon^{\prime}).$

We say that $\epsilon\in I_{N}$ if

(6.14) $\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{3})\|_{2}^{2}>C^{\prime\prime}(1+\kappa^{-1/2}C^{\prime})\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}^{2}.$

If $\epsilon\in I_{N}$, then we have by (6.11), and then (6.13)–(6.14) that

(6.15)
$\sup_{\alpha\epsilon\leq\epsilon^{\prime}\leq\alpha^{2}\epsilon}\omega(\epsilon^{\prime})\leq
C^{\prime}\epsilon\|\widetilde{N}(\nabla
u|\Psi_{\epsilon}^{9})\|_{2}^{2}<\inf_{\epsilon/\alpha^{2}\leq\epsilon^{\prime}\leq\epsilon/\alpha}\omega(\epsilon^{\prime}).$

Step 3: We want to prove that the point $0$ is in the closure of

(6.16) $(0,+\infty)\setminus(I_{\mathcal{B}}\cup I_{N}).$

Indeed, we take $\kappa$ (that depends on $\lambda$, $d$, and $n$) such that
$2C_{M}\kappa C^{\prime\prime}(1+\kappa^{-1/2}C^{\prime})\leq 1$, and then we
have (6.6) for any value of
$\epsilon\in(0,+\infty)\setminus(I_{\mathcal{B}}\cup I_{N})$. If the claim
(6.16) is true, then we can find values of
$\epsilon\in(0,+\infty)\setminus(I_{\mathcal{B}}\cup I_{N})$ arbitrary close
to 0, and for those values, (6.5) self-improves to the desired bound (6.4).

The argument is a bit technical because we have to combine (6.10) - which
shows that $\omega(\epsilon)$ ‘continuously increases’ as $\epsilon\to 0$ \-
with (6.15) - which implies that $\omega(\epsilon)$ increases by ‘jumps’ when
$\epsilon$ get closer to 0. The quantity $\omega(\epsilon)$ may not be
increasing as $\epsilon\to 0$, but a subsequence will be increasing, and that
is enough for us.

We decided the write the (simpler) arguments that show that zero is in the
closure of both $(0,+\infty)\setminus I_{\mathcal{B}}$ and
$(0,+\infty)\setminus I_{N}$. These simpler arguments are not necessary for
the proof, but we hope it will help the reader understand later what we are
doing when we look at $(0,+\infty)\setminus(I_{\mathcal{B}}\cup I_{N})$.

Step 3(a): We claim that $0$ is in the closure of $(0,+\infty)\setminus
I_{\mathcal{B}}$. Indeed, if it is not the case, then there exists
$\epsilon_{0}$ such that $(0,\epsilon_{0}]\in I_{\mathcal{B}}$. The bound
(6.10) implies then that $\omega(\epsilon)\geq\omega(\epsilon_{0}/2)>0$ for
all $\epsilon\in(0,\epsilon_{0}/2)$, which contradicts (6.8).

Step 3(b): We claim that $0$ is in the closure of $(0,+\infty)\setminus
I_{N}$. Indeed, if it is not the case, then there exists $\epsilon_{0}$ such
that $(0,\epsilon_{0}]\in I_{N}$. The bound (6.15) implies then that
$\omega(\epsilon/\alpha^{2k+1})\geq\omega(\epsilon_{0}/\alpha)>0$ for all
$k\in\mathbb{N}$, which contradicts (6.8).

Step 4: The argument is similar to the one done in the proof of Lemma 7.8 from
[FMZ21], but we try to give a clearer presentation.

We assume that $(0,\epsilon_{0}]\subset I_{\mathcal{B}}\cup I_{N}$, and we
want to prove that (6.8) does not hold. Because of Step 3(a), we can also
assume that zero is in the closure of $(0,\infty)\setminus I_{\mathcal{B}}$,
meaning that for any $0<\epsilon<\epsilon_{0}$, we have

(6.17) $\inf\\{\epsilon^{\prime}>0,\,(\epsilon^{\prime},\epsilon)\subset
I_{\mathcal{B}}\\}>0.$

We write

$\delta(\epsilon):=\inf_{\epsilon\leq\epsilon^{\prime}\leq\alpha\epsilon}\omega(\epsilon^{\prime})$

and we want to construct $\epsilon_{k}$ such that that
$\epsilon_{k+1}\leq\epsilon_{k}/\alpha^{2}$ and
$\delta(\epsilon_{k+1})>\delta(\epsilon_{k})$.

Induction step. We have $(0,\epsilon_{k}]\subset I_{\mathcal{B}}\cup I_{N}$.

  1. (1)

If we have $(\epsilon_{k}/\alpha^{2},\epsilon_{k})\subset I_{\mathcal{B}}$,
then by (6.17), we have

$0<\epsilon_{k+1}:=\inf\\{\epsilon,\,(\epsilon,\epsilon_{k})\subset
I_{B}\\}\leq\epsilon_{k}/\alpha^{2}.$

Now, thanks to (6.10), we also get

$\delta(\epsilon_{k+1}):=\inf_{\epsilon_{k+1}\leq\epsilon^{\prime}\leq\alpha\epsilon_{k+1}}\omega(\epsilon^{\prime})=\omega(\alpha\epsilon_{k+1})>\omega(\epsilon_{k})\geq\delta(\epsilon_{k}).$

  2. (2)

If (1) is false, then take again
$\tilde{\epsilon}:=\inf\\{\epsilon,\,(\epsilon,\epsilon_{0})\subset
I_{B}\\}>\epsilon_{0}/\alpha^{2}$. Since $I_{\mathcal{B}}$ is open, we
necessarily have $\tilde{\epsilon}\notin I_{\mathcal{B}}$, which forces
$\tilde{\epsilon}\in I_{N}$. Since
$\epsilon_{k}/\alpha^{2}<\tilde{\epsilon}\leq\epsilon_{k}$, the intersection
$[\alpha\tilde{\epsilon}_{,}\alpha^{2}\tilde{\epsilon}]\cap[\epsilon_{k},\alpha\epsilon_{k}]$
is not empty and contains $\tilde{\tilde{\epsilon}}$. So if we choose
$\epsilon_{k+1}=\tilde{\epsilon}/\alpha^{2}$ we have by (6.15) that

$\delta(\epsilon_{k+1})>\sup_{\alpha\tilde{\epsilon}\leq\epsilon^{\prime}\leq\alpha^{2}\tilde{\epsilon}}\omega(\epsilon^{\prime})\geq\omega(\tilde{\tilde{\epsilon}})\geq\delta(\epsilon_{k}).$

By construction, the value of $\omega(\epsilon_{k})$ will be bigger than
$\delta(\epsilon_{1})>0$ for any $k\geq 1$, which means that the convergence
(6.8) does not hold. To summarize, we established that if zero is not in the
closure of $(0,+\infty)\setminus(I_{\mathcal{B}}\cup I_{N})$, which means that
there exists $\epsilon_{0}>0$ such that $(0,\epsilon_{0}]\subset
I_{\mathcal{B}}\cup I_{N}$, then (6.8) fails. By contraposition, the
convergence (6.8) - which holds because $u\in W$ \- implies that zero is in
the closure of $(0,+\infty)\setminus(I_{\mathcal{B}}\cup I_{N})$. The theorem
follows. ∎

##  7\. Approximation by operators with Lipschitz coefficients.

With Theorem 6.4, we get closer to Theorem 1.5, which is our objective. We
“just” need to prove that if $u_{g}\in W$ is the solution given by Lemma 6.1
for $g\in H$ satisfying $\|\nabla g\|_{L^{2}(\mathbb{R}^{d})}<\infty$, we have

(7.1) $\limsup_{\epsilon\to
0}\int_{\mathbb{R}^{d}}\fint_{\epsilon/2\leq|s|\leq\epsilon}|\nabla_{x}u|^{2}\,ds\,dy\leq
C\int_{\mathbb{R}^{d}}|\nabla g|^{2}\,dy.$

However, the above convergence is not a simple fact. In some sense, it is a
weaker version of Theorem 1.5 that only consider the values of $u$ as close as
the boundary as we want. The strategy consists of smoothing the coefficients
of $\mathcal{A}$ in a small tube close to the boundary while satisfying
($\mathcal{H}$) with uniform constants. For those operators, the convergence
(7.1) hence Theorem 1.5 will hold with uniform constants. But since the
coefficients are modified only an a small enough set, the solutions to the
modified operators will converge to the solution of the initial operator, and
we eventually are able to prove Theorem 1.5.

First, we show that the weak solutions are in $W^{2,2}_{loc}$ (see Proposition
7.1), which means that taking second derivative is allowed, and so the square
functional $S$ and its local version make sense. We should have given this
argument long ago, but this result is already well known, and we decided to
write it just as an introduction for Proposition 7.3. In Proposition 7.3, we
establish a global bound for the tangential derivatives: we show that if the
boundary data and the coefficients of $\mathcal{A}$ are smooth enough, any
weak solution satisfies $\nabla_{x}u\in W$. As a corollary (Proposition 7.5)
we prove a technical lemma stating that the “approximation of the trace” of a
solution converges to the actual trace.

In the end of the section (Theorem 7.6), we establish that any elliptic
operator satisfying ($\mathcal{H}$) can be approximated by operators with
smooth enough coefficients (so that the global estimates given in Proposition
7.3 apply) that satisfy ($\mathcal{H}$) with constants controlled by the ones
of the approximated operator.

In the next section, we establish the convergence of solutions of
approximating operators and then we combine Theorem 6.4, Theorem 7.6, and
Proposition 7.5 to obtain Theorem 1.5.

######  Proposition 7.1.

Let $L:=-\operatorname{div}[A\nabla]$ be an elliptic operator defined on $D$
such that $A$ satisfies (1.1) and $\nabla A\in L^{\infty}_{loc}(D)$. Then any
weak solution $u\in W^{1,2}_{loc}(D)$ to $Lu=0$ in $D$ also lies in
$W^{2,2}_{loc}(D)$.

######  Remark 7.2.

Observe that the quantities $\nabla\partial_{x}$, $\nabla\partial_{\varphi}$
and $\nabla\partial_{r}$ are a linear combination of second order derivatives
and first order derivatives, so those derivatives are (locally in $L^{2}$)
well defined for any weak solution to $Lu=0$ for which $L$ satisfies
($\mathcal{H}$).

###### Proof.

The proof is classical, but we could not pinpoint a good reference, so since
the proof is quite simple and will be a good introduction to the global
analogue, we decided to write it.

Take $E\Subset D$ and then $\Psi\in C^{\infty}_{0}(D)$ such that $\Psi\equiv
1$ on $E$ and $0\leq\Psi\leq 1$. Pick a unit vector $e\in\mathbb{R}^{n}$.
Define when $0<h<\operatorname{dist}(\operatorname{supp}\Psi,D^{c})/2$ the
operator $\Delta_{e}^{h}$ as

$\Delta_{e}^{h}u:=\frac{u(X+he)-u(X)}{h}\in W^{1,2}_{loc}(D)$

We want to prove that

(7.2) $I_{h}=\iint_{D}|\nabla(\Delta^{h}_{e}u)|^{2}\Psi^{2}\,dX\leq C_{E,u},$

with a bound $C_{E,u}$ independent of $h$. Indeed, once the claim (7.2) is
established, by the weak compactness of the unit ball in $L^{2}$ we can
extract a sequence $h_{m}\in(0,1)$ such that $\nabla(\Delta_{e}^{h_{m}}u)$
converges weakly in $L^{2}(E)$ (and thus in the sense of distribution). But we
know that $\nabla(\Delta_{e}^{h_{m}}u)$ has to converge to
$\nabla\partial_{e}u$ in the sense of distribution - where of course
$\partial_{e}$ is the derivative in the direction $e$ \- so the weak limit of
the $\nabla(\Delta_{e}^{h_{m}}u)$ is $\nabla\partial_{e}u$, which is now in
$L^{2}(E)$ and satisfies $\int_{E}|\nabla\partial_{e}u|^{2}\,dX\leq C_{E,u}$.
Since we have the bound for every compact subset $E$ and any direction $e$, we
conclude that $W^{2,2}_{loc}(D)$ as desired.

So it remains to prove (7.2). Since $u$ is a weak solution to
$-\operatorname{div}A\nabla u=0$ in $D$, and that $X\to\Psi(X-he)$ still lies
in $C^{\infty}_{0}(D)$, we deduce

$\begin{split}0&=\iint_{D}\frac{A(X+he)\nabla u(X+he)-A(X)\nabla
u(X)}{h}\cdot\nabla[\Psi^{2}\Delta_{h}^{e}u]\,dX\\\
&=\iint_{D}(\Delta_{e}^{h}A)\nabla
u(X+he)\cdot\nabla[\Psi^{2}\Delta_{h}^{e}u]\,dX+\iint_{D}A\nabla\Delta_{e}^{h}u\cdot\nabla[\Psi^{2}\Delta_{h}^{e}u]\,dX\end{split}$

We use the product rule to write
$\nabla[\Psi^{2}\Delta_{h}^{e}u]=2\nabla\Psi\,(\Psi\Delta_{h}^{e}u)+\nabla\Delta_{h}^{e}u\,(\Psi^{2})$
we obtain

$I_{h}=-2\iint_{D}A\nabla\Delta_{e}^{h}u\cdot\nabla\Psi\,(\Psi\Delta_{h}^{e}u)\,dX-2\iint_{D}(\Delta_{e}^{h}A)\nabla
u(X+he)\cdot\nabla\Psi\,(\Psi\Delta_{h}^{e}u)\,dX\\\
-\iint_{D}(\Delta_{e}^{h}A)\nabla
u(X+he)\cdot\nabla\Delta_{h}^{e}u\,(\Psi^{2})\,dX$

We use the identity $ab\leq\epsilon a^{2}+b^{2}/4\epsilon$ on each of the
three terms of the right-hand side above, and we group the similar terms.
Afterwards, we get that

$I_{h}\leq\frac{1}{2}I_{h}+C\left((1+\|A\|_{\infty}^{2})\iint_{D}|\Delta_{h}^{e}u|^{2}|\nabla\Psi|^{2}\,dX+\|\nabla
A\|_{L^{\infty}(F)}^{2}\iint_{D}|\nabla u(X+he)|^{2}\Psi^{2}\,dX\right)$

where $F\Subset D$ is the set of points at distance at most
$\operatorname{dist}(\operatorname{supp}\Psi,D^{c})/2$ from
$\operatorname{supp}\Psi$. Since $I_{h}$ is finite, we hide the term
$\frac{1}{2}I_{h}$ in the right-hand side, as for the two other terms, we
observe that they are easily bounded - up to a constant that depends only on
$A$ and $\Psi$ \- by $\iint_{F}|\nabla u|^{2}\,dX$. The claim (7.2) and the
proposition follows. ∎

The global analogue of the previous proposition is the following result.

######  Proposition 7.3.

Let $L:=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ be an elliptic
operator that satisfies the uniform ellipticity condition (1.1). Suppose that
$u\in W$ is a weak solution of $Lu=0$ and $\operatorname{Tr}(u)=g\in
C^{\infty}_{0}(\partial\Omega)$. If
$\|\nabla\mathcal{A}_{x}\|_{L^{\infty}(\Omega)}<\infty$, then $\nabla_{x}u\in
W$. More precisely,

(7.3)
$\iint_{\Omega}|\nabla\nabla_{x}u|^{2}\frac{dt}{|t|^{n-d-1}}dx\lesssim\int_{\partial\Omega}|\nabla
g|^{2}dx+\int_{\partial\Omega}|\nabla\nabla g|^{2}dx+\iint_{\Omega}|\nabla
u|^{2}\frac{dtdx}{|t|^{n-d-1}},$

where the implicit constant depends on elliptic constant $\lambda$,the
dimensions $d$ and $n$, and $\|\nabla\mathcal{A}\|_{L^{\infty}(\Omega)}$.

######  Remark 7.4.

In the codimension 1 case, where we do not have angular derivatives, we can
deduce a bound on the full set of second derivatives by using the equation
(see Proposition 2.3, that allow us to write the second order radial
derivative $\partial_{r}^{2}$ of a solution as a linear combination of first
order derivatives and $\nabla\nabla_{x}$). However, we did not succeed to
bound globally the angular derivatives, so we did not succeed to show that
$\nabla u\in W$.

###### Proof.

Let $v(x,t):=u(x,t)-e^{-|t|}g(x)$. We first notice that $e^{-|t|}g\in W$ and
$\operatorname{Tr}(e^{-|t|}g)=g$, hence $v\in W_{0}$. Consider the difference
quotient in the tangential direction $e_{i}\in\mathbb{R}^{d}$ such that

$\displaystyle\Delta_{i}^{h}v(x,t)=\frac{v(x+he_{i},t)-v(x,t)}{h},\ \ \ h\neq
0.$

We can easily see that $\Delta_{i}^{h}v\in W_{0}$ for each fixed $h\neq 0$. In
particular, the quantity $J(v)$ defined as

$\displaystyle
J(v):=\iint_{\Omega}|\nabla\Delta^{h}_{i}v|^{2}\frac{dt}{|t|^{n-d-1}}dx.$

is finite. We turn to the bound of $J(v)$. By uniform ellipticity of
$\mathcal{A}$,

(7.4) $\displaystyle
J(v)\lesssim\iint_{\Omega}\mathcal{A}\nabla(\Delta^{h}_{i}v)\cdot\nabla(\Delta^{h}_{i}v)\frac{dt}{|t|^{n-d-1}}dx.$

Since $\Delta_{i}^{h}v\in W_{0}$ and $u$ is a weak solution to the equation
$Lu=0$, we have,

$0=\iint_{\Omega}\frac{\mathcal{A}(x+he_{i},t)\nabla
u(x+he_{i},t)-\mathcal{A}(x,t)\nabla
u(x,t)}{h}\cdot\nabla(\Delta^{h}_{i}v)\frac{dt}{|t|^{n-d-1}}dx\\\
=\iint_{\Omega}(\Delta_{i}^{h}\mathcal{A})\nabla
u(x+he_{i},t)\cdot\nabla(\Delta^{h}_{i}v)\frac{dt}{|t|^{n-d-1}}dx\\\
+\iint_{\Omega}\mathcal{A}(x,t)\Big{(}\Delta_{i}^{h}\nabla
u(x,t)\Big{)}\cdot\nabla(\Delta^{h}_{i}v)\frac{dt}{|t|^{n-d-1}}dx.$

Since two operators $\Delta_{i}^{h},\nabla$ commute and
$u(x)=v(x)+e^{-|t|}g(x)$, the identity above implies that

$J=-\iint_{\Omega}\mathcal{A}\nabla(e^{-|t|}\Delta_{i}^{h}g)\cdot\nabla(\Delta_{i}^{h}v)\frac{dt}{|t|^{n-d-1}}dx\\\
-\iint_{\Omega}(\Delta^{h}_{i}\mathcal{A})\nabla
u(x+he_{i},t)\cdot\nabla(\Delta^{h}_{i}v)\frac{dt}{|t|^{n-d-1}}dx=J_{1}+J_{2}.$

The term $J_{1}$ can be bounded by

$J_{1}\leq\frac{\|\mathcal{A}\|^{2}_{L^{\infty}(\Omega)}}{4\epsilon}\iint_{\Omega}|\nabla(e^{-|t|}\Delta_{i}^{h}g)|^{2}\frac{dt}{|t|^{n-d-1}}dx+\epsilon
J(v).$

and

$J_{2}\leq\epsilon
J(v)+\frac{\|\Delta^{h}_{i}\mathcal{A}\|^{2}_{L^{\infty}(\Omega)}}{4\epsilon}\iint_{\Omega}|\nabla
u(x+he_{i},t)|^{2}\frac{dt}{|t|^{n-d-1}}dx.$

The term $\epsilon J(v)$ can be hidden to the left-thand side of (7.4) by
choosing $\epsilon$ small enough. Moreover, the mean value inequality infers
that
$\|\Delta^{h}_{i}\mathcal{A}\|_{\infty}\leq\|\nabla_{x}\mathcal{A}\|_{\infty}$.
Consequently,

(7.5)
$J(v)\lesssim\iint_{\Omega}|\nabla(e^{-|t|}\Delta_{i}^{h}g)|^{2}\frac{dtdx}{|t|^{n-d-1}}+\iint_{\Omega}|\nabla
u(x+he_{i},t)|^{2}\frac{dtdx}{|t|^{n-d-1}}:=J_{3}+J_{4}.$

After a change of variable, $J_{4}$ is just
$\|u\|_{W}^{2}=\iint_{\Omega}|\nabla u|^{2}|t|^{d+1-n}dtdx$. The term $J_{3}$
is bounded brutally by using cylindrical coordinate as follows

(7.6) $\displaystyle
J_{3}\lesssim\left(\int_{0}^{\infty}e^{-2r}dr\right)\int_{\mathbb{R}^{d}}\big{(}|\Delta^{h}_{i}g|^{2}+|\nabla\Delta^{h}_{i}g|^{2}\big{)}dx\lesssim\int_{\mathbb{R}^{d}}(|\nabla
g|^{2}+|\nabla\nabla g|^{2})\,dx.$

We just prove (7.3) with the rate of change $\Delta^{h}_{i}$ instead of the
derivative $\partial_{x_{i}}$, and with a constant independent of $h$. So by
the same compactness argument as the one given in Proposition 7.1, (7.3)
follows. ∎

We study the sufficient conditions to define $\operatorname{Tr}(\nabla_{x}u)$
in the next proposition.

######  Proposition 7.5.

Let $L:=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ be an elliptic
operator that satisfies the uniform ellipticity condition (1.1). Suppose that
$u\in W$ is a weak solution of $Lu=0$ and $\operatorname{Tr}(u)=g\in
C^{\infty}_{0}(\partial\Omega)$. If
$\|\nabla_{x}\mathcal{A}\|_{L^{\infty}(\Omega)}<\infty$, then
$\operatorname{Tr}(\nabla_{x}u)=\nabla g$ almost everywhere, and in particular

(7.7) $\lim_{\epsilon\to
0}\int_{\mathbb{R}^{d}}\fint_{\epsilon/2<|t|\leq\epsilon}|\nabla_{x}u|^{2}\,dt\,dx=\int_{\mathbb{R}^{d}}|\nabla
g|^{2}\,dx.$

###### Proof.

The exact definition of trace is not always the same (but it is well known
that the different definitions are equivalent, as we shall show). We prove the
result in the case $d<n-1$, which has way less background, with the trace
introduced in [DFM21b]. We let the reader check that proof in the case $d=n-1$
is analogous with any reasonable notion of trace.

The trace of a function $u$ in $W$ is defined as in [DFM21b] by

$g(x)=\operatorname{Tr}(u)(x):=\lim_{\epsilon\to
0}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{B(x,\epsilon)}u\,dy\,dt.$

The definition is valid because, $|B(x,\epsilon)\cap\Omega^{c}|=0$ when
$d<n-1$, but that does not matter much, because in the case $d=n-1$, we can
simply extend $u\in W$ from $\mathbb{R}^{n}_{+}$ to $\mathbb{R}^{n}$ by
symmetry. If we set
$g_{\epsilon}(x):=\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{B(x,\epsilon)}u\,dy\,dt$
then (3.24) in [DFM21b] shows that

(7.8)
$\|g_{\epsilon}-g\|_{L^{2}(\mathbb{R}^{d}}\lesssim\epsilon^{\alpha}\|u\|_{W}\quad\text{
for any $\alpha\in(0,1/2)$}.$

We pick now a smooth nonnegative function $\theta\in
C^{\infty}_{0}(\mathbb{R}^{d+1})$ such that $\theta\equiv 0$ outside
$B(0,\frac{1}{4})$ and $\int_{\mathbb{R}^{n}}\theta=1$. We define

$\theta_{x,\epsilon}(y,t):=\epsilon^{-n}\theta\Big{(}\frac{y-x}{\epsilon},\frac{|s|-3\epsilon/4}{\epsilon}\Big{)}$

and then

$\operatorname{Tr}_{\epsilon}(u)(x):=\iint_{\mathbb{R}^{n}}\theta_{x,\epsilon}(y,t)u(y,t)\,dt\,dy.$

We show that $\operatorname{Tr}_{\epsilon}(u)$ is a good substitute of
$g_{\epsilon}$. Indeed, since $\fint\theta_{x,\epsilon}=1$ and
$\theta_{x,\epsilon}$ is supported in $B(x,\epsilon)$, we have

$|\operatorname{Tr}_{\epsilon}(u)(x)-g_{\epsilon}(x)|\leq\iint_{\mathbb{R}^{n}}\theta_{x,\epsilon}|u(y,t)-g_{\epsilon}(x)|\,dy\,dt\lesssim\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{B(x,\epsilon)}|u(y,t)-g_{\epsilon}(x)|\,dy\,dt\\\
\lesssim\left(\epsilon^{2}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{B(x,\epsilon)}|\nabla
u(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}\lesssim\left(\epsilon^{1-d}\iint_{B(x,\epsilon)}|\nabla
u(y,t)|^{2}\,dy\,\frac{dt}{|t|^{n-d-1}}\right)^{\frac{1}{2}}$

by using the $L^{2}$-Poincaré inequality and then (2.13) in [DFM21b]. So we
have by Fubini’s theorem

(7.9)
$\|\operatorname{Tr}_{\epsilon}(u)-g_{\epsilon}\|_{2}^{2}=\epsilon^{1-d}\int_{\mathbb{R}^{d}}\iint_{B(x,\epsilon)}|\nabla
u(y,t)|^{2}\,\frac{dt\,dy}{|t|^{n-d-1}}\,dx\\\
\lesssim\epsilon\iint_{\mathbb{R}^{n}}|\nabla
u(y,t)|^{2}\,\frac{dt\,dy}{|t|^{n-d-1}}=\epsilon\|u\|_{W}.$

Together with (7.8), we deduce

(7.10)
$\|\operatorname{Tr}_{\epsilon}(u)-\operatorname{Tr}(u)\|_{L^{2}(\mathbb{R}^{d}}\lesssim\epsilon^{\alpha}\|u\|_{W}\quad\text{
for any $\alpha\in(0,1/2)$}.$

The above inequality shows that $\operatorname{Tr}_{\epsilon}(u)$ converges to
$\operatorname{Tr}(u)$ in $L^{2}$, so also in the sense of distribution.
Therefore $\nabla_{x}\operatorname{Tr}_{\epsilon}(u)$ converges to
$\nabla\operatorname{Tr}(u)$ in the sense of distributions. Moreover, since
$\nabla_{x}u\in W$, we similarly have that
$\operatorname{Tr}_{\epsilon}(\nabla_{x}u)$ converges to
$\operatorname{Tr}(\nabla_{x}u)$ in the sense of distributions. But since we
easily have by definition of $\operatorname{Tr}_{\epsilon}(u)$ that

(7.11)
$\nabla_{x}\operatorname{Tr}_{\epsilon}(u)=-\iint_{\mathbb{R}^{n}}\nabla_{x}\Big{[}\epsilon^{-n}\theta\Big{(}\frac{y-x}{\epsilon},\frac{|s|-3\epsilon/4}{\epsilon}\Big{)}\Big{]}u(y,t)\,dt\,dy\\\
=-\iint_{\mathbb{R}^{n}}\nabla_{y}\theta_{x,\epsilon}(y,t)u(y,t)\,dt\,dy=\iint_{\mathbb{R}^{n}}\theta_{x,\epsilon}(y,t)\nabla_{y}u(y,t)\,dt\,dy=\operatorname{Tr}_{\epsilon}(\nabla_{x}u),$

then by uniqueness of the limit, we deduce that
$\operatorname{Tr}(\nabla_{x}u)=\nabla\operatorname{Tr}(u)=g$.

It remains to prove (7.7). Observe that

$\int_{\mathbb{R}^{d}}\fint_{\epsilon/2<|t|\leq\epsilon}|\nabla_{x}u|^{2}\,dt\,dx\leq\int_{\mathbb{R}^{d}}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(x,\epsilon)}|\nabla_{x}u|^{2}\,dt\,dy\,dx.$

The function
$h_{\epsilon}(x):=\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(x,\epsilon)}\nabla_{x}u\,dt\,dy$
is similar to $\operatorname{Tr}_{\epsilon}(\nabla_{x}u)$, so we can repeat
the argument used to obtain (7.10), and we have

$\lim_{\epsilon\to
0}\int_{\mathbb{R}^{d}}\left|\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(x,\epsilon)}\nabla_{x}u\,dt\,dy-\nabla
g\right|^{2}\,dx=\lim_{\epsilon\to
0}\int_{\mathbb{R}^{d}}\left|\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(x,\epsilon)}\nabla_{x}u\,dt\,dy-\operatorname{Tr}(\nabla_{x}u)\right|^{2}\,dx=0$

The combination of the two last computations easily implies (7.7), which ends
the proof of the proposition. ∎

######  Theorem 7.6.

Let $L=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ be an elliptic
operator that satisfies ($\mathcal{H}$)λ,κ. Then there exists a sequence of
$\\{\mathcal{A}^{j}\\}_{j\in\mathbb{N}}$ such that

  1. (a)

the convergence $\mathcal{A}^{j}\to\mathcal{A}$ holds uniformly on compact
sets of $\Omega$;

  2. (b)

$\|\nabla\mathcal{A}^{j}\|_{L^{\infty}(\Omega)}\leq Cj\kappa^{\frac{1}{2}}$;

  3. (c)

the operator $L_{j}:=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}^{j}\nabla)$
satisfies ($\mathcal{H}$)λ,Cκ.

where in both cases, $C>0$ is a constant that depends only on $d$ and $n$.

######  Remark 7.7.

We adapt the construction from Lemma 7.12 in [KP93] to the higher co-
dimensional boundaries, that is a construction that smoothens the coefficients
of $\mathcal{A}$ while preserving the form of the matrix, the (constant of
the) Carleson measure conditions on the coefficients, and the ellipticity
constant of $\mathcal{A}$. Note that the construction does not rely on the
specific structure (1.26).

###### Proof.

Suppose that $\psi\in C_{0}^{\infty}(\mathbb{R}_{+})$ with $0\leq\psi\leq 1$
such that $\psi\equiv 1/2$ on $[2,\infty]$ and $\psi\equiv 0$ on $[0,1]$. For
$j\in\mathbb{N}$, set $\psi_{j}(t):=\psi(j|t|)$. We construct the matrix
$\mathcal{A}^{j}$ as follows:

(7.12)
$\displaystyle\mathcal{A}^{j}(x,t)=\psi_{j}(t)\mathcal{A}(x,t)+(1-\psi_{j}(t))\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(x,\frac{1}{j})}\mathcal{A}(x^{\prime},t^{\prime})\,dx^{\prime}\,dt^{\prime}.$

From the construction above, we observe that
$\mathcal{A}^{j}\rightarrow\mathcal{A}$ uniformly on compact sets of $\Omega$,
which is a direct consequence of the uniform convergence $\psi_{j}\rightarrow
1$ on compact sets. The fact that the structure (1.26) is transferred to
$\mathcal{A}_{j}$ and the ellipticity bound (1.23) on $\mathcal{A}_{j}$ (with
the same constant as $\mathcal{A}$) is an immediate consequence of the fact
that each coefficient in $\mathcal{A}_{j}$ is an average of some value of the
same coefficient in $\mathcal{A}$.

It remains to estimate $\nabla\mathcal{A}^{j}$, we want to show that
$|t||\nabla_{x,\varphi}\mathcal{A}^{j}|\in CM(C\kappa)$,
$|t||\partial_{r}\mathcal{A}^{j}|\in CM(CM)$, and
$|\nabla\mathcal{A}^{j}|\lesssim j$. Since $\psi_{j}$ is $x$-independent, we
have

$\nabla_{x,\varphi}\mathcal{A}^{j}(x,t)=\psi_{j}(t)\nabla_{x,\varphi}\mathcal{A}(x,t)+(1-\psi_{j}(t))\nabla_{x}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(x,\frac{1}{j})}\mathcal{A}(x^{\prime},t^{\prime})dx^{\prime}dt^{\prime}\Big{)}\\\
=:\textup{I}_{1}+\textup{I}_{2}.$

According to (1.8) and the fact that $|t||\nabla_{x,\varphi}\mathcal{A}|\in
CM(\kappa)$, we have
$\|t\nabla_{x,\varphi}\mathcal{A}\|^{2}_{\infty}\lesssim\kappa$ and thus

(7.13) $\|{\mathds{1}}_{|t|\geq
1/2j}\nabla_{x,\varphi}\mathcal{A}\|^{2}_{L^{\infty}(\Omega)}\lesssim
j^{2}\kappa.$

But for $h$ small enough, we have

(7.14)
$\frac{1}{|h|}\Big{|}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W_{a}(x+h,\frac{1}{j})}\mathcal{A}(x^{\prime},t^{\prime})dx^{\prime}dt^{\prime}-\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W_{a}(x,\frac{1}{j})}\mathcal{A}(x^{\prime},t^{\prime})dx^{\prime}dt^{\prime}\Big{|}\\\
=\Big{|}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W_{a}(x,\frac{1}{j})}\frac{\mathcal{A}(x^{\prime}+h,t^{\prime})-\mathcal{A}(x^{\prime},t^{\prime})}{|h|}dx^{\prime}dt^{\prime}\Big{|}\lesssim\|{\mathds{1}}_{|t|\geq
j}\nabla_{x}\mathcal{A}\|_{L^{\infty}(\Omega)}\lesssim j\kappa^{1/2},$

where the righthand side above is independent of $h$. By taking the limit
$h\rightarrow 0$ in (7.14), we obtain that

$\nabla_{x}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W_{a}(x,\frac{1}{j})}\mathcal{A}(x^{\prime},t^{\prime})dx^{\prime}dt^{\prime}\Big{)}\lesssim
j\kappa^{1/2}$

and then

(7.15) $|\textup{I}_{2}|\lesssim(j\kappa^{1/2}){\mathds{1}}_{|t|\leq 2/j}.$

We deduce

(7.16)
$\|\nabla_{x,\varphi}\mathcal{A}^{j}\|_{\infty}\leq\|(\nabla_{x,\varphi}\mathcal{A}){\mathds{1}}_{|t|\geq
1/j}\|_{\infty}+\|\textup{I}_{2}\|_{\infty}\lesssim j\kappa^{1/2}$

by (7.13) and (7.15), and

(7.17)
$|t||\nabla_{x,\varphi}\mathcal{A}^{j}|\leq|t||\textup{I}|+|t||\textup{I}_{2}|\lesssim|t||\nabla_{x,\varphi}\mathcal{A}|+\frac{|t|}{j}\kappa^{\frac{1}{2}}{\mathds{1}}_{|t|\leq
2/j}\in CM(C\kappa)$

because $|t||\nabla_{x,\varphi}\mathcal{A}|\in CM(\kappa)$ and a simple
computation shows that $\frac{|t|}{j}\kappa^{\frac{1}{2}}{\mathds{1}}_{|t|\leq
2/j}\in CM(4\kappa c_{n-d})$, where $c_{n-d}$ is the surface of the unit
sphere in $\mathbb{R}^{n-d}$.

We have the two desired bounds (7.16) and (7.17) on
$\nabla_{x,\varphi}\mathcal{A}^{j}$, and it remains to prove the analogue
estimates on $\partial_{r}\mathcal{A}^{j}$. We have that

$|\partial_{r}\mathcal{A}^{j}|=\left|\psi_{j}\partial_{r}\mathcal{A}+\partial_{r}\psi_{j}\Big{(}\mathcal{A}-\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(x,\frac{1}{j})}\mathcal{A}(x^{\prime},t^{\prime})dx^{\prime}dt^{\prime}\Big{)}\right|\\\
\leq{\mathds{1}}_{|t|\geq 1/j}|\partial_{r}\mathcal{A}|+j{\mathds{1}}_{1\leq
j|t|\leq 2}\|{\mathds{1}}_{|t|\geq 1/2j}\nabla\mathcal{A}\|.$

And since, similarly to (7.13), we have $\|{\mathds{1}}_{|t|\geq
j}\nabla\mathcal{A}\|^{2}_{\infty}\lesssim j^{2}\kappa$. So we easily conclude
that $\|\partial_{r}\mathcal{A}^{j}\|_{\infty}\leq j\kappa^{\frac{1}{2}}$ and

$|t||\partial_{r}\mathcal{A}^{j}|\lesssim|t||\partial_{r}\mathcal{A}|+j|t|{\mathds{1}}_{1\leq
j|t|\leq 2}(M+\kappa)^{\frac{1}{2}}\in CM(C\kappa)$

because $j|t|{\mathds{1}}_{1\leq j|t|\leq 2}\in CM(C)$. The lemma follows. ∎

##  8\. Proof of Theorem 1.5: The Regularity Problem for a Reduced Class of
Operators.

In Theorem 8.1, we study the convergence of the solutions $u_{j}$ of the
approximating operators constructed in Theorem 7.6 to the solution $u$ of the
initial operator. Then we solve the Regularity problem for smooth boundary
data in Theorem 1.5, using the bound obtained in Theorem 6.4, the convergence
of trace provided by Proposition 7.5, and of course the convergence of
solutions established in Theorem 8.1. It is important to understand that we
have two convergences (one on the trace given by Proposition 7.5 and one on
the solutions given by Theorem 8.1) and the uniqueness of the double limit is
only guaranteed by the uniform convergence of the traces, which is given by
(6.3). That is, we can prove the identity

$\lim_{j\to
0}\int_{\mathbb{R}^{d}}|\operatorname{Tr}(\nabla_{x}u_{j})|^{2}\,dx=\lim_{\epsilon\to
0}\int_{\mathbb{R}^{d}}\fint_{\epsilon/2<|t|\leq\epsilon}|\nabla_{x}u|^{2}\,dt\,dx$

only when the assumptions of Theorem 6.4 are satisfied.

######  Theorem 8.1.

Let $L=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ be a uniformly
elliptic operator satisfying (1.1). Let $\mathcal{A}^{j}$ be a sequence of
matrices that converges pointwise to $\mathcal{A}$ and for which each
$\mathcal{A}^{j}$ satisfies (1.1) with the same constant as $\mathcal{A}$.

If $u$ and $u_{j}$ to be weak solutions in $W$ of respectively $Lu=0$ and
$L_{j}u_{j}=0$ with the same trace, i.e. $u_{j}-u\in W_{0}$, then
$\|u_{j}-u\|_{W}$ converges to 0.

###### Proof.

According to Lemma 6.1, we have

(8.1) $\displaystyle\|u\|_{W}+\|u_{j}\|_{W}\lesssim\|g\|_{H}$

where $g$ is the common trace
$g=\operatorname{Tr}(u)=\operatorname{Tr}(u_{j})$ and the constant is
independent of $j$. Since $u-u_{j}\in W_{0}$ and $Lu=L_{j}u_{j}=0$, we have

(8.2) $\displaystyle\iint_{\Omega}\mathcal{A}\nabla
u\cdot\nabla(u-u_{j})\frac{dtdx}{|t|^{n-d-1}}=\iint_{\Omega}\mathcal{A}^{j}\nabla
u_{j}\cdot\nabla(u-u_{j})\frac{dtdx}{|t|^{n-d-1}}=0.$

By the uniform ellipticity of matrix $\mathcal{A}^{j}$ and (8.2), we have

$\|u-u_{j}\|^{2}_{W}=\iint_{\Omega}|\nabla(u-u_{j})|^{2}\frac{dtdx}{|t|^{n-d-1}}\\\
\lesssim\iint_{\Omega}\mathcal{A}^{j}\nabla(u-u_{j})\cdot\nabla(u-u_{j})\frac{dtdx}{|t|^{n-d-1}}=\iint_{\Omega}\mathcal{A}^{j}\nabla
u\cdot\nabla(u-u_{j})\frac{dtdx}{|t|^{n-d-1}}\\\
=\iint_{\Omega}(\mathcal{A}^{j}-\mathcal{A})\nabla
u\cdot\nabla(u-u_{j})\frac{dtdx}{|t|^{n-d-1}}.$

Furthermore, we apply the Cauchy-Schwarz inequality to obtain that

(8.3)
$\|u-u_{j}\|^{2}_{W}\lesssim\Big{(}\iint_{\Omega}|\nabla(u-u_{j})|^{2}\frac{dtdx}{|t|^{n-d-1}}\Big{)}^{1/2}\Big{(}\iint_{\Omega}|\mathcal{A}^{j}-\mathcal{A}|^{2}|\nabla
u|^{2}\frac{dtdx}{|t|^{n-d-1}}\Big{)}^{1/2}\\\
=\|u-u_{j}\|_{W}\Big{(}\iint_{\Omega}|\mathcal{A}^{j}-\mathcal{A}|^{2}|\nabla
u|^{2}\frac{dtdx}{|t|^{n-d-1}}\Big{)}^{1/2},$

hence

$\|u-u_{j}\|_{W}^{2}\lesssim\iint_{\Omega}|\mathcal{A}^{j}-\mathcal{A}|^{2}|\nabla
u|^{2}\frac{dtdx}{|t|^{n-d-1}}$

Since $\mathcal{A}$ and $\mathcal{A}_{j}$ are bounded by a uniform constant,
the functions $|\mathcal{A}^{j}-\mathcal{A}|^{2}|\nabla u|^{2}|t|^{d+1-n}$ are
bounded (uniformly in $j$) by $(2\lambda)^{2}|\nabla u|^{2}|t|^{d+1-n}$ which
is integrable on $\Omega$. So by the Lebesgue’s dominated convergence theorem,

$\lim_{j\to\infty}\|u-u_{j}\|_{W}^{2}\lesssim\iint_{\Omega}\lim_{j\to
0}|\mathcal{A}^{j}-\mathcal{A}|^{2}|\nabla u|^{2}\frac{dtdx}{|t|^{n-d-1}}=0$

since $\mathcal{A}^{j}$ converges pointwise to $\mathcal{A}$. The theorem
follows. ∎

######  Corollary 8.2.

Under the hypotheses of Theorem 8.1, if $\|\widetilde{N}(\nabla
u_{j})\|_{L^{2}(\mathbb{R}^{d})}\lesssim\|\nabla g\|_{L^{2}(\mathbb{R}^{d})}$
for all $j\in\mathbb{N}$, where the implicit constant is independent of $j$,
then

$\displaystyle\|\widetilde{N}(\nabla
u)\|_{L^{2}(\mathbb{R}^{d})}\lesssim\|\nabla g\|_{L^{2}(\mathbb{R}^{d})}.$

###### Proof.

Let $\epsilon>0$. We first notice that:

(8.4) $\|\widetilde{N}(\nabla
u|\Psi_{\epsilon})\|_{L^{2}}\leq\|\widetilde{N}(\nabla
u_{j}|\Psi_{\epsilon})\|_{L^{2}}+\|\widetilde{N}(\nabla(u-u_{j})|\Psi_{\epsilon})\|_{L^{2}}\\\
\lesssim\|\nabla
g\|_{L^{2}}+\|\widetilde{N}(\nabla(u-u_{j})|\Psi_{\epsilon})\|_{L^{2}}$

by assumption. Yet, we have
$\|\widetilde{N}(\nabla(u-u_{j})|\Psi_{\epsilon})\|_{2}\leq
C\epsilon^{-1/2}\|u-u_{j}\|_{W}\rightarrow 0$ by Lemma 6.2 and then Theorem
8.1. So by taking the limit as $j$ goes to infinity, (8.4) becomes

$\|\widetilde{N}(\nabla u|\Psi_{\epsilon})\|_{L^{2}}\lesssim\|\nabla
g\|_{L^{2}}$

The corollary follows then from the monotone convergence theorem. ∎

We conclude our section with the proof of Theorem 1.5

Proof of Theorem 1.5. Pick $\lambda>0$. Let $C_{0}$ (that depends only on
$\lambda$, $d$ and $n$) be the constant in Theorem 7.6, and then let
$\kappa_{0}<1$ (that depends on $d$, $n$ and $\lambda$) be the “kappa” value
provided by Theorem 6.4 for $\lambda$. We pick then
$\kappa:=\kappa_{0}/C_{0}$.

According to Theorem 7.6, there exists a sequence of
$\\{\mathcal{A}^{j}\\}_{j\in\mathbb{N}}$ such that
$\mathcal{A}^{j}\rightarrow\mathcal{A}$ pointwise as $j\rightarrow\infty$.
Each $\mathcal{A}^{j}$ satisfies the following conditions,

  1. (a)

$\|\nabla\mathcal{A}^{j}\|_{\infty}\leq Cj\kappa^{1/2}$,

  2. (b)

the operator $L_{j}:=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}^{j}\nabla)$
satisfies ($\mathcal{H}$)${}_{\lambda,\kappa_{0}}$.

Let $u_{j}\in W$ be the solution to $L_{j}u_{j}$ with
$\operatorname{Tr}(u_{j})=\operatorname{Tr}(u)=g\in H$ provided by Lemma 6.1.
Our choice of $\kappa_{0}$ is small enough to have the inequality (6.3) for
each $u_{j}$. So Theorem 6.4 and then Proposition 7.5 infer that

$\|\widetilde{N}(\nabla u_{j})\|^{2}_{2}\lesssim\limsup_{\epsilon\to
0}\int_{\mathbb{R}^{d}}\fint_{\epsilon/2\leq|s|\leq\epsilon}|\nabla_{x}u_{j}|^{2}\,ds\,dy=\|\nabla
g\|_{2}^{2},$

with a constant that depends only on $d$, $n$ and $\lambda$ (in particular is
independent of $j$). The theorem follows now from Corollary 8.2. $\square$

##  9\. Proof of Theorem 1.1

First, the solvability of the Regularity problem is stable under Carleson
perturbations.

######  Theorem 9.1 (Theorem 2.1 in [KP93], Theorem 1.3 in [DFM21]).

Let

$L_{0}=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}_{0}\nabla]\quad\mbox{and}\quad
L_{1}=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}_{1}\nabla]$

be two uniformly elliptic operators that satisfy (1.1) with the same constant
$\lambda$. Assume that:

  1. (1)

the Regularity problem for the operator $L_{0}$ is solvable in $L^{p_{0}}$,
that is there exists a constant $C_{0}$ such that for any $g\in
C^{\infty}_{0}(\Omega)$, the solution $u_{0}$ to $L_{0}u_{0}=0$ constructed as
in (1.3) (or equivalently as in Lemma 6.1) verifies

(9.1) $\displaystyle\|\widetilde{N}(\nabla
u)\|_{L^{p_{0}}(\mathbb{R}^{d})}\leq C_{0}\|\nabla
g\|_{L^{p_{0}}(\mathbb{R}^{d})},$

  2. (2)

the disagreement $\mathcal{A}_{1}-\mathcal{A}_{0}$ satisfies the Carleson
measure condition with the constant $M$ \- i.e.
$|\mathcal{A}_{1}-\mathcal{A}_{0}|\in CM(M)$.

Then the Regularity problem for the operator $L_{1}$ is solvable in
$L^{p_{1}}$ for some $p_{1}>1$, more precisely there exists
$p_{1}\in(1,p_{0}]$ and $C_{1}$ both depending only on $\lambda$, $d$, $n$,
$p_{0}$, $C_{0}$, and $M$, such that for any $g\in C^{\infty}_{0}(\Omega)$,
the solution $u_{1}$ to $L_{1}u_{1}=0$ constructed in (1.3) verifies

(9.2) $\displaystyle\|\widetilde{N}(\nabla
u_{1})\|_{L^{p_{1}}(\mathbb{R}^{d})}\leq C_{1}\|\nabla
g\|_{L^{p_{1}}(\mathbb{R}^{d})}.$

Furthermore, if $M>0$ is small enough (depending on $\lambda$, $n$, $p_{0}$,
and $C_{0}$), then we can take $p_{1}=p_{0}$ in (9.2).

The second result shows that any operator as in Theorem 1.1 can be compared to
a Carleson perturbation of an operator satisfying ($\mathcal{H}$).

######  Proposition 9.2.

Let $L_{0}=-\operatorname{div}[|t|^{d+1-n}\mathcal{B}\nabla]$ be such that
$\mathcal{B}$ satisfies (1.1) and can be written as a block matrix in the form

$\mathcal{B}=\begin{pmatrix}B_{1}&B_{2}\frac{t}{|t|}\\\
\frac{t^{T}}{|t|}B_{3}&b_{4}I\end{pmatrix}$

where $b_{4}I$ is the product of the identity matrix of order $n-d$ with a
scalar function, $B_{2}$ is a $d$-dimensional vertical vertor 555Recall that
$t$ is a horizontal vector, so $B_{2}\frac{t}{|t|}$ is a valid matrix
product., $B_{3}$ is a $d$-dimensional horizontal vector 666Since $t^{T}$ is a
vertical vector, so $\frac{t^{T}}{|t|}B_{3}$ is a $(n-d)\times d$-matrix., and

$|t||\nabla B_{1}|+|t||\nabla B_{2}|+|t||\nabla B_{3}|+|t||\nabla b_{4}|\in
CM(\kappa).$

If $\kappa>0$ is small enough, there exists a bi-Lipschitz change of variable
$\rho$ from $\overline{\Omega}$ to $\overline{\Omega}$ such that

  1. (1)

$\rho(x)=x$ for any $x\in\mathbb{R}^{d}=\partial\Omega$;

  2. (2)

there exists $C_{\lambda}$ such that for any weak solution $u$ to $L_{0}u=0$,
the function $u\circ\rho$ is a weak solution to $L_{\rho}(u\circ\rho)=0$ where
the operator $L_{\rho}$ satisfies (1.1) with constant $C_{\lambda}$ and can be
written as

$L_{\rho}=-\operatorname{div}[|t|^{d+1-n}(\mathcal{B}_{\rho}+\mathcal{C}_{\rho})\nabla]$

where $\mathcal{C}_{\rho}\in CM(2\kappa)$ and
$L_{\rho,0}:=-\operatorname{div}[|t|^{d+1-n}\mathcal{B}_{\rho}\nabla]$
satisfies ($\mathcal{H}$)λ/2,2κ.

Moreover, if $\mathcal{B}$ is symmetric, then $\mathcal{B}_{\rho}$ and
$\mathcal{C}_{\rho}$ are symmetric.

######  Remark 9.3.

The assumption that $\kappa$ is small can actually be removed, but will make
the proof longer. The proposition is a variant of the method presented in
[Fen21], and we refer any reader that wants to remove the condition on the
smallness of $\kappa$ to the later article.

###### Proof.

As we just said, the proposition is a variant of the result given in [Fen21].
We will try to keep it light and refer to [Fen21] for the details that we
skipped.

Step 1: Change of variables to cancel the bottom left corner of $\mathcal{B}$.
We write $v$ for the $d\times n$ matrix function $B_{3}/b_{4}$. We define
$\rho_{v}$ as

$\rho_{v}:=\rho(x+|t|v(x,t),t).$

Observe that $\rho_{v}$ maps $\Omega$ to $\Omega$ and is the identity on
$\mathbb{R}^{d}$. Its Jacobian matrix is

$\operatorname{Jac}_{v}=\begin{pmatrix}I+|t|\nabla_{x}v&0\\\
\frac{t^{T}}{|t|}v+|t|\nabla_{t}v&I\end{pmatrix}=\begin{pmatrix}I&0\\\
\frac{t^{T}}{|t|}v&I\end{pmatrix}+\mathcal{O}(|t||\nabla B_{3,4}|)$

where $\mathcal{O}(h)$ denotes a quantity bounded by $Ch$, and $B_{3,4}$
denotes the the couple $(B_{3},b_{4})$. We have $|t||\nabla B_{3,4}|\in
CM(\kappa)$, which implies $|t||\nabla B_{3,4}|\leq C\kappa$ by (1.8), and
since $\kappa$ is small, we deduce that $\operatorname{Jac}_{v}$ is invertible
and $|\operatorname{Jac}_{v}|+|\operatorname{Jac}^{-1}_{v}|$ are bounded by
$1.1\lambda$. In addition

$\operatorname{Jac}_{v}^{-1}=\begin{pmatrix}I&0\\\
-\frac{t^{T}}{|t|}v&I\end{pmatrix}+\mathcal{O}(|t||\nabla B_{3,4}|)\ \text{
and }\ \det(\operatorname{Jac}_{v})=1+\mathcal{O}(|t||\nabla B_{3,4}|).$

We define the conjugate777By conjugate, we mean that $L_{v}(u\circ\rho_{v})=0$
whenever $Lu=0$. operator
$L_{v}=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}_{v}\nabla]$ where

(9.3)
$\mathcal{A}_{v}:=\left(\frac{\operatorname{dist}(\rho_{v}(x,t),\mathbb{R}^{d})}{|t|}\right)^{d+1-n}\det(\operatorname{Jac}_{v})\operatorname{Jac}^{-T}_{v}(\mathcal{B}\circ\rho_{v})\operatorname{Jac}^{-1}_{v}.$

We check that $\operatorname{dist}(\rho_{v}(x,t),\mathbb{R}^{d})=|t|$ and
hence

$\mathcal{A}_{v}=\begin{pmatrix}*&[B_{2}\circ\rho_{v}-(b_{4}\circ\rho_{v})v^{T}]\frac{t}{|t|}\\\
\frac{t^{T}}{|t|}[B_{3}\circ\rho_{v}-(b_{4}\circ\rho_{v})v]&(b_{4}\circ\rho_{v})I\end{pmatrix}+\mathcal{O}(|t||\nabla
B_{3/4}|)\\\ =\begin{pmatrix}*&[B_{2}-(B_{3})^{T}]\frac{t}{|t|}\\\
0&b_{4}I\end{pmatrix}+\mathcal{O}(|t||\nabla
B_{1,2,3,4}|+|B_{1,2,3,4}\circ\rho_{v}-B_{1,2,3,4}|)$

because $B_{3}-b_{4}v=0$ with our choice of $v$. We did not compute the upper
left corner in the matrix above to lighten the notation, but we can have
$B_{1}-B_{2}B_{3}/b_{4}$. We write $\mathcal{B}^{v}$ for the matrix in the
right-hand side above which has 0 in the bottom left corner, and
$\mathcal{C}^{v}$ for $\mathcal{A}^{v}-\mathcal{B}^{v}$. We have that

$|\mathcal{C}^{v}|\lesssim|t||\nabla
B_{1,2,3,4}|+|B_{1,2,3,4}\circ\rho_{v}-B_{1,2,3,4}|\in CM(C\kappa)$

because $|B_{1,2,3,4}\circ\rho_{v}-B_{1,2,3,4}|(x,t)$ is bounded by the
supremum of $|t||\nabla B_{1,2,3,4}|$ in a Whitney region around $(x,t)$, so
satisfies Carleson estimate as long as $|t||\nabla B_{1,2,3,4}|$ does. The
matrix $\mathcal{B}^{v}$ has the form

$\mathcal{B}^{v}=\begin{pmatrix}B_{1}^{v}&B_{2}^{v}\frac{t}{|t|}\\\
\frac{t^{T}}{B_{3}^{v}}&b_{4}^{v}I\end{pmatrix}=\begin{pmatrix}B_{1}^{v}&B_{2}^{v}\frac{t}{|t|}\\\
0&b_{4}I\end{pmatrix}$

and the Carleson bound $|t||\nabla B^{v}_{1,2,3,4}|\in CM(C\kappa)$ is
consequence of the fact that the coefficients of $\mathcal{B}^{v}$ are
product, quotient, difference, and sums of coefficients of $\mathcal{B}$ (we
actually have $|t||\nabla B^{v}_{1,2,3,4}|\leq C|t||\nabla B_{1,2,3,4}|$).

Step 2: Change of variables to reduce the bottom right corner of
$\mathcal{B}_{v}$ to $I$. The strategy is very similar to what we did in Step
1. Set $h:=b_{4}$ and define $\rho_{h}$ as

$\rho_{h}:=\rho(x,th(x,t)).$

As before, observe that $\rho_{h}$ maps $\Omega$ to $\Omega$ and is the
identity on $\mathbb{R}^{d}$. Its Jacobian matrix is

$\operatorname{Jac}_{h}=\begin{pmatrix}I&t\nabla_{x}h\\\
0&hI+(\nabla_{t}h)t\end{pmatrix}=\begin{pmatrix}I&0\\\
0&hI\end{pmatrix}+\mathcal{O}(|t||\nabla b_{4}|).$

Since $\kappa$ is small (depending only on $d$, $n$ and $\lambda$), the matrix
$\operatorname{Jac}_{h}$ is invertible and
$|\operatorname{Jac}_{h}|+|\operatorname{Jac}^{-1}_{h}|$ are bounded by a
constant that depends only on $\lambda$. In this case, we have

$\operatorname{Jac}_{h}^{-1}=\begin{pmatrix}I&0\\\
0&h^{-1}\end{pmatrix}+\mathcal{O}(|t||\nabla b_{4}|)\ \text{ and }\
\det(\operatorname{Jac}_{v})=h^{n-d}+\mathcal{O}(|t||\nabla b_{4}|).$

Note that $\operatorname{dist}(\rho_{h}(x,t),\mathbb{R}^{d})=|t|h(x,t)$.
Therefore, the conjugate operator
$L_{h}=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}_{v}\nabla]$ of $L_{v}$ by
$\rho$ is such that

$\mathcal{A}_{h}:=\left(\frac{\operatorname{dist}(\rho_{h}(x,t),\mathbb{R}^{d})}{|t|}\right)^{d+1-n}\det(\operatorname{Jac}_{h})\operatorname{Jac}^{-T}_{h}(\mathcal{A}_{v}\circ\rho_{h})\operatorname{Jac}^{-1}_{h}\\\
=\begin{pmatrix}*&[(B_{2}-B_{3}^{T})\circ\rho_{h}]\frac{t}{|t|}\\\
0&h(b_{4}\circ\rho_{h})I\end{pmatrix}+\mathcal{O}(|t||\nabla
b_{4}|+|\mathcal{C}_{v}\circ\rho_{h}|)\\\
=\begin{pmatrix}*&[B_{2}-B_{3}^{T}]\frac{t}{|t|}\\\
0&I\end{pmatrix}+\mathcal{O}(|t||\nabla
b_{4}|+|\mathcal{C}_{v}\circ\rho_{h}|+|\mathcal{B}\circ\rho_{h}-\mathcal{B}|)$

with our choice for $h$. We denote the matrix in the right-hand side above as
$\mathcal{B}_{h}$, and $\mathcal{C}_{h}$ is $\mathcal{A}_{h}-\mathcal{B}_{h}$.
The matrix $\mathcal{B}_{h}$ has the desired form, and
$L_{\rho,0}:=-\operatorname{div}[|t|^{d+1-n}\mathcal{B}_{\rho}\nabla]$
satisfies ($\mathcal{H}$)${}_{C_{\lambda},C_{\lambda}\kappa}$. By definition

$|\mathcal{C}_{h}|\lesssim|t||\nabla
b_{4}|+|\mathcal{C}_{v}\circ\rho_{h}|+|\mathcal{B}\circ\rho_{h}-\mathcal{B}|$

and the right-hand side above easily satisfies the Carleson measure condition
with constant $C\kappa$ for the same reasons as in Step 1 (and the fact that
bi-Lipschitz changes of variable preserve the Carleson measure condition).

Conclusion. The change of variables is $\rho:=\rho_{v}\circ\rho_{h}$, which is
bi-Lipschitz because $\rho_{v}$ and $\rho_{h}$ are bi-Lipschitz. The conjugate
of $L$ by $\rho$ is $L_{h}$, and the ellipticity constant of $L_{\rho}$ is
controlled by the ellipticity constant of $L$ (because Jacobian matrices of
$\operatorname{Jac}_{v}$, $\operatorname{Jac}_{h}$, and their inverses are
bounded by constants that depends only on $\lambda$). The top left corner of
$\mathcal{B}_{h}$ does not really matter, but one can check that we (can) have

$\mathcal{B}_{h}=\begin{pmatrix}b_{4}B_{1}-B_{2}B_{3}&[B_{2}-B_{3}^{T}]\frac{t}{|t|}\\\
0&I\end{pmatrix}$

so $\mathcal{B}_{h}$ easily satisfies
($\mathcal{H}$)${}_{C_{\lambda},C_{\lambda}\kappa}$. At last, notice that all
our operations on the coefficients preserve the symmetry of matrix
coefficients, which means that $\mathcal{B}_{h}$ and $\mathcal{C}_{h}$ are
symmetric as long as $\mathcal{B}$ is symmetric. The proposition follows. ∎

We are now ready for the proof of our main theorem.

Proof of Theorem 1.1. We consider the elliptic operator
$L^{\prime}=-\operatorname{div}\mathcal{B}\nabla$, then we construct from it
the change of variable from Proposition 9.2. The conjugated operator of
$L^{\prime}$ by $\rho$ is in the form
$L^{\prime}_{\rho}:=-\operatorname{div}[|t|^{d+1-n}(\mathcal{B}_{\rho}+\mathcal{C}_{\rho})\nabla]$,
where
$L^{\prime}_{\rho,0}:=-\operatorname{div}[|t|^{d+1-n}\mathcal{B}_{\rho}\nabla]$
satisfies ($\mathcal{H}$)${}_{C_{\lambda},C_{\lambda}\kappa}$ and
$\mathcal{C}_{\rho}\in CM(C\kappa)$. Therefore, if $\kappa:=\kappa(\lambda,n)$
is small enough we can apply Theorem 1.5 to say that the Regularity problem
for the operator $L^{\prime}_{\rho,0}$ is solvable in $L^{2}$ .

The operator $L^{\prime}_{\rho}$ is a small Carleson perturbation of
$L^{\prime}_{\rho,0}$, so Theorem 9.1 gives that the Regularity problem for
the operator $L^{\prime}_{\rho}$ is solvable in the same space $L^{2}$. Since
$L^{\prime}$ and $L^{\prime}_{\rho}$ are the same operator up to a bi-
Lipschitz change of variable, the Regularity problem is also solvable for
$L^{\prime}$ in $L^{2}$. Now, $L$ is a small Carleson perturbation of
$L^{\prime}$, so we use Theorem 9.1 again to obtain that the Regularity
problem is still solvable for $L$ in $L^{2}$. $\square$

##  10\. A complement of a Lipschitz graph

The definition of cones, Whitney regions, non-tangential maximal function, and
other objects given in the introduction was adapted to the that fact that the
domain $\mathbb{R}^{n}\setminus\mathbb{R}^{d}$ is the product space
$\mathbb{R}^{d}\times(\mathbb{R}^{n-d}\setminus\\{0\\})$. But we did so only
for convenience, and equivalent definition can be given in general spaces.

If the domain is more general $\Omega$, we define the cones in $\Omega$ with
vertex in $x\in\partial\Omega$ as

(10.1)
$\Gamma(x):=\\{X\in\Omega,|X-x|<2\operatorname{dist}(X,\partial\Omega)\\}.$

We can change the ‘aperture’ of the cone by replacing the value 2 by any
$\alpha>0$. The Whitney box $W(X)$ is defined as

$W(X):=B(X,\operatorname{dist}(X,\partial\Omega)/2).$

The definition of the cones and Whitney boxes given here are just example, as
many variants exist. From there, we define the averaged non-tangential maximal
function as

$\widetilde{N}(u)(x):=\sup_{X\in\Gamma(x)}\left(\iint_{W(X)}|u(Y)|^{2}dY\right)^{\frac{1}{2}}.$

From now on, we need a doubling measure on $\partial\Omega$, that we call
$\sigma$. When $\partial\Omega$ is the graph of a Lipschitz function as in
Corollary 1.2 \- $\sigma$ will simply be the $d$-dimensional Hausdorff
measure. The Carleson measure condition, that is the substitute of (1.7) is

$f\in
CM(M)\Longleftrightarrow\sup_{x\in\partial\Omega,r>0}\iint_{B(x,r)\cap\Omega}\sup_{W(X)}|f|^{2}\frac{dX}{\operatorname{dist}(X,\partial\Omega)}\lesssim
M\sigma(B(x,r)\cap\partial\Omega).$

Moreover, we say that the Regularity problem is solvable in $L^{p}$ if, for
any $g\in C_{0}^{\infty}(\mathbb{R}^{d})$, we have

$\|\widetilde{N}(\nabla u)\|_{L^{p}(\partial\Omega,\sigma)}\leq
C\|\nabla_{\partial\Omega}g\|_{L^{p}(\partial\Omega,\sigma)}.$

The gradient $\nabla_{\partial\Omega}$ is a gradient on $\partial\Omega$,
which needs to be defined. In the simple case where $\partial\Omega$ is the
graph of a Lipschitz function, as in Corollary 1.2, $\nabla_{\partial\Omega}$
is simply the classical gradient (that can be defined almost everywhere).

Proof of Corollary 1.2.  We can construct a bi-Lipschitz change of variable
$\rho$ (with Lipschitz constants close to 1) such that, for any weak solution
$u$ to $L_{\varphi}u=0$ in $\Omega_{\varphi}$, the function $u\circ\rho$ is a
solution to $L_{\rho}(u\circ\rho)=0$ in
$\mathbb{R}^{n}\setminus\mathbb{R}^{d}$, where the operator $L_{\rho}$
satisfies the assumptions of Theorem 1.1. The construction of such change of
variable, and the properties of the conjugate operator $L_{\rho}$ are the main
purpose of the article [DFM19].

The fact that $\rho$ is bi-Lipschitz and Theorem 1.1 entail then that

$\|\widetilde{N}([\nabla u_{g}]\circ\rho)\|_{L^{2}(\mathbb{R}^{d})}\leq
2\|\widetilde{N}(\nabla[u_{g}\circ\rho])\|_{L^{2}(\mathbb{R}^{d})}\leq
C\|\nabla[g\circ\rho]\|_{L^{2}(\mathbb{R}^{d})}.$

The fact that
$\|\widetilde{N}(v\circ\rho)\|_{L^{2}(\mathbb{R}^{d})}\approx\|\widetilde{N}(v)\|_{L^{2}(\rho(\mathbb{R}^{d}),\sigma)}$
is a consequence of the fact that $\rho$ change the shape the regions
$W(z,r)$, but preserves the fact that they are Whitney regions. Similarly,
$\rho$ change the shape of the cones, but not the fact that they are the union
of Whitney regions for a same point at all scale - i.e. a weaker version of
“cones” variant to (10.1) - and we know from [Ste93, Chapter II, § 2.5.1] and
the various definitions of cones does not change the $L^{p}$-boundedness of
the non-tangential maximal functions $N$ and $\widetilde{N}$. The fact that
$\rho$ is bi-Lipschitz also infers the equivalence
$\|\nabla[g\circ\rho]\|_{L^{2}(\mathbb{R}^{d})}\approx\|\nabla
g\|_{L^{2}(\partial\Omega_{\varphi})}$. The corollary follows. $\square$

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