# Weak anisotropic Hardy inequality: essential self-adjointness of drift-
diffusion operators on domains in $\mathbb{R}^{d}$, revisited

Gheorghe Nenciu  Gheorghe Nenciu
Institute of Mathematics “Simion Stoilow” of the Romanian Academy
21, Calea Griviţei
010702-Bucharest, Sector 1
Romania<EMAIL_ADDRESS>and  Irina Nenciu  Irina Nenciu
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
851 S. Morgan Street
Chicago, IL and Institute of Mathematics “Simion Stoilow” of the Romanian
Academy
21, Calea Griviţei
010702-Bucharest, Sector 1
Romania<EMAIL_ADDRESS>
###### Abstract.

We consider the problem of essential self-adjointness of the drift-diffusion
operator $H=-\frac{1}{\rho}\nabla\cdot\rho\mathbb{D}\nabla+V$ on domains
$\Omega\subset{\mathbb{R}}^{d}$ with ${\mathcal{C}}^{2}$-boundary
$\partial\Omega$ and for large classes of coefficients $\rho,\;{\mathbb{D}}$
and $V$. We give criteria showing how the behavior as
$x\rightarrow\partial\Omega$ of these coefficients balances to ensure
essential self-adjointness of $H$. On the way we prove a weak anisotropic
Hardy inequality which is of independent interest.

###### Contents

  1. 1 Introduction
  2. 2 The setting
  3. 3 Weak anisotropic Hardy inequality
  4. 4 Essential self-adjointness of drift-diffusion operators
  5. 5  Comments, extensions, open problems

##  1\. Introduction

In this paper we revisit the problem of essential self-adjointness for a
drift-diffusion operator on domains in ${\mathbb{R}}^{d}$. More precisely, we
consider the operator

$H=-\frac{1}{\rho}\nabla\cdot\rho\mathbb{D}\nabla+V$ (1.1)

on a domain $\Omega\in{\mathbb{R}}^{d}$, where
$\rho:\Omega\rightarrow(0,\infty)$,
${\mathbb{D}}:\Omega\rightarrow{\mathbb{R}}^{d\times d}$ is a strictly
positive definite matrix valued function, and
$V:\Omega\rightarrow{\mathbb{R}}$.

The self-adjointness problem for second order elliptic partial differential
operators on domains in ${\mathbb{R}}^{d}$ (or more general on Riemannian
manifolds) has a long and ramified history. In the 1-dimensional case, the
problem is well understood due to the powerful tools of ordinary differential
equations theory, especially Weyl limit point/circle theory [26], [30]. The
multi-dimensional case has been also much studied and we send the reader to
[9], [10], [12], [17], [26] for references, heuristics, and a detailed
account. For more recent developments see [6], [7], [11], [20], [23], [24],
[25], [27], [28].

Assuming enough regularity of the coefficients, $H$ is symmetric in
$L^{2}\big{(}\Omega;\rho\,dx\big{)}$ on ${\mathcal{C}}^{\infty}_{0}(\Omega)$
and as it is well known (see e.g. [11]) the essential self-adjointness of $H$
is dictated only by their behavior near the boundary, $\partial\Omega$ of
$\Omega$. Hence the problem boils down to finding conditions on the behavior
of its coefficients as $x\rightarrow\partial\Omega$ ensuring essential self-
adjointness of $H$. The case which is well understood is when ${\mathbb{D}}$
is strongly degenerate, i.e. it decays sufficiently quickly as
$x\rightarrow\partial\Omega$ so that $\Omega$ endowed with the distance

$ds^{2}=\sum_{j,k=1}^{d}{\mathbb{D}}(x)^{-1}_{j,k}dx_{j}dx_{k}$ (1.2)

is a complete metric space (e.g. in the case when
$\|{\mathbb{D}}(x)\|\leqslant\operatorname{const}.\delta(x)^{2}$ where
$\delta(x)=\operatorname{dist}\big{(}x,\partial\Omega\big{)}$). One particular
result in this case, which follows from the general theory in [9], is that
$H_{0}$ is essentially self-adjoint irrespective of the geometry of $\Omega$
and the behavior of $\rho$. The situation when ${\mathbb{D}}$ is not strongly
degenerate is much more involved and after more than five decades of research
(see e.g. the line of research following the famous paper of Wienholtz [32],
[33], [34], the Cordes approach [12] and the Brusentev theory [10]) the
picture is far from being complete. More recently, for the particular case of
Schrödinger operator, $H=-\Delta+V$ on bounded, simply connected domains with
smooth boundary we refined the previously known results on the behavior of
$V(x)$ as $x\rightarrow\partial\Omega$ ensuring essential self-adjointness
[23]. The key point of the method in [23] is that essential self-adjointness
is ensured by lower bounds of
$\big{(}\varphi,H\varphi\big{)},\;\varphi\in{\mathcal{C}}^{\infty}_{0}$:

$\big{(}\varphi,\big{(}-\Delta+V\big{)}\varphi\big{)}\geqslant\int_{\Omega}\big{(}L(x)+V(x)\big{)}\big{|}\varphi(x)\big{|}^{2}dx$
(1.3)

via Agmon-type exponential estimates for weak solutions of partial
differential equations [1]. The heuristics behind is that if the ”barrier”
$L(x)+V(x)\rightarrow\infty$ sufficiently fast as $x\rightarrow\partial\Omega$
the quantum particle cannot reach $\partial\Omega$, hence no boundary
conditions are needed to specify the dynamics. It turns out that the method in
[23] works as well in more general settings and in particular, very recently,
criteria for essential self-adjointness of some classes of drift-diffusion
operators were obtained in [24] and (by merging the approaches in [24], [27])
in [28].

From the above discussion it is clear that the main ingredient which is needed
in this approach is a (as ”optimal” as possible) lower bound (aka Hardy
inequality) for the general drift-diffusion operator
$H_{0}=-\frac{1}{\rho}\nabla\cdot\rho\mathbb{D}\nabla$

$\big{(}\varphi,H_{0}\varphi\big{)}\geqslant\int_{\Omega}L(x)\big{|}\varphi(x)\big{|}^{2}\rho(x)dx.$
(1.4)

If $\rho=1,\;{\mathbb{D}}=\mathds{1}$ then (1.4) is provided by the well known
weak Hardy inequality (see e.g. [8], [13])

$L(x)=C+\frac{1}{4\delta(x)^{2}},\;C>-\infty,\;\delta(x)=\operatorname{dist}(x,\partial\Omega).$
(1.5)

More generally if ${\mathbb{D}}(x)$ is isotropic i.e.
${\mathbb{D}}(x)=d(x)\mathds{1}=a(x)\delta(x)^{\beta}\mathds{1}$ then $L(x)$
is provided by weighted Hardy inequalities (see [29] and references given
there). Weighted Hardy inequalities can be used also for anisotropic
${\mathbb{D}}(x)$ by noticing that

$\big{(}\varphi,H_{0}\varphi\big{)}\geqslant\int_{\Omega}\big{|}\nabla\varphi(x)\big{|}^{2}d(x)\rho(x)dx$
(1.6)

if $d(x)$ satisfies ${\mathbb{D}}(x)\geqslant d(x)\mathds{1}$ but (1.6) might
be very poor if ${\mathbb{D}}(x)$ is not in a ”neighborhood” of an isotropic
one.

The aim of this paper is to extend the results in [24], [28] to larger classes
of $\rho,\;{\mathbb{D}}$ and $V$ in order to allow the degree of degeneracy of
${\mathbb{D}}$ to vary with the point on $\partial\Omega$ and to dispense of
isotropy like conditions. On the way we prove a weak anisotropic Hardy
inequality which might be of independent interest.

The content of the paper is as follows. In Section 2 we spell the setting for
Sections 3 and 4. Section 3 is devoted to the proof of our weak anisotropic
Hardy inequality. In section 4 combining the result in Section 3 with the
method in [23], [24] we provide criteria for essential self-adjointness oh $H$
for classes of coefficients described in Section 2. While in order not to
obscure the main ideas in Sections 3 and 4 we restricted ourselves to the
setting in Section 2, the method of proof works as well (adding the necessary
technicalities) in more general settings, and we give in Section 5 some
refinements and extensions of the results in Sections 3 and 4. Comments about
relation with previous results, optimality, as well as an application to the
essential self-adjointness of Laplace-Beltrami operators on two dimensional
almost-Riemannian structures [5], [6] are also given.

##  2\. The setting

Through the paper, if not otherwise stated, $\Omega$ denotes a bounded simple
connected domain in ${{\mathbb{R}}}^{d}$ whose boundary $\partial\Omega$ is a
${\mathcal{C}}^{2}$ manifold of codimension 1. Let

$\delta(x)=\operatorname{dist}\big{(}x,\partial\Omega\big{)};\quad
x\in{\Omega}$ (2.1)

be the distance to the boundary and

$\Gamma_{\nu}=\big{\\{}x\in\Omega|{\delta}(x)<\nu\big{\\}};\quad\nu>0.$ (2.2)

The main properties of the function $\delta(x)$ we need are given by the
following:

######  Lemma 2.1.

[14, Ch. 14.6] There exist $\nu_{\Omega}>0$ and $C_{\Delta}<\infty$ such that

$\delta\in{\mathcal{C}}^{2}(\Gamma_{\nu_{\Omega}});\quad\big{|}\nabla\delta(x)\big{|}=1;\quad\big{|}\Delta\delta(x)\big{|}\leqslant
C_{\Delta}.$ (2.3)

Now let

$\rho\in{\mathcal{C}}^{1}\big{(}\Omega;(0,\infty)\big{)},$ (2.4)

${\mathbb{D}}(x)$ a matrix valued function:

${\mathbb{D}}\in{\mathcal{C}}^{1}\big{(}\Omega;{\mathbb{R}}^{d\times
d}\big{)};\quad{\mathbb{D}}(x)>0$ (2.5)

and

$V\in L^{\infty}_{loc}\big{(}\Omega;{\mathbb{R}}\big{)}.$ (2.6)

With these we consider the following symmetric operator in
$L^{2}\big{(}\Omega;\rho(x)dx\big{)}$:

$H=-\frac{1}{\rho}\nabla\cdot\rho\mathbb{D}\nabla+V;\quad{\mathcal{D}}(H)={\mathcal{C}}^{\infty}_{0}(\Omega).$
(2.7)

By $H_{0}$ we denote the above operator with $V=0$ i.e.

$H_{0}=-\frac{1}{\rho}\nabla\cdot\rho\mathbb{D}\nabla.$ (2.8)

The corresponding quadratic forms will be denoted by
$h\big{[}\cdot,\cdot\big{]}$ and $h_{0}\big{[}\cdot,\cdot\big{]}$
respectively.

Next we spell the assumptions on $\rho$ and ${\mathbb{D}}$ which will be used
through the paper:

Assumption A

Let

$s_{\beta}\leqslant s<1;\quad
0<\nu_{0}<\text{min}\big{\\{}\nu_{{\Omega}},e^{-1}\big{\\}}.$ (2.9)

$\mathbf{A_{\rho}}$ :

For $x\in\Gamma_{\nu_{0}}$,

$\rho(x)=r(x)\delta(x)^{\gamma(x)};\quad
r\in{\mathcal{C}}^{1}\big{(}\Gamma_{\nu_{0}};(0,\infty)\big{)};\quad\gamma\in{\mathcal{C}}^{1}\big{(}\Gamma_{\nu_{0}};{\mathbb{R}}\big{)},$
(2.10)

$\sup_{x\in\Gamma_{\nu_{0}}}\big{(}\big{|}\nabla\text{\rm
ln}r(x)\big{|}+\big{|}\nabla\gamma(x)\big{|}\big{)}\delta(x)^{s}<\infty.$
(2.11)

$\mathbf{A_{D}}$:

For $x\in\Gamma_{\nu_{0}}$ let ${\mathbb{P}}_{x}$ be the orthogonal projection
in ${\mathbb{R}}^{d}$ along $\nabla\delta(x)$. Write ${\mathbb{D}}(x)$ as a
$2\times 2$ block matrix according to the decomposition
${\mathbb{R}}^{d}={\mathbb{P}}_{x}{\mathbb{R}}^{d}\oplus(1-{\mathbb{P}}_{x}){\mathbb{R}}_{d}$:

$\displaystyle{\mathbb{D}}(x)$
$\displaystyle={\mathbb{P}}_{x}{\mathbb{D}}(x){\mathbb{P}}_{x}+{\mathbb{P}}_{x}{\mathbb{D}}(x)(1-{\mathbb{P}}_{x})+(1-{\mathbb{P}}_{x}){\mathbb{D}}(x){\mathbb{P}}_{x}+{\mathbb{P}}_{x}{\mathbb{D}}(x){\mathbb{P}}_{x}$
$\displaystyle=:\left(\begin{matrix}d(x)&d_{12}(x)\\\
d_{21}(x)&d_{22}(x)\end{matrix}\right).$ (2.12)

Assume that

$d(x)=a(x)\delta(x)^{\beta(x)};\quad
a\in{\mathcal{C}}^{1}\big{(}\Gamma_{\nu_{0}};(0,\infty)\big{)};\quad\beta\in{\mathcal{C}}^{1}\big{(}\Gamma_{\nu_{0}};{\mathbb{R}}\big{)},$
(2.13) $\sup_{x\in\Gamma_{\nu_{0}}}\big{|}\nabla\text{\rm
ln}a(x)\big{|}\delta(x)^{s}<\infty;\quad\sup_{x\in\Gamma_{\nu_{0}}}\big{|}\nabla\beta(x)\big{|}\delta(x)^{s_{\beta}}<\infty.$
(2.14)

For further use we notice a direct consequence of Assumption A. More precisely
since $\Omega$ is bounded, $\partial\Omega$ is smooth and $s_{\beta}\leqslant
s<1$ from (2.11) and (2.14) it follows that

$\beta,\;\gamma,\;\text{\rm ln}a,\;\text{\rm ln}r\in
L^{\infty}(\Gamma_{\nu_{0}}).$ (2.15)

In particular

$a,\;\frac{1}{a},\;r,\;\frac{1}{r}\in L^{\infty}(\Gamma_{\nu_{0}}).$ (2.16)

Since ${\mathbb{D}}(x)>0$, ${\mathbb{D}}(x)^{-1}$ exists,
${\mathbb{D}}(x)^{-1}>0$ hence (see (2)) $d(x)>0,\;d_{22}(x)>0$. By Schur
complement lemma [15] (we use its form used by physicists, see e.g. Lemma 2.3
in [16]) ${\mathbb{D}}(x)^{-1}$ written as a $2\times 2$ block matrix reads:

$\displaystyle{\mathbb{D}}^{-1}=$
$\displaystyle\left(\begin{matrix}\tilde{d}&-\tilde{d}d_{12}d_{22}^{-1}\\\
&\\\
-d_{22}^{-1}d_{21}\tilde{d}&d_{22}^{-1}+d_{22}^{-1}d_{21}\tilde{d}d_{12}d_{22}^{-1}\end{matrix}\right),$
(2.17)

with

$\tilde{d}=d^{-1}\big{(}1-q\big{)}^{-1}$ (2.18)

where

$q=d^{-1}d_{12}d_{22}^{-1}d_{21}\,.$ (2.19)

Since $d(x),\;\tilde{d}(x),\;d_{22}^{-1}(x)$ are all strictly positive it
follows that

$0\leqslant q(x)<1.$ (2.20)

Assumption Q

$\sup_{x\in\Gamma_{\nu_{0}}}\big{|}\nabla\text{\rm
ln}\big{(}1-q(x)\big{)}\big{|}\delta(x)^{s}<\infty.$ (2.21)

As in the case of $a,r$ from (2.21) one has

$\frac{1}{1-q}\in L^{\infty}\big{(}\Gamma_{\nu_{0}}\big{)}.$ (2.22)

##  3\. Weak anisotropic Hardy inequality

######  Theorem 3.1.

Suppose that Assumptions A and Q hold true. Then there exists
$0<\nu_{1}<\frac{\nu_{0}}{2}$ such that for all $0<\nu<\nu_{1}$ and
$\varphi\in{\mathcal{C}}^{\infty}_{0}(\Omega)$:

$\displaystyle
h_{0}\big{[}\varphi,\varphi\big{]}=\int_{\Omega}\overline{\nabla\varphi(x)}\cdot\big{(}{\mathbb{D}}(x)\nabla\varphi(x)\big{)}\rho(x)dx$
$\displaystyle\geqslant-
c_{1}(\nu)\|\chi_{\Omega\setminus\Gamma{\nu}}\varphi\|^{2}+\int_{\Gamma_{\nu}}{\mathcal{H}}_{0}(x)|\varphi(x)|^{2}\rho(x)dx$
(3.1)

where $c_{1}(\nu)<\infty$, $\chi_{\Omega\setminus\Gamma{\nu}}$ is the
characteristic function of $\Omega\setminus\Gamma{\nu}$ and

${\mathcal{H}}_{0}(x)=\frac{1}{4}\big{(}1-q(x)\big{)}a(x)\delta(x)^{\beta(x)-2}\Big{[}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\frac{1}{2}\Big{(}\text{\rm
ln}\tfrac{1}{\delta(x)}\Big{)}^{-2}\Big{]}.$ (3.2)

In particular if $\varphi\in{\mathcal{C}}^{\infty}_{0}(\Gamma_{\nu})$ then

$h_{0}[\varphi,\varphi]\geqslant\int_{\Gamma_{\nu}}{\mathcal{H}}_{0}(x)|\varphi(x)|^{2}\rho(x)dx.$
(3.3)

###### Proof.

As in the proof of Lemma 5.4 in [24] we use the so called ”vector field”
approach to Hardy inequalities see e.g. [10], [2], [21], [18] :

######  Lemma 3.2.

[10, Theorem 4.1] Let ${\boldsymbol{X}}\in
C^{1}\big{(}\Omega;\mathbb{R}^{d}\big{)}$ be a differentiable real vector
field on $\Omega$. Then

$h_{0}[\varphi,\varphi]\geqslant\int_{\Omega}\left(\nabla\cdot{\boldsymbol{X}}(x)-{\boldsymbol{X}}(x)\cdot\big{(}\rho(x)\mathbb{D}(x)\big{)}^{-1}{\boldsymbol{X}}(x)\right)\,\big{|}\varphi(x)\big{|}^{2}\,dx\,,$
(3.4)

for all $\varphi\in C_{0}^{\infty}(\Omega)$.

With this the proof consists in making an appropriate Ansatz,
${\boldsymbol{X}}_{0}(x)$, for ${\boldsymbol{X}}(x)$ and estimating the r.h.s.
of (3.4).

Let for $x\in\Gamma_{\nu_{0}}$ (recall that $\nu_{0}<\frac{1}{e}$ see (2.9)):

$\widetilde{\boldsymbol{X}}_{0}=\frac{1}{2}\big{(}1-q(x)\big{)}a(x)r(x)\delta(x)^{\beta(x)+\gamma(x)-1}\big{[}\beta(x)+\gamma(x)-1+f\big{(}\delta(x)\big{)}\big{]}\nabla\delta(x)$
(3.5)

with

$f(t)=\frac{1}{\text{\rm ln}\frac{1}{t}}$ (3.6)

and then

${\boldsymbol{X}}_{0}(x)=\Psi(x)\widetilde{\boldsymbol{X}}_{0}(x)$ (3.7)

where $\Psi:\Omega\rightarrow[0,1],\;\Psi\in{\mathcal{C}}^{1}(\Omega)$, and

$\Psi=\begin{cases}1&\quad\text{for }x\in\Gamma_{\frac{\nu_{0}}{2}}\\\
0&\quad\text{for }x\not\in\Gamma_{\frac{3\nu_{0}}{4}}\end{cases}\,.$ (3.8)

By construction $\sup_{x\in\Omega}\big{|}\nabla\Psi(x)\big{|}<\infty$, hence
from Assumption A for all $0<\nu<\frac{\nu_{0}}{2}$

$\sup_{x\in\Omega\setminus\Gamma_{\nu}}\frac{1}{\rho(x)}\big{|}\nabla\cdot{\boldsymbol{X}}_{0}(x)-{\boldsymbol{X}}_{0}(x)\cdot\big{(}\rho(x){\mathbb{D}}(x)\big{)}^{-1}{\boldsymbol{X}}_{0}(x)\big{|}=:M_{0,\nu}<\infty.$
(3.9)

On $\Gamma_{\nu}$, since $\nu<\frac{\nu_{0}}{2}$, from (3.7) and (3.8)
${\boldsymbol{X}}_{0}(x)=\tilde{\boldsymbol{X}}_{0}(x)$ hence we have to
compute
$\nabla\cdot\widetilde{\boldsymbol{X}}_{0}(x)-\widetilde{\boldsymbol{X}}_{0}(x)\cdot\big{(}\rho(x){\mathbb{D}}(x))^{-1}\widetilde{\boldsymbol{X}}_{0}(x)$.

By direct (a bit tedious) computation, from (3.5) one obtains for
$x\in\Gamma_{\frac{\nu_{0}}{2}}$:

$\displaystyle\nabla\cdot{\boldsymbol{X}}_{0}(x)=\frac{1}{2}\big{(}1-q(x)\big{)}a(x)\rho(x)\delta(x)^{\beta(x)-2}$
$\displaystyle\Big{[}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\big{(}\beta(x)+\gamma(x)-1\big{)}f\big{(}\delta(x)\big{)}+\delta(x)f^{\prime}\big{(}\delta(x)\big{)}+R_{0}(x)\Big{]}$
(3.10)

with

$\displaystyle
R_{0}(x)=\frac{\delta(x)}{2}\Big{\\{}\big{[}1+\big{(}\beta(x)+\gamma(x)-1+f\big{(}\delta(x)\big{)}\big{)}\text{\rm
ln}\delta(x)\big{]}$
$\displaystyle\big{(}\nabla\beta(x)+\nabla\gamma(x)\big{)}+\big{(}\beta(x)+\gamma(x)-1+\big{(}\delta(x)\big{)}\big{)}\big{(}\nabla\big{(}1-q(x)\big{)}a(x)r(x)\big{)}\Big{\\}}\cdot\nabla\delta(x)$
$\displaystyle+\frac{\beta(x)+\gamma(x)-1+f\big{(}\delta(x)\big{)}}{2}\delta(x)\Delta\delta(x).$
(3.11)

Taking into account (2.3), (2.11), (2.14) and the fact that
$\lim_{\delta(x)\rightarrow 0}\delta(x)^{\frac{1-s}{2}}\text{\rm
ln}\delta(x)=0$ one obtains that uniformly for $s_{\beta}\leqslant s$

$\sup_{x\in\Gamma_{\frac{\nu_{0}}{2}}}\big{|}R_{0}(x)\big{|}\delta(x)^{\frac{s-1}{2}}=:R_{0,\nu_{0}}<\infty.$
(3.12)

We compute now
${\boldsymbol{X}}_{0}(x)\cdot\big{(}\rho(x)\mathbb{D}(x)\big{)}^{-1}{\boldsymbol{X}}_{0}(x))$.
From (2.13), (2), (2.18), (3.5) and (3.7),

$\displaystyle{\boldsymbol{X}}_{0}(x)\cdot\big{(}\rho(x)\mathbb{D}(x)\big{)}^{-1}{\boldsymbol{X}}_{0}(x))=\frac{1}{4}\big{(}1-q(x)\big{)}a(x)\rho(x)\delta(x)^{\beta(x)-2}$
$\displaystyle\Big{[}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+2\big{(}\beta(x)+\gamma(x)-1\big{)}f\big{(}\delta(x)\big{)}+f\big{(}\delta(x)\big{)}^{2}\Big{]}.$
(3.13)

Putting together (3) and (3) one obtains

$\displaystyle\nabla\cdot{\boldsymbol{X}}_{0}(x)-{\boldsymbol{X}}_{0}(x)\cdot\big{(}\rho(x)\mathbb{D}(x)\big{)}^{-1}{\boldsymbol{X}}_{0}(x))=\big{(}1-q(x)\big{)}a(x)\rho(x)\delta(x)^{\beta(x)-2}$
$\displaystyle\Big{[}\frac{1}{4}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\frac{1}{2}\big{(}\delta(x)f^{\prime}\big{(}\delta(x)\big{)}-\frac{1}{4}f\big{(}\delta(x)\big{)}^{2}+R_{0}(x)\Big{]}.$
(3.14)

Further from (3.6)

$tf^{\prime}(t)-\frac{1}{2}f(t)^{2}=\frac{1}{2}\Big{(}\frac{1}{\text{\rm
ln}\frac{1}{t}}\Big{)}^{2}.$ (3.15)

Finally from (3) and (3.15) (on $\Gamma_{\frac{\nu_{0}}{2}}$)

$\displaystyle\nabla\cdot{\boldsymbol{X}}_{0}(x)-{\boldsymbol{X}}_{0}(x)\cdot\big{(}\rho(x)\mathbb{D}(x)\big{)}^{-1}{\boldsymbol{X}}_{0}(x))=\big{(}1-q(x)\big{)}a(x)\rho(x)\delta(x)^{\beta(x)-2}$
$\displaystyle\Big{[}\frac{1}{4}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\frac{1}{4}\Big{(}\frac{1}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+R_{0}(x)\Big{]}.$ (3.16)

From (3.12) there exists $0<\nu_{1}<\frac{\nu_{0}}{2}$ such that for
$0<\nu<\nu_{1}$

$\frac{1}{8}\Big{(}\frac{1}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+R_{0}(x)\geqslant 0$ (3.17)

hence on $\Gamma_{\nu_{1}}$

$\displaystyle\nabla\cdot{\boldsymbol{X}}_{0}(x)-{\boldsymbol{X}}_{0}(x)\cdot\big{(}\rho(x)\mathbb{D}(x)\big{)}^{-1}{\boldsymbol{X}}_{0}(x))\geqslant$
$\displaystyle\frac{1}{4}\big{(}1-q(x)\big{)}a(x)\rho(x)\delta(x)^{\beta(x)-2}\Big{[}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\frac{1}{2}\frac{1}{\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{2}}\Big{]}.$ (3.18)

Then (3.1) with $c_{1}(\nu)=M_{0,\nu}$ and ${\mathcal{H}}_{0}$ given by (3.2)
follows from Lemma 3.4, (3.9) and (3). ∎

##  4\. Essential self-adjointness of drift-diffusion operators

In this section we give criteria for essential self-adjointness of operator
$H$ (see (2.7)) in terms of $\rho$, ${\mathbb{D}}$, $V$ in the setting given
in Section 2. The strategy of the proof of these criteria is the one outlined
in Section 3 in of [24]. It relies on the following inequality.

######  Lemma 4.1.

[24, Lemma 3.4] Assume that there exist $E_{0}>-\infty$ and a function
$B\,:\,\Omega\,\rightarrow\,[0,\infty)$ such that all $\varphi\in
C_{0}^{\infty}(\Omega)$, it holds that:

$(\varphi,(H-E_{0})\varphi)\geqslant\int_{\Omega}|\varphi(x)|^{2}B(x)\rho(x)\,dx\,.$
(4.1)

Let $\psi_{E}$ be a weak solution of $\big{(}H-E\big{)}\psi_{E}=0$ for some
$E<E_{0}$, and let $g$ be a real-valued, Lipschitz continuous function on
$\Omega$ satisfying (a.e):

$\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla g(x)\big{)}\leqslant
B(x)+\frac{|E-E_{0}|}{2}\,.$ (4.2)

Then

$\big{(}\psi_{E},f^{2}\psi_{E}\big{)}\leqslant\frac{2}{|E-E_{0}|}\big{(}\psi_{E},|m|\psi_{E}\big{)}\,,$
(4.3)

where $f=e^{g}\phi$ with $\phi\,:\,\Omega\rightarrow[0,1]$,
$\phi\in{\mathcal{C}}^{1}_{0}(\Omega)$, and

$\displaystyle|m|(x)=e^{2g(x)}\Big{\\{}2\big{[}\nabla
g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla g(x)\big{)}\big{]}^{\frac{1}{2}}$
$\displaystyle\big{[}\nabla\phi(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\phi(x)\big{)}\big{]}^{\frac{1}{2}}+\nabla\phi(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\phi(x)\big{)}\Big{\\}}.$
(4.4)

The choice of $\phi$ to be used in Lemma 4.1 is as follows (see [24]). Let for
$l=1,2,...$

$r_{l}=\frac{r_{1}}{2^{l-1}}\,,\quad r_{1}<\frac{\nu_{0}}{2},$ (4.5)
$k_{l}(t)\in{\mathcal{C}}^{1}\big{(}(0,\infty)\big{)};\quad\operatorname{supp}k^{\prime}_{l}\subset(r_{l+1},r_{l});\quad
k_{l}(t)=\begin{cases}0\,,&\quad t>r_{l+1}\\\ 1\,,&\quad t<r_{l},\end{cases}$
(4.6) $\big{|}k^{\prime}_{l}(t)\big{|}\leqslant\frac{2}{r_{l}-r_{j+l}}$ (4.7)

and set

$\phi_{l}(x)=k_{l}(\delta(x)),$ (4.8)

$B_{l}=\big{\\{}x\in\Omega\big{|}r_{l+1}<\delta(x)<r_{l}\big{\\}}.$ (4.9)

Let $|m_{l}|(x)$ as given by (4.1) with $f=e^{2g(x)}\phi_{l}(x)$. By
construction $B_{l}$ are disjoint and

$\operatorname{supp}|m_{l}|\subset\operatorname{supp}\nabla\phi_{l}\subset
B_{l}.$ (4.10)

The next lemma gives the essential self-adjointness of $H$ providing the
following estimation, which has to be verified for each particular case, holds
true.

######  Lemma 4.2.

Suppose that (uniformly in $l$)

$\sup_{x\in\Omega}|m_{l}|(x)\leqslant C_{m}l,\quad C_{m}<\infty.$ (4.11)

Then $H$ is essentially self-adjoint.

###### Proof.

Since, by assumption (see Lemma 4.1), $g(x)$ is continuous on $\Omega$ for any
compact $K\subset\Omega$

$\inf_{x\in K}e^{2g(x)}=c_{K}>0.$ (4.12)

Fix now arbitrarily a compact $K$ in $\Omega$. Since
$\Omega\setminus\Gamma_{r_{l}}$ exhaust $\Omega$ as $l\rightarrow\infty$ there
exists an integer $L(K)$ such that
$K\subset\big{\\{}x\in\Omega\big{|}\delta(x)>r_{L(K)}\big{\\}}$. By
construction (see (4.6), (4.8))

$\phi_{l}\big{|}_{K}=1\;\text{for}\;l\geqslant L(K).$ (4.13)

Then from (4.12), (4.8), (4.2) and Lemma (4.1) for all $l\geqslant L(K)$

$\int_{K}\big{|}\Psi_{E}(x)\big{|}^{2}\rho(x)dx\leqslant\frac{1}{c_{K}}\int_{K}\big{|}\Psi_{E}(x)\big{|}^{2}e^{2g(x)}\phi_{l}(x)^{2}\rho(x)dx\leqslant\frac{2C_{m}l}{c_{K}(E_{0}-E)}\int_{B_{l}}\big{|}\Psi_{E}(x)\big{|}^{2}\rho(x)dx.$
(4.14)

Summing (4.14) over $l$ from $L(K)$ to $N$ and taking into account that
$B_{l}$ are disjoint one obtains

$\frac{c_{K}\big{(}E_{0}-E\big{)}}{2C_{m}}\sum_{l=L(K)}^{N}\frac{1}{l}\int_{K}\big{|}\Psi_{E}(x)\big{|}^{2}\rho(x)dx\leqslant\|\Psi_{E}\|^{2}$
(4.15)

which in the limit $N\rightarrow\infty$ gives

$\int_{K}\big{|}\Psi_{E}(x)\big{|}^{2}\rho(x)dx=0.$ (4.16)

Since $K$ was arbitrary, $\Psi_{E}=0$ and the proof is finished by invoking
the basic criterion for (essential) self-adjointness [31, Proposition 3.8]. ∎

Summing up, for a given class of $\rho,\;{\mathbb{D}},\;V$ the problem of
essential self-adjointness of $H$ boils down to the problem of finding $g(x)$
satisfying the conditions of Lemma 4.1 and Lemma 4.2 .

As a warming up we begin with the simple case when
${\mathbb{P}}_{x}{\mathbb{D}}{\mathbb{P}}_{x}$ is strongly degenerate at the
boundary of $\Omega$ (i.e.
${\mathbb{P}}_{x}{\mathbb{D}}{\mathbb{P}}_{x}\rightarrow 0$ quickly enough as
$\delta(x)\rightarrow 0$) and $V$ is bounded from below.

######  Theorem 4.3.

Suppose that there exists $\mu<\frac{\nu_{0}}{2}$ such that

$\sup_{x\in\Gamma_{\mu}}\big{\|}{\mathbb{P}}_{x}{\mathbb{D}}(x){\mathbb{P}}_{x}\big{\|}\delta^{-\beta(x)}<\infty;\quad\inf_{x\in\Gamma_{\mu}}\beta(x)\geqslant
2;\quad\inf_{x\in\Omega}V(x)>-\infty.$ (4.17)

Then $H$ is essentialy self-adjoint.

######  Remark 4.4.

The fact that strong degeneracy of ${\mathbb{D}}$ ensure the essential self-
adjointness of $H_{0}$ is well known. In particular the fact that if
$\sup_{x\in\Gamma_{\mu}}\big{\|}{\mathbb{D}}(x)\big{\|}\delta^{-\beta(x)}<\infty$
and $\inf_{x\in\Gamma_{\mu}}\beta(x)\geqslant 2$ then, without any conditions
on $\rho$ and on regularity of $\partial\Omega$, $H_{0}$ is essentially self-
adjoint follows from general results on essential self-adjointness of Laplace-
Beltrami operators on complete weighted Riemannian manifolds (see e.g. [9],
[24]).

###### Proof.

In this case the verification of Lemma 4.1 and Lemma 4.2 is immediate. Without
restricting the generality one can suppose that $V(x)\geqslant 0$ on $\Omega$.
In this case we choose $g=0$ so that the condition(4.2) is trivially satisfied
with $E_{0}=0$ and $B(x)=0$. The condition (4.1) is also trivially satisfied
since $H_{0}\geqslant 0,\;V\geqslant 0$. It remains to verify Lemma 4.2. We
take $r_{1}=\mu$ in (4.5). From (4.1) (recall that $g=0$)

$|m_{l}|(x)=\nabla\phi_{l}(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\phi_{l}(x)\big{)}$
(4.18)

hence from (4.7), (4.8) and (4.17) (recall that
$\big{|}\nabla\delta(x)\big{|}=1$) one obtains that

$\sup_{x\in\Omega}|m_{l}|(x)=:C_{3}<\infty.$ (4.19)

and the proof is finished. ∎

We turn now to the results concerning essential self-adjointness of $H$ when
$\inf_{x\in\Gamma_{\nu_{0}}}\beta(x)\geqslant 2$ does not hold true. Consider
first the case when $\nabla\beta(x)=0$.

######  Theorem 4.5.

Suppose that Assumption A and Assumption Q hold true, $\beta(x)=\beta<2$ and
in addition there exists

$0<\mu<\frac{\nu_{0}}{2}$ (4.20)

such that on $\Gamma_{\mu}$

$V(x)\geqslant\delta(x)^{\beta-2}\Big{[}v(x)-w_{\mu}\delta(x)^{\frac{1-s}{2}}\Big{]};\;0\leqslant
w_{\mu}<\infty;\;v\in L^{\infty}(\Gamma_{\mu};{\mathbb{R}}).$ (4.21)

Then:

i. If

$\inf_{\Gamma_{\mu}}\frac{\big{(}1-q(x)\big{(}\beta+\gamma(x)-1\big{)}^{2}+\frac{4v(x)}{a(x)}}{\big{(}\beta-2\big{)}^{2}}\geqslant
1$ (4.22)

$H$ is essentially self-adjoint.

ii. If

$\sup_{x\in\Gamma_{\frac{\nu_{0}}{2}}}q(x)\delta(x)^{\frac{s-1}{2}}=:Q_{\nu_{0}}<\infty$
(4.23)

and

$\inf_{\Gamma_{\mu}}\frac{\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\frac{4v(x)}{a(x)}}{\big{(}\beta(x)-2\big{)}^{2}}\geqslant
1,$ (4.24)

$H$ is essentially self-adjoint

###### Proof.

i. As in the proof of Theorem 4.3 one uses Lemma 4.1 but in this case one
needs a nontrivial $g(x)$. The chioce of $g(x)$ is as follows:

$g(x)=\Big{(}\frac{2-\beta}{2}\text{\rm ln}\delta(x)\Big{)}\Psi(x)$ (4.25)

where $\Psi$ (see (3.8)) is the same as in the proof of Theorem 3.1 . From the
fact that $g\in{\mathcal{C}}^{1}(\Omega;{\mathbb{R}}^{d})$,
${\mathbb{D}}\in{\mathcal{C}}(\Omega;{\mathbb{R}}^{d\times d})$ it follows
that for all $\nu>0$

$\sup_{\Omega\setminus\Gamma_{\nu}}\nabla
g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla g(x)\big{)}=:g_{\nu}<\infty.$ (4.26)

Next, we compute $\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla g(x)\big{)}$on
$\Gamma_{\mu}$. For $\delta(x)<\frac{\nu_{0}}{2}$, $\Psi(x)=1$ hence for
$x\in\Gamma_{\mu}$

$\nabla g(x)=\frac{2-\beta}{2}\delta(x)^{-1}\nabla\delta(x).$ (4.27)

From (2) and (2.13)

$\nabla\delta(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\delta(x)\big{)}=a(x)\delta(x)^{\beta},$
(4.28)

which together with (5.53) leads to

$\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla
g(x)\big{)}=a(x)\Big{(}\frac{2-\beta}{2}\Big{)}^{2}\delta(x)^{\beta-2}.$
(4.29)

The next step is to show that using Theorem 3.1 and (4.29) one can choose
$B(x);\;E_{0},\;E$ such that (4.1) and (4.2) hold true. From (2.6) it follows
that for all $\nu>0$

$\text{ess-
sup}_{x\in\Omega\setminus\Gamma_{\nu}}\big{|}V(x)\big{|}=:V_{\nu}<\infty$
(4.30)

and from (2.16) for $0<\nu<\mu$

$0<a_{\mu}:=\inf_{\Gamma_{\mu}}a(x)\leqslant\inf_{\Gamma_{\nu}}a(x).$ (4.31)

From (4.21), (4.30), (4.31) and Theorem 3.1, for $0<\nu<\min\\{\nu_{1},\mu\\}$
(see Theorem 3.1 for $\nu_{1}$ and recall that on $\Gamma_{\mu}$,
$\delta(x)<1$)

$\displaystyle
h\big{[}\varphi,\varphi\big{]}\geqslant-\big{(}c_{1}(\nu)+V_{\nu}\big{)}\|\varphi\|^{2}+\int_{\Gamma_{\nu}}a(x)\delta(x)^{\beta-2}\Big{[}\big{(}1-q(x)\big{)}\Big{(}\frac{\beta+\gamma(x)-1}{2}\Big{)}^{2}+$
$\displaystyle\frac{v(x)}{a(x)}+$
$\displaystyle\frac{1-q(x)}{8}\frac{1}{\text{\rm
ln}\Big{(}\frac{1}{\delta(x)}\Big{)}^{2}}-\frac{w_{\mu}}{a_{\mu}}\delta(x)^{\frac{1-s}{2}}\Big{]}\big{|}\varphi(x)\big{|}^{2}\rho(x)dx.$
(4.32)

Since $\frac{1}{\text{\rm ln}\frac{1}{\delta(x)}}\rightarrow 0$ as
$\delta(x)\rightarrow 0$ slower than any (strictly) positive power of
$\delta(x)$ and $\frac{1}{1-q}\in L^{\infty}(\Gamma_{\mu})$ there exists

$0<\nu_{2}<\min\\{\mu,\nu_{1}\\}$ (4.33)

such that on $\Gamma_{\nu_{2}}$

$\frac{1-q(x)}{8}\frac{1}{\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{2}}-\frac{w_{\mu}}{a_{\mu}}\delta(x)^{\frac{1-s}{2}}\geqslant
0,$ (4.34)

hence on $\Gamma_{\nu_{2}}$,

$\displaystyle
h\big{[}\varphi,\varphi\big{]}\geqslant-\big{(}c_{1}(\nu_{2})+V_{\nu_{2}}\big{)}\|\varphi\|^{2}+\int_{\Gamma_{\nu_{2}}}a(x)\delta(x)^{\beta-2}$
$\displaystyle\Big{[}\big{(}1-q(x)\big{)}\Big{(}\frac{\beta+\gamma(x)-1}{2}\Big{)}^{2}+\frac{v(x)}{a(x)}\Big{]}\big{|}\varphi(x)\big{|}^{2}\rho(x)dx.$
(4.35)

Now, from (4), (4.1) holds true by choosing:

$E_{0}=-\big{(}c_{1}(\nu_{2})+V_{\nu_{2}}\big{)}$ (4.36)

and

$B(x)=\begin{cases}a(x)\delta(x)^{\beta-2}\Big{[}\big{(}1-q(x)\big{)}\Big{(}\frac{\beta+\gamma(x)-1}{2}\Big{)}^{2}+\frac{v(x)}{a(x)}\Big{]}&\text{for}\;x\in\Gamma_{\nu_{2}}\\\
0&\text{otherwise}\end{cases}.$ (4.37)

Then by choosing

$\frac{E_{0}-E}{2}=g_{\nu_{2}},$ (4.38)

(4.2) follows from (4.22), (4.26) (4.29) and (4.37). The last step in the
proof of Theorem 4.5 is to verify (4.2) in Lemma 4.2 where in the definition
of $\phi_{l}$ we take $r_{1}<\nu_{2}$.

From (4.7), (4.8) and Assumption A:

$\nabla\phi_{l}(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\phi_{l}(x)\big{)}=\big{(}k_{l}^{\prime}(\delta(x)\big{)}^{2}\nabla\delta(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\delta(x)\big{)}\leqslant\frac{4}{r^{2}_{l+1}}a(x)\delta(x)^{\beta(x)}$
(4.39)

hence uniformly in $l$:

$\sup_{x\in
B_{l}}\nabla\phi_{l}(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\phi_{l}(x)\big{)}\delta(x)^{2-\beta}\leqslant
c_{\phi}<\infty.$ (4.40)

Further, from (4.29) (again uniformly in $l$)

$\sup_{x\in B_{l}}\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla
g(x)\big{)}\delta(x)^{2-\beta}\leqslant c_{g}<\infty.$ (4.41)

On the other hand from (4.25) on $\Gamma_{\mu}$:

$e^{2g(x)}=\delta(x)^{2-\beta}.$ (4.42)

Plugging (4.40),(4.41), (4.42) into (4.1) one obtains (recall that
$\operatorname{supp}|m|_{l}\subset B_{l}\subset\Gamma_{\mu}$) (4.11) and the
proof of Theorem 4.5 i. is finished.

ii. Rewrite (4) as

$\displaystyle
h\big{[}\varphi,\varphi\big{]}\geqslant-\big{(}c_{1}(\nu)+V_{\nu}\big{)}\|\varphi\|^{2}+\int_{\Gamma_{\nu}}a(x)\delta(x)^{\beta-2}\Big{[}\Big{(}\frac{\beta+\gamma(x)-1}{2}\Big{)}^{2}+\frac{v(x)}{a(x)}-$
$\displaystyle-q(x)\Big{(}\frac{\beta+\gamma(x)-1}{2}\Big{)}^{2}+\frac{1-q(x)}{8}\frac{1}{\text{\rm
ln}\Big{(}\frac{1}{\delta(x)}\Big{)}^{2}}-\frac{w_{\mu}}{a_{\mu}}\delta(x)^{\frac{1-s}{2}}\Big{]}\big{|}\varphi(x)\big{|}^{2}\rho(x)dx.$
(4.43)

From (4.23) (recall that $\gamma\in L^{\infty}(\Gamma_{\nu_{0}}$) it follows
again that there exists $0<\nu_{2}<\min\\{\mu,\nu_{1}\\}$ such that on
$\Gamma_{\nu_{2}}$

$-q(x)\Big{(}\frac{\beta+\gamma(x)-1}{2}\Big{)}^{2}+\frac{1-q(x)}{8}\frac{1}{\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{2}}-\frac{w_{\mu}}{a_{\mu}}\delta(x)^{\frac{1-s}{2}}\geqslant
0.$ (4.44)

The rest of the argument goes unchanged.

∎

We state now our main result concerning essential self-adjointness oh $H$ in
the case when neither $\inf_{x\in\Gamma_{\nu_{0}}}\beta(x)\geqslant 2$ nor
$\beta(x)=constant$ holds true. For $\eta<\nu_{0}$ let

$\Gamma_{\eta,-}=\big{\\{}x\in\Gamma_{\eta}\big{|}\beta(x)<2\big{\\}};\quad\Gamma_{\eta,+}=\big{\\{}x\in\Gamma_{\eta}\big{|}\beta(x)\geqslant
2\big{\\}}.$ (4.45)

######  Theorem 4.6.

Suppose Assumptions A and Q hold true, and in addition there exists

$0<\mu<\frac{\nu_{0}}{2}$ (4.46)

such that

$\sup_{x\in\Gamma_{\mu,-}}\|{\mathbb{D}}(x)\|\delta(x)^{-\big{(}\beta(x)+s_{\beta}-s\big{)}}<\infty,$
(4.47) $V(x)\geqslant\begin{cases}0\quad&\text{for}\;x\in\Gamma_{\mu,+}\\\
V(x)\geqslant\delta(x)^{\beta-2)}\\\
\Big{[}v(x)-w_{\mu}\delta(x)^{\frac{1-s}{2}}\Big{]};\;0\leqslant
w_{\mu}<\infty;\;v\in
L^{\infty}(\Gamma_{\mu};{\mathbb{R}})&\text{for}\;x\in\Gamma_{\mu,-}\end{cases}.$
(4.48)

Then if

$\inf_{\Gamma_{\mu,-}}\frac{\big{(}1-q(x)\big{)}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\frac{4v(x)}{a(x)}}{\big{(}\beta(x)-2\big{)}^{2}}\geqslant
1$ (4.49)

$H$ is essentially self-adjoint.

Proof The proof of Theorem 4.6 follows the proof of Theorem 4.5. As in the
proof of Theorem 4.5 we use Lemma 4.1 and Lemma 4.2 but in this case we need a
more elaborate choice of $g(x)$. The chioce of $g(x)$ is as folloows:

$g(x)=\min\big{\\{}0,\tilde{g}(x)\big{\\}}$ (4.50)

with

$\tilde{g}(x)=\frac{2-\beta(x)}{2}\Big{(}\text{\rm
ln}\delta(x)+\lambda\text{\rm ln}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}\Psi(x)$ (4.51)

where $\Psi$ is the same as in the proof of Theorem 3.1 (see (3.8)) and

$0<\lambda\leqslant\min\Big{\\{}1,\frac{1}{1+\sup_{x\in\Gamma_{\nu_{0}}}|\beta(x)-2|}\Big{\\}}.$
(4.52)

On $\Gamma_{\frac{\nu_{0}}{2}}$ one has

$g(x)=\begin{cases}0\quad&\text{for}\;x\in\Gamma_{\frac{\nu_{0}}{2},+}\\\
\frac{2-\beta(x)}{2}\Big{(}\text{\rm ln}\delta(x)+\lambda\text{\rm
ln}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}&\text{for}\;x\in\Gamma_{\frac{\nu_{0}}{2},-}\end{cases}.$
(4.53)

Indeed (recall that $\nu_{0}\leqslant e^{-1}$ and
$\Psi\big{|}_{\Gamma_{\frac{\nu_{0}}{2}}}=1$) since $\text{\rm
ln}\delta(x)+\lambda\text{\rm ln}\text{\rm ln}\frac{1}{\delta(x)}<0$ it
follows that $g(x)<0$ if and only if $\beta(x)-2<0$. Moreover since $g$ is
Lipschitz continuous, by Rademacher theorem $\nabla g(x)$ exists a.e. and,
whenever it exists, equals $\nabla\tilde{g}(x)$ if $\beta(x)-2<0$ and vanishes
otherwise. The estimation involving $g(x)$ we need is provided by the next
lemma.

######  Lemma 4.7.

There exist $0\leqslant c_{4}(\nu_{0}),\;c_{5}(\nu_{0})<\infty$ such that on
$\Gamma_{\frac{\nu_{0}}{2}}$:

$\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla
g(x)\big{)}\begin{cases}=0\quad&\text{for}\;x\in\Gamma_{\frac{\nu_{0}}{2},+}\\\
\leqslant
c_{5}(\nu_{0})+\big{(}\frac{\beta(x)-2}{2}\big{)}^{2}a(x)\delta(x)^{\beta(x)-2}\\\
\Big{[}\big{(}1-\frac{\lambda}{\text{\rm
ln}\frac{1}{\delta(x)}}\big{)}^{2}+c_{4}(\nu_{0})\delta(x)^{\frac{1-s}{2}}\Big{]}&\text{for}\;x\in\Gamma_{\frac{\nu_{0}}{2},-}\end{cases}.$
(4.54)

###### Proof.

We estimate $\nabla g\cdot\big{(}{\mathbb{D}}\nabla g\big{)}$ for
$x\in\Gamma_{\frac{\nu_{0}}{2},-}$. From (4.53) on
$\Gamma_{\frac{\nu_{0}}{2},-}$:

$\nabla g(x)=\frac{2-\beta(x)}{2}\Big{(}1-\frac{\lambda}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}\delta(x)^{-1}\nabla\delta(x)-\frac{1}{2}\Big{(}\text{\rm
ln}\delta(x)+\lambda\text{\rm ln}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}\nabla\beta(x)$ (4.55)

hence from (2.13) and Assumption A (recall that
$\big{|}\nabla\delta(x)\big{|}=1$)

$\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla
g(x)\big{)}=\Big{(}\frac{2-\beta(x)}{2}\Big{)}^{2}a(x)\Big{(}1-\frac{\lambda}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}\delta(x)^{\beta(x)-2}+T(x)$ (4.56)

with

$\displaystyle T(x)=$
$\displaystyle-\frac{2-\beta(x)}{4}\Big{(}1-\frac{\lambda}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}\Big{(}\text{\rm ln}\delta(x)+\lambda\text{\rm
ln}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}\delta(x)^{-1}\Big{[}\nabla\delta(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\beta(x)\big{)}+$
$\displaystyle\nabla\beta(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\delta(x)\big{)}\Big{]}+\frac{1}{4}\Big{(}\text{\rm
ln}\delta(x)+\lambda\text{\rm ln}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{2}\nabla\beta(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\beta(x)\big{)}.$
(4.57)

Further, one has to estimate $T$. Consider first the case

$\beta(x)-2\geqslant\frac{s-1}{2}.$ (4.58)

From Assumption A, (2.15), (4.47) and (4), counting the powers of $\delta(x)$
one arrives at the conclusion that

$\sup_{x\in\Gamma_{\frac{\nu_{0}}{2}};\;0>\beta(x)-2\geqslant\frac{s-1}{2}}\big{|}T(x)\big{|}=:c_{5}(\nu_{0})<\infty,$
(4.59)

hence from (4.56)

$\nabla g(x)\cdot({\mathbb{D}}(x)\nabla
g(x))\leqslant\Big{(}\frac{2-\beta(x)}{2}\Big{)}^{2}a(x)\Big{(}1-\frac{\lambda}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}\delta(x)^{\beta(x)-2}+c_{5}(\nu_{0}).$
(4.60)

Consider now the alternative case

$\beta(x)-2<\frac{s-1}{2}$ (4.61)

which implies

$\big{|}\beta(x)-2\big{|}>\frac{1-s}{2}$ (4.62)

In this case from (4.62), taking into account that $\frac{1}{a}\in
L^{\infty}\big{(}\Gamma_{\frac{\nu_{0}}{2}}\big{)}$, by counting the powers of
$\delta(x)$ as before one obtain that

$\sup_{x\in\Gamma_{\frac{\nu_{0}}{2}};\;\beta(x)-2<\frac{s-1}{2}}\Big{(}\frac{2}{2-\beta(x)}\Big{)}^{2}\frac{1}{a(x)}\delta(x)^{2-\beta(x)}\delta(x)^{\frac{s-1}{2}}\big{|}T\big{|}=:c_{4}(\nu_{0})<\infty.$
(4.63)

which together with(4.56)

$\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla
g(x)\big{)}\leqslant\Big{(}\frac{2-\beta(x)}{2}\Big{)}^{2}a(x)\delta(x)^{(\beta(x)-2)}\Big{[}\Big{(}1-\frac{\lambda}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+c_{4}(\nu_{0})\Big{]}.$ (4.64)

Finally, (4.55) follows from (4.60) and (4.64). ∎

We estimate now $h\big{[}\varphi,\varphi\big{]}$ from below. From (4.30),
(4.48) and Theorem 3.1 for all $0<\nu<\min\\{\nu_{1},\mu\\}$

$\displaystyle
h\big{[}\varphi,\varphi\big{]}\geqslant-\big{(}c_{1}(\nu)+V_{\nu}\big{)}\|\varphi\|^{2}+\int_{\Gamma_{\nu,-}}a(x)\delta(x)^{\beta(x)-2}$
$\displaystyle\Big{[}\big{(}1-q(x)\big{)}\Big{(}\frac{\beta+\gamma(x)-1}{2}\Big{)}^{2}+\frac{v(x)}{a(x)}+\frac{1-q(x)}{8}\frac{1}{\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\big{)}^{2}}-\frac{w_{\nu}}{a(x)}\delta(x)^{\frac{1-s}{2}}\Big{]}\big{|}\varphi(x)\big{|}^{2}\rho(x)dx.$
(4.65)

Since $\frac{1}{\text{\rm ln}\frac{1}{\delta(x)}}\rightarrow 0$ as
$\delta(x)\rightarrow 0$ slower than any (strictly) positive power of
$\delta(x)$ there exists

$0<\nu_{3}\leqslant\min\\{\nu_{1},\mu\\}$ (4.66)

such that on $\Gamma_{\nu_{3}}$

$\frac{1-q(x)}{8}\frac{1}{\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\big{)}^{2}}-\frac{w_{\nu}}{a(x)}\delta(x)^{\frac{1-s}{2}}\geqslant
0$ (4.67)

and

$\Big{(}1-\frac{\lambda}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+c_{4}(\nu_{0})\delta(x)^{\frac{1-s}{2}}\leqslant
1.$ (4.68)

.

From (4) and (4.67), (4.1) holds true by choosing:

$E_{0}=-\big{(}c_{1}(\nu_{3})+V_{\nu_{3}}\big{)},$ (4.69)
$B(x)=\begin{cases}a(x)\delta(x)^{\beta(x)-2}\Big{[}\big{(}1-q(x)\big{)}\big{(}\frac{\beta(x)+\gamma(x)-1}{2}\big{)}^{2}+\frac{v(x)}{a(x)}\Big{]}&\text{for}\;x\in\Gamma_{\nu_{3},-}\\\
0&\text{otherwise}\end{cases}$ (4.70)

Further, let

$\sup_{x\in\Omega\setminus\Gamma_{\nu_{3}}}\nabla
g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla g(x)\big{)}=:g_{\nu_{3}}<\infty.$ (4.71)

Then by choosing

$\frac{E_{0}-E}{2}=c_{5}(\nu_{0})+g_{\nu_{3}},$ (4.72)

(4.2) follows from (4.49), (4.54), (4.68) and (4.70).

Finally we have to verify Lemma 4.2. The choice of $\phi_{l}$ is as in the
proof of Theorem 4.5 with $r_{1}\leqslant\nu_{3}$. From (4.7), (4.8) and
Assumption A (recall also that by construction on $B_{l}$, $\delta(x)>r_{l}$
and $a\in L^{\infty}\big{(}\Gamma_{\nu_{0}}\big{)}$):

$\displaystyle\nabla\phi_{l}(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\phi_{l}(x)\big{)}=\big{(}k_{l}^{\prime}(\delta(x)\big{)}^{2}\nabla\delta(x)\cdot\big{(}{\mathbb{D}}(x)\nabla\delta(x)\big{)}\leqslant$
$\displaystyle\frac{4}{r^{2}_{l+1}}a(x)\delta(x)^{\beta(x)}\leqslant
16a(x)\delta^{\beta(x)-2}\leqslant C_{\phi}\delta^{\beta(x)-2}.$ (4.73)

Let $B_{l,+}=\big{\\{}x\in B_{l}\big{|}\beta(x)\geqslant
2\big{\\}},\;B_{l,-}=\big{\\{}x\in B_{l}\big{|}\beta(x)<2\big{\\}}$.

From Lemma 4.7 and (4)

$\big{|}m_{l}\big{|}\Big{|}_{B_{l,+}}=\nabla\phi_{l}\cdot\big{(}{\mathbb{D}}\nabla\phi_{l}\big{)}\Big{|}_{B_{l,+}}\leqslant
C_{\phi}.$ (4.74)

Further, for $x\in B_{l,-}$, from Lemma 4.7 there exists
$c_{6}(\nu_{0})<\infty$ such that

$\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla g(x)\big{)}\leqslant
c_{5}(\nu_{0})+c_{6}(\nu_{0})\delta(x)^{\beta(x)-2},$ (4.75)

and from (4.50), (4.52)

$e^{2g(x)}\leqslant e^{2-\beta(x)}\text{\rm ln}\frac{1}{\delta(x)}.$ (4.76)

Plugging (4), (4.74), (4.75), (4.76) into (4.1) and taking into account that
on $B_{l}$, $\delta(x)\geqslant\nu_{1}2^{1-l}$ (see (4.5) and (4.9)) one
obtains (4.11) and the proof of Theorem 4.6 is finished.

##  5\.  Comments, extensions, open problems

In this section besides comments and applications we give some extensions of
the results in previous sections. We would like to stress that the proofs of
these extensions follow closely the proofs in Section 3 and Section 4. What
has to be added in each case is the choice of the vector field,
${\boldsymbol{X}}(x)$ in Lemma 3.4 and/or the choice of the function $g(x)$ in
the Agmon type exponential estimation as given Lemma 4.1 leading via Lemma 4.2
to essential self-adjointness. The choices for $g(x)$ in this paper are
extensions to the case at hand of those used in [23] for the case
${\mathbb{D}}=\mathds{1},\;\rho=1$.

1\. For the sake of discussion let us fix (remind) the terminology. Let
$\Omega\subset{\mathbb{R}}^{d},\;H_{0}$ as given in Section 2 and
${\mathcal{H}}:\Omega\rightarrow[0,\infty),\;{\mathcal{H}}\in
L^{\infty}_{loc}(\Omega)$. $H_{0}$ is said to satisfy a ${\mathcal{H}}$-weak
Hardy inequality if there exists $M>-\infty$ such that for all
$\varphi\in{\mathbb{C}}_{0}^{\infty}(\Omega)$

$h\big{[}\varphi,\varphi\big{]}\geqslant
M\int_{\Omega}\big{|}\varphi(x)\big{|}^{2}\rho(x)dx+\int_{\Omega}{\mathcal{H}}(x)\big{|}\varphi(x)\big{|}^{2}\rho(x)dx,$
(5.1)

and if in addition $M\geqslant 0$, $H_{0}$ is said to satisfy a
${\mathcal{H}}$-strong Hardy inequality. $H_{0}$ is said to satisfy a
${\mathcal{H}}$-boundary layer Hardy inequality if there exists $\nu>0$ (see
Section 2 for $\Gamma_{\nu}$) such that for all
$\varphi\in{\mathbb{C}}_{0}^{\infty}(\Gamma_{\nu})$ (see [29] for relations
between week and boundary layer Hardy inequalities)

$h\big{[}\varphi,\varphi\big{]}\geqslant\int_{\Gamma_{\nu}}{\mathcal{H}}(x)\big{|}\varphi(x)\big{|}^{2}\rho(x)dx.$
(5.2)

Notice that the ${\mathcal{H}}$-weak Hardy inequality for $H_{0}$ is
equivalent with the ${\mathcal{H}}$-weak Hardy inequality for $H_{0}-M$. Since
essential self-adjointness is stable against addition of constants when using
Hardy inequalities to ensure essential self-adjointness of $H$, weak Hardy
inequalities (which are easier to come by) are sufficient. The function
${\mathcal{H}}$ is sometimes called ”Hardy barrier” and in order to provide
nontrivial lower bound for $h_{0}\big{[}\varphi,\varphi\big{]}$ it must blow
up as $x\rightarrow\partial\Omega$

With this terminology, Theorem 3.1 provides a general
${\mathcal{H}}_{0}$-weak/ boundary layer Hardy inequality for $H_{0}$. The
formula (3.2) gives the height or the Hardy barrier. We would like to stress
that ${\mathcal{H}}_{0}(x)$ depends locally on ${\mathbb{D}}$ and $\rho$ i.e.
the Hardy barrier is a local quantity.

In the particular case when $\rho=1$ and in addition ${\mathbb{D}}$ is
isotropic (hence $q=0$) i.e.
${\mathbb{D}}(x)=d(x)\mathds{1}=a(x)\delta(x)^{\beta(x)}\mathds{1}$, Theorem
3.1 gives (for simply connected, bounded domains with smooth boundary and
sufficiently small $\nu$) a weak/boundary layer weighted Hardy inequality
generalizing previous results in [29]

$\displaystyle\int_{\Omega}a(x)\delta(x)^{\beta(x)}\big{|}\nabla\varphi(x)\big{|}^{2}dx\geqslant$
$\displaystyle-
c_{1}(\nu)\int_{\Omega}\big{|}\chi_{\Omega\setminus\Gamma_{\nu}}(x)\varphi(x)\big{|}^{2}dx+\frac{1}{4}\int_{\Gamma_{\nu}}a(x)\big{(}\beta(x)-1\big{)}^{2}\delta(x)^{\beta(x)-2}\big{|}\varphi(x)\big{|}^{2}dx$
(5.3)

with $c_{1}(\nu)<\infty$. What (5) adds to the previous results is that the
degree of degeneracy of ${\mathbb{D}}$ can vary in $\Gamma_{\nu}$ along
$\partial\Omega$. In the ”classical” case ${\mathbb{D}}=\mathds{1};\rho=1$ one
recovers the famous constant $\frac{1}{4}$ [8]:

$\displaystyle\int_{\Omega}\big{|}\nabla\varphi(x)\big{|}^{2}dx\geqslant$
$\displaystyle-
c_{1}(\nu_{1})\|\chi_{\Omega\setminus\Gamma_{\nu_{1}}}\varphi\|^{2}+\frac{1}{4}\int_{\Gamma_{\nu_{1}}}\frac{1}{\delta(x)^{2}}\big{|}\varphi(x)\big{|}^{2}dx\geqslant$
$\displaystyle-C\|\varphi\|^{2}+\frac{1}{4}\int_{\Omega}\frac{1}{\delta(x)^{2}}\big{|}\varphi(x)\big{|}^{2}dx,\;C<\infty$
(5.4)

which is known to be optimal in the sense that there exist simply connected,
bounded domains with ${\mathcal{C}}^{2}$ boundary for which $C$ cannot be
negative [19].

There is a large body of literature on weak weighted Hardy inequalities on
domains in ${\mathbb{R}}^{d}$ with smooth/rough boundaries which can be partly
traced from references in [29]. However the optimality problem for the case of
rough boundaries is far from being solved. In particular an interesting open
problem is whether Theorem 3.1 remains true if $\partial\Omega$ is
${\mathcal{C}}^{1+\epsilon}$, $0<\epsilon<1$.

2\. As a particular case of a more general result, for $H_{0}=-\Delta$,
$\rho=1$ , $\Omega$ bounded and convex G. Barbatis, S. Filippas, and A.
Tertikas (see [3] and [22] for more references and related developments)
refined the strong Hardy inequality by adding an infinite series of
logarithmic sub-leading terms to the Hardy barrier. By using the approach in
[3] one can refine Theorem 3.1 and give a hierarchy of weak anisotropic Hardy
inequalities by adding logarithmic subleading terms to ${\mathcal{H}}_{0}(x)$.

The technical tool is a class of functions related to iterated logarithms.
Given $a\in(0,\infty)$ and $p=1,2,...$ let $t_{a,p}$ and
$L_{p}:(0,t_{a,p}]\rightarrow(0,1]$ given recurrently by

$t_{a,1}=\frac{a}{e},\;t_{a,p}=\frac{a}{e^{\frac{1}{t_{a,p-1}}}},$ (5.5)
$L_{a,1}(t)=\frac{1}{\text{\rm
ln}\frac{a}{t}};\;L_{a,p}(t)=L_{a,1}\big{(}L_{a,p-1}(t)\big{)}$ (5.6)

and then

$M_{a,p}(t)=\prod_{k=1}^{p}L_{a,k}(t);\;f_{a,p}(t)=\sum_{l=1}^{p}M_{a,l}(t).$
(5.7)

By direct computation the solution of recurrence (5.6) is

$L_{a,p}(t)=\frac{1}{\text{\rm ln}a\text{\rm ln}a...\text{\rm
ln}a\frac{1}{t}}$ (5.8)

and from (5.8) it is not hard to see that $L_{a,p}$ are well defined on
$(0,t_{a,p}]$ and take values in $(0,1]$. We also notice that
$t_{1,p}=\frac{1}{e^{e^{\cdot^{\cdot^{e}}}}}$, $t_{e,p}=1$ and

$t_{a,p}\begin{cases}<1;\;\lim_{p\rightarrow\infty}t_{a,p}=0&\text{for}\;a<e\\\
>1;\;\lim_{p\rightarrow\infty}t_{a,p}=\infty&\text{for}\;a>e\end{cases}$ (5.9)

The magic of these functions lies in the following lemma ( see formula (2.4)
in [3] where the case $a=e$ was considered).

######  Lemma 5.1.

For N=1,2,…

$tf^{\prime}_{a,N}(t)-\frac{1}{2}f_{a,N}(t)^{2}=\frac{1}{2}\sum_{k=1}^{N}M_{a,k}(t)^{2}.$
(5.10)

###### Proof.

See [3]. However for the sake of the reader we give it below. The proof is by
induction. For $N=1$, $f_{a,1}=L_{a,1}$ and (5.10) follows by direct
computation. Suppose (5.10) holds true for $f_{a,N}$. By direct computation:

$L_{a,l}^{{}^{\prime}}=\frac{1}{t}L_{a,l}M_{a,l};\;M_{a,N+1}^{{}^{\prime}}(t)=\frac{1}{t}M_{a,N+1}(t)f_{a,N+1}(t).$
(5.11)

Further from (5.7)

$f_{a,N+1}=f_{a,N}+M_{a,N+1}$ (5.12)

hence from (5.11)

$f^{\prime}_{a,N+1}(t)=f^{\prime}_{a,N}(t)+\frac{1}{t}M_{a,N+1}(t)f_{a,N+1}(t)$
(5.13)

which together with (5.12) and (5.10) leads to

$tf^{\prime}_{a,N+1}(t)-\frac{1}{2}f_{a,N+1}(t)^{2}=\frac{1}{2}\sum_{k=1}^{N+1}M_{a,k}(t)^{2}.$
(5.14)

∎

For definiteness we formulate the refinement of Theorem 3.1 in the case $a=e$
and omit the subscript $e$ i.e. we write $M_{p}(t),\;f_{p}(t)$ for
$M_{e,p}(t)$ and $f_{e,p}(t)$ respectively.

######  Theorem 5.2.

Suppose that Assumptions A and Q hold true. Then for $N=1,2,...$ there exists
$0<\nu_{N}<\frac{\nu_{0}}{2}$ such that for all $0<\nu<\nu_{N}$ and
$\varphi\in{\mathcal{C}}^{\infty}_{0}(\Omega)$:

$h_{0}\big{[}\varphi,\varphi\big{]}\geqslant-
c_{N}(\nu)\|\chi_{\Omega\setminus\Gamma{\nu}}\varphi\|^{2}+\int_{\Gamma_{\nu}}{\mathcal{H}}_{N}(x)|\varphi(x)|^{2}\rho(x)dx$
(5.15)

where $c_{N}(\nu)<\infty$, $\chi_{\Omega\setminus\Gamma{\nu}}$ is the
characteristic function of $\Omega\setminus\Gamma{\nu}$ and

${\mathcal{H}}_{N}(x)=\frac{1}{4}\big{(}1-q(x)\big{)}a(x)\delta(x)^{\beta(x)-2}\Big{[}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\sum_{k=1}^{N}M_{k}\big{(}\delta(x)\big{)}^{2}\Big{]}.$
(5.16)

In particular if $\varphi\in{\mathcal{C}}^{\infty}_{0}(\Gamma_{\nu})$

$h_{0}[\varphi,\varphi]\geqslant\int_{\Gamma_{\nu}}{\mathcal{H}}_{N}(x)|\varphi(x)|^{2}\rho(x)dx.$
(5.17)

###### Proof.

Repeat the proof of Theorem 3.1. Replace $\widetilde{\boldsymbol{X}}_{0}$
given by (3.5) with $\widetilde{\boldsymbol{X}}_{N}$ given by the same formula
with $f\big{(}\delta(x)\big{)}$ replaced by $f_{N+1}\big{(}\delta(x)\big{)}$.
The same computation as in the proof of Theorem 3.1 together with Lemma 5.1
leads to:

$\displaystyle\nabla\cdot{\boldsymbol{X}}_{N}(x)-{\boldsymbol{X}}_{N}(x)\cdot\big{(}\rho(x)\mathbb{D}(x)\big{)}^{-1}{\boldsymbol{X}}_{N}(x))=\big{(}1-q(x)\big{)}a(x)\rho(x)\delta(x)^{\beta(x)-2}$
$\displaystyle\Big{[}\frac{1}{4}\big{(}\beta(x)+\gamma(x)-1\big{)}^{2}+\frac{1}{4}\sum_{k=1}^{N}M_{k}\big{(}\delta(x)\big{)}^{2}+\frac{1}{4}M_{N+1}\big{(}\delta(x)\big{)}^{2}+R_{N+1}(x)\Big{]}.$
(5.18)

with $R_{N}(x)$ given by (3) where $f\big{(}\delta(x)\big{)}$ is replaced by
$f_{N+1}\big{(}\delta(x)\big{)}$. $R_{N+1}(x)$ satisfies (3.12) hence there
exists $0<\nu_{N}<\frac{\nu_{0}}{2}$ such that on $\Gamma_{\nu_{N}}$

$\frac{1}{4}M_{N+1}\big{(}\delta(x)\big{)}^{2}+R_{N+1}(x)\geqslant 0.$ (5.19)

∎

The subleading term, $\sum_{k=1}^{N}M_{k}\big{(}\delta(x)\big{)}^{2}$, in
(5.16) becomes relevant when the leading term vanishes. An example where this
is the case is the ”standard” weak weighted/ boundary level Hardy inequality
i.e. ${\mathbb{D}}(x)=\delta(x)^{\beta}\mathds{1},\;\rho(x)=1$ when (3.2)
becomes (see also [29])

$h_{0}\big{[}\varphi,\varphi\big{]}\geqslant-
c_{N}(\nu)\|\chi_{\Omega\setminus\Gamma{\nu}}\varphi\|^{2}+\Big{(}\frac{\beta-1}{2}\Big{)}^{2}\int_{\Gamma_{\nu}}\delta(x)^{\beta-2}|\varphi(x)|^{2}\rho(x)dx.$
(5.20)

Notice that for $\beta=1$ (5.20) gives no information while Theorem 5.2 gives
(for sufficient small $\nu$)

$h_{0}\big{[}\varphi,\varphi\big{]}\geqslant-
c_{N}(\nu)\|\chi_{\Omega\setminus\Gamma{\nu}}\varphi\|^{2}+\frac{1}{4}\int_{\Gamma_{\nu}}\frac{1}{\delta(x)}\big{(}\sum_{k=1}^{N}M_{k}\big{(}\delta(x)\big{)}^{2}\big{)}|\varphi(x)|^{2}\rho(x)dx.$
(5.21)

3\. Theorem 3.1 can be generalised to the case when logarithmic factors are
added in the behaviour of $d(x)$ (see (2)) as $x\rightarrow\partial\Omega$ and
we give below such an extension.

######  Theorem 5.3.

Suppose that Assumptions A and Q with (2.13) replaced with

$d(x)=a(x)\delta(x)^{\beta(x)}\big{(}\text{\rm
ln}\delta(x)\big{)}^{\alpha};\quad
a\in{\mathcal{C}}^{1}\big{(}\Gamma_{\nu_{0}};(0,\infty)\big{)};\quad\beta\in{\mathcal{C}}^{1}\big{(}\Gamma_{\nu_{0}};{\mathbb{R}}\big{)},\;\alpha\in{\mathbb{R}}.$
(5.22)

hold true. Then there exists $0<\nu_{\alpha}<\frac{\nu_{0}}{2}$ such that for
for all $0<\nu<\nu_{\alpha}$ and
$\varphi\in{\mathcal{C}}^{\infty}_{0}(\Omega)$:

$h_{0}\big{[}\varphi,\varphi\big{]}\geqslant-
c_{1}(\nu)\|\chi_{\Omega\setminus\Gamma{\nu}}\varphi\|^{2}+\int_{\Gamma_{\nu}}{\mathcal{H}}_{\alpha}(x)|\varphi(x)|^{2}\rho(x)dx$
(5.23)

where $c_{1}(\nu)<\infty$, $\chi_{\Omega\setminus\Gamma{\nu}}$ is the
characteristic function of $\Omega\setminus\Gamma{\nu}$ and

${\mathcal{H}}_{\alpha}(x)=\frac{1}{4}\big{(}1-q(x)\big{)}a(x)\delta(x)^{\beta(x)-2}\Big{[}\big{(}\beta(x)+\gamma(x)-1-\frac{\alpha}{\text{\rm
ln}\frac{1}{\delta(x)}}\big{)}^{2}+\frac{1-2\alpha-\alpha^{2}}{\text{\rm
ln}\frac{1}{\delta(x)}^{2}}\Big{]}.$ (5.24)

In particular if $\varphi\in{\mathcal{C}}^{\infty}_{0}(\Gamma_{\nu})$

$h_{0}[\varphi,\varphi]\geqslant\int_{\Gamma_{\nu}}{\mathcal{H}}_{\alpha}(x)|\varphi(x)|^{2}\rho(x)dx.$
(5.25)

###### Proof.

Let $\tilde{\nu}_{0}=\min\\{\nu_{0},\frac{1}{e^{e}}\\}$ and on
$\Gamma_{\tilde{\nu}_{0}}$

$\widetilde{\boldsymbol{X}}_{\alpha}=\frac{1}{2}\big{(}1-q(x)\big{)}a(x)r(x)\delta(x)^{\beta(x)+\gamma(x)-1}\big{(}\text{\rm
ln}\frac{1}{\delta(x)}\big{)}^{\alpha}\big{[}\beta(x)+\gamma(x)-1+f_{1,2}\big{(}\delta(x)\big{)}\big{]}\nabla\delta(x)$
(5.26)

where (see (5.6), (5.7))

$f_{1,2}(t)=\frac{1}{\text{\rm ln}\frac{1}{t}}+\frac{1}{\big{(}\text{\rm
ln}\frac{1}{t}\big{)}\big{(}\text{\rm ln}\text{\rm ln}\frac{1}{t}\big{)}}.$
(5.27)

Defining ${\boldsymbol{X}}_{\alpha}$ in terms of
$\widetilde{\boldsymbol{X}}_{\alpha}$ as in the proof of Theorem 3.1 with
$\nu_{0}$ replaced by $\tilde{\nu}_{0}$, repeating the computations and using
the Lemma 5.1 one obtains that on $\Gamma_{\frac{\tilde{\nu}_{0}}{2}}$

$\displaystyle\nabla\cdot{\boldsymbol{X}}_{\alpha}(x)-{\boldsymbol{X}}_{\alpha}(x)\cdot\big{(}\rho(x)\mathbb{D}(x)\big{)}^{-1}{\boldsymbol{X}}_{\alpha}(x))=\big{(}1-q(x)\big{)}a(x)\rho(x)\delta(x)^{\beta(x)-2}\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{\alpha}$
$\displaystyle\Big{\\{}\Big{[}\frac{1}{4}\Big{(}\beta(x)+\gamma(x)-1-\alpha\frac{1}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+\frac{1-2\alpha-\alpha^{2}}{\text{\rm
ln}\frac{1}{\delta(x)}^{2}}+$ $\displaystyle\Big{(}\frac{1}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}\frac{1}{\text{\rm ln}\text{\rm
ln}\frac{1}{\delta(x)}}\Big{(}1-\alpha\frac{1}{\text{\rm ln}\text{\rm
ln}\frac{1}{\delta(x)}}\big{)}\Big{]}+R_{\alpha}(x)\Big{\\}}.$ (5.28)

with

$\sup_{x\in\Gamma_{\frac{\tilde{\nu}_{0}}{2}}}\big{|}R_{\alpha}(x)\big{|}\delta(x)^{1-s}<\infty.$
(5.29)

Then (5.23) with (5.24) follows choosing ${\nu_{\alpha}}$ small enough as to
have on $\Gamma_{\frac{\tilde{\nu}_{0}}{2}}$

$\frac{1}{4}\Big{(}\frac{1}{\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}\frac{1}{\text{\rm ln}\text{\rm
ln}\frac{1}{\delta(x)}}\Big{(}1-\alpha\frac{1}{\text{\rm ln}\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}+R_{\alpha}(x)\geqslant 0.$ (5.30)

∎

4\. For the sake of simplicity we assumed that $\Omega$ is bounded, simply
connected and with smooth boundary. It is not hard to generalise Theorems 3.1
and 5.2 to the case when boundedness and simply connectedness conditions are
relaxed. Still in order to control the behaviour at infinity uniformity
conditions are to be imposed. We follow [10] at this point. More precisely,
let $\Gamma\subset{\mathbb{R}}^{d}$ be a $k$-dimensional,
$k=0,1,2,...,d-1$ differentiable manifolds without boundary. To each point
$x^{*}\in\Gamma$ we associate a Cartesian system of coordinates in which the
first $k$ coordinate axes are in the tangent space to $\Gamma$ at $x^{*}$. For
given $0<r,\;C<\infty$ we say that $\Gamma\in{\mathcal{F}}_{r,C}$ if for every
$x^{*}\in\Gamma$ there exists a sphere a radius $r$ with centre at $x^{*}$
such that in this system of coordinates
$\Gamma\cap\big{\\{}\big{|}x-x^{*}\big{|}<r\big{\\}}$ is given by the
equations $x_{j}=f_{j}\big{(}x_{1},...,x_{k}\big{)};\;j=k+1,...,d$ where
$f_{j}\in{\mathcal{C}}\big{(}{\mathbb{R}}^{d};{\mathbb{R}}\big{)}$ and
$\big{|}D^{\alpha}f_{j}(x_{1},...,x_{k})\big{|}\leqslant C$ for all
multiindices $\alpha,\;|\alpha|\leqslant 2$.

For example, in this setting Theorem 3.1 takes the form:

######  Theorem 5.4.

Let $\Omega\subset{\mathbb{R}}^{d}$ such that
$\partial\Omega=\cup_{j=1}^{N}\Gamma_{j},\;N\leqslant\infty;\;\Gamma_{j}\in{\mathcal{F}}_{r,C};\;\text{dist}\big{(}\Gamma_{j},\Gamma_{k}\big{)}\geqslant
d_{0}>0$.

Suppose that Assumption A and Q hold true with
$\nu_{\Omega}\leqslant\frac{d_{0}}{4}$ , and in addition

$(1-q)^{-1},\;a,\;r,\;\frac{1}{a},\;\frac{1}{r},\;\beta,\;\gamma\in
L^{\infty}\big{(}\Gamma_{\nu_{0}}\big{)}.$ (5.31)

Then there exists $0<\nu_{1}<\frac{\nu_{0}}{2}$ such that for all
$0<\nu<\nu_{1}$ and $\varphi\in{\mathcal{C}}^{\infty}_{0}(\Omega)$:

$\displaystyle
h_{0}\big{[}\varphi,\varphi\big{]}=\int_{\Omega}\overline{\nabla\varphi(x)}\cdot\big{(}{\mathbb{D}}(x)\nabla\varphi(x)\big{)}\rho(x)dx$
$\displaystyle\geqslant-
c_{1}(\nu)\|\chi_{\Omega\setminus\Gamma_{\nu}}\varphi\|^{2}+\sum_{j=1}^{N}\int_{\Gamma_{j,\nu}}{\mathcal{H}}_{0,j}(x)|\varphi(x)|^{2}\rho(x)dx$
(5.32)

where

$\Gamma_{j,\nu}=\big{\\{}x\in\Omega\big{|}\operatorname{dist}\big{(}x,\Gamma_{j}\big{)}<\nu\big{\\}}$
(5.33)

and

${\mathcal{H}}_{0,j}(x)=\frac{1}{4}\big{(}1-q(x)\big{)}a(x)\delta(x)^{\beta(x)-2}\Big{[}\big{(}\beta(x)+\gamma(x)+d-k_{j}-2\big{)}^{2}+\frac{1}{2}\frac{1}{\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{2}}\Big{]}.$ (5.34)

In addition if $\varphi\in{\mathcal{C}}^{\infty}_{0}(\Gamma_{j,\nu})$

$h_{0}[\varphi,\varphi]\geqslant\int_{\Gamma_{j,\nu}}{\mathcal{H}}_{0,j}(x)|\varphi(x)|^{2}\rho(x)dx.$
(5.35)

Outline of proof. With the following changes the proof of Theorem 5.4 mimics
closely the proof of Theorem 3.1.

i. Instead of Lemma 2.1 one has to use its generalisation (see Lemma 6.1 and
Lemma 6.2 in [10], see also Theorem 3.2 in [4])

######  Lemma 5.5.

There exists $0<\nu_{\Omega}\leqslant\frac{d_{0}}{4}$ and $C_{\Delta}<\infty$
such that $\delta\in{\mathcal{C}}^{2}\big{(}\Gamma_{\nu_{\Omega}}\big{)}$ and
for $x\in\Gamma_{j,\nu_{\Omega},},$

$\big{|}\nabla\delta(x)\big{|}=1;\quad\big{|}\Delta\delta(x)-\frac{d-k_{j}-1}{\delta(x)}\big{|}\leqslant
C_{\Delta}.$ (5.36)

where $k_{j}$ is the dimension of $\Gamma_{j}$.

ii. The choice of $\widetilde{\boldsymbol{X}}_{0}(x)$ in (3.5) is now as
follows.

For $x\in\Gamma_{j,\nu_{0}},$:

$\widetilde{\boldsymbol{X}}_{0}=\frac{1}{2}\big{(}1-q(x)\big{)}a(x)\rho(x)\delta(x)^{\beta(x)-1}\big{(}\beta(x)+\gamma(x)+d-k_{j}-2+f\big{(}\delta(x)\big{)}\big{)}\nabla\delta(x).$
(5.37)

Accordingly, for $x\in\Gamma_{j,\nu_{0}}\;R_{0}(x)$ in (3) rewrites as:

$R_{0}(x)=\frac{1}{2}\big{(}\beta(x)+\gamma(x)+d-k_{j}-1\big{)}+\mathcal{O}\big{(}\delta(x)^{\frac{1-s}{2}}\big{)}.$
(5.38)

5\. Together with (3.2) giving the Hardy barrier, the inequalities (4.22),
(4.49) which, for domains with ${\mathcal{C}}^{2}$ boundary, shows how the
coefficients of $H$ merge together to ensure essential self-adjointness, are
the main results of this paper. As in the case of Hardy barrier the essential
self-adjointness condition is local in the sense that it is expressed in terms
of the infimum over boundary layers of local quantities. Notice that one can
have essential self-adjointness of $H$ even for negative (deconfining!)
potentials and essential self-adjointness of $H_{0}$ when $\beta(x)$ is
negative i.e. ${\mathbb{D}}(x)$ blows up as $x\rightarrow\partial\Omega$. What
our results add to the previous ones is on one hand that the degree,
$\beta(x)$, of degeneracy of ${\mathbb{D}}(x)$ is allowed to vary over
$\partial\Omega$ thus giving a positive answer to a question in [27] and on
the other hand the isotropy of ${\mathbb{D}}(x)$ in the limit
$x\rightarrow\partial\Omega$ is weakened to the assumption that it has near
$\partial\Omega$ a $x$-dependent $2\times 2$ block structure dominated by the
diagonal (see (2), (2.19) and (2.21)).

6\. Having the Hardy barrier from Theorem 5.4 one can extend also the criteria
for essential self-adjointness to unbounded multiply connected domains with
${\mathcal{C}}^{2}$ boundary. For simplicity we only give the analog of
Theorem 4.5 for $V=0$.

######  Theorem 5.6.

In the setting of Theorem 5.4 suppose suppose that

$\big{(}1+|\cdot|\big{)}^{2}{\mathbb{D}}\in
L^{\infty}\big{(}\Omega\setminus\Gamma_{\nu_{0}}\big{)},$ (5.39)

and in addition there exists

$0<\mu<\frac{\nu_{0}}{2}$ (5.40)

such that on $\Gamma_{\mu,j}$ either:

i.

$\beta(x)=\beta_{j}\geqslant 2$ (5.41)

or

ii

$\beta(x)=\beta_{j}<2,\quad\inf_{x\in\Gamma_{\mu,j}}\big{(}1-q(x)\big{)}\Big{(}\frac{\beta_{j}+\gamma(x)+d-k_{j}-2}{\beta_{j}-2}\Big{)}^{2}\geqslant
1.$ (5.42)

Then $H_{0}$ is essentially self-adjoint.

###### Proof.

The verification of Lemma 4.1 and Lemma 4.2 follows the one in Theorem 4.3 in
the case i. and the one in Theorem 4.5 in the case ii. respectively. Since for
unbounded domains $\phi_{l}$ as defined in (4.8) do not have compact support
one has to replace (4.8) with

$\phi_{l}(x)=k_{l}\big{(}\delta(x)\big{)}k_{\infty}\big{(}\frac{|x|}{l}\big{)}$
(5.43)

where $k_{l}$ is given by (4.6) and

$k_{\infty}(t)\in{\mathcal{C}}^{1}\big{(}(0,\infty)\big{)};\quad\operatorname{supp}k^{\prime}_{\infty}\subset(1,2);\quad
k_{\infty}(t)=\begin{cases}0\,,&\quad t>2\\\ 1\,,&\quad t<1,\end{cases}$
(5.44) $\big{|}k^{\prime}_{\infty}(t)\big{|}\leqslant 2.$ (5.45)

Now, notice that as $|x|\rightarrow\infty$, $\big{|}\nabla
k_{\infty}\big{(}\frac{|x|}{l}\big{)}\big{|}\sim\frac{1}{l}$ which together
with (5.39) implies the boundedness of
$\nabla\phi_{l}\cdot\big{(}{\mathbb{D}}\nabla\phi_{l}\big{)}$ as
$|x|\rightarrow\infty$.

∎

7\. Building on methods and results in [24] and [27] D. W. Robinson [28] made
a thorough study of essential self-adjointness of $H_{0}$ for the case
$\rho=1$, $\beta=const$, ${\mathbb{D}}\in
L^{\infty}(\Omega:{\mathbb{R}}^{d\times d})$ and is isotropic in the limit
$\delta(x)\rightarrow 0$:

$\inf_{r\in(0,r_{0}]}\sup_{\Gamma_{r}}\|{\mathbb{D}}(x)\delta(x)^{-\beta}-d(x)\mathds{1}\|=0.$
(5.46)

For the sake of simplicity of the discussion below we state his main result
for the particular case when $\Omega$ is bounded, simply connected and with
${\mathcal{C}}^{2}$ boundary.

######  Theorem 5.7.

Suppose that ${\mathbb{D}}$ is Lipschitz continuous and there exists $r_{0}>0$
such that (5.46) holds true with $d$ Lipschitz continuous
$d,\;\frac{1}{d},\;|\nabla d|\in L^{\infty}\big{(}\Gamma_{r_{0}}\big{)}$.
Then:

i. For $\beta>\frac{3}{2}$ $H_{0}$ is essentially self-adjoint.

ii. If in addition

$\sup_{\Gamma_{r_{0}}}\big{|}\big{(}\nabla{\mathbb{D}}\delta\big{)}(x)\cdot\nabla\delta(x)\big{|}<\infty,$
(5.47)

where

$\big{(}\nabla{\mathbb{D}}\delta^{-\beta}\big{)}_{j}:=\sum_{k=1}^{d}\partial_{k}\big{(}\nabla{\mathbb{D}}\delta^{-\beta}\big{)}_{kj},$
(5.48)

$H_{0}$ is not essentially self-adjoint for $\beta<\frac{3}{2}$.

Theorem 5.7 provides the answer (at the level of power behaviour) to the
question of dependence of essential self-adjointness as a function of the
degree of degeneracy of ${\mathbb{D}}(x)$ as $x\rightarrow\partial\Omega$,
except for the critical value, $\beta=\frac{3}{2}$, which remains undecided.
At the price of a quantitative version of (5.46) the following proposition
fills this gap.

######  Proposition 5.8.

Let $\Omega$ is bounded, simply connected and with ${\mathcal{C}}^{2}$
boundary. Suppose that
${\mathbb{D}}\in{\mathcal{C}}^{1}(\Omega;{\mathbb{R}}^{d\times d})$ and there
exists $r_{0}>0,\;s<1$ such that on $\Gamma_{r_{0}}$

$\|{\mathbb{D}}(x)\delta(x)^{-\beta}-d(x)\mathds{1}\|\leqslant
c_{r_{0}}\delta(x)^{1-s}$ (5.49)

with $d\in{\mathcal{C}}^{1}(\Omega)$ and $d,\;\frac{1}{d},\;|\nabla d|\in
L^{\infty}\big{(}\Gamma_{r_{0}}\big{)}$. Then $H_{0}$ is essentially self-
adjoint for $\beta\geqslant\frac{3}{2}$.

###### Proof.

Since the case $\beta\geqslant 2$ is covered by Theorem 4.3 one can assume
that $\beta<2$. Remark first that from (5.56) and $d,\;\frac{1}{d},\;\in
L^{\infty}\big{(}\Gamma_{r_{0}}\big{)}$ it follows that if
$\widetilde{\mathbb{D}}=d\delta^{\beta}$ then

${\mathbb{D}}=\widetilde{\mathbb{D}}+\delta^{\beta}{\mathcal{O}}\big{(}\delta^{1-s}\big{)}$
(5.50)

and for sufficiently small $\delta(x)$,

${\mathbb{D}}^{-1}=\widetilde{\mathbb{D}}^{-1}+\delta^{-\beta}{\mathcal{O}}\big{(}\delta^{1-s}\big{)}.$
(5.51)

Repeat the proof of Theorem 3.1 with $\widetilde{\boldsymbol{X}}_{0}$
corresponding to $\widetilde{\mathbb{D}}$. i.e.
$\widetilde{\boldsymbol{X}}_{0}=\frac{1}{2}\big{(}1-q(x)\big{)}d(x)\delta(x)^{\beta-1}\big{(}\beta-1+f\big{(}\delta(x)\big{)}\big{)}\nabla\delta(x)$.
Notice that the ”perturbation” ${\mathcal{O}}\big{(}\delta^{1-s}\big{)}$ enter
only in the computation
$\widetilde{\boldsymbol{X}}_{0}\cdot\big{(}{\mathbb{D}}^{-1}\widetilde{\boldsymbol{X}}_{0}\big{)}=\widetilde{\boldsymbol{X}}_{0}\cdot\big{(}\widetilde{\mathbb{D}}^{-1}\widetilde{\boldsymbol{X}}\big{)}+\delta^{\beta-2}{\mathcal{O}}\big{(}\delta^{1-s}\big{)}$
hence one obtains (3.2) with $q=0$. As concerning $g$ due to the perturbation
term in (5.50) we need we need a better choice than in Theorem 4.5. More
precisely we take

$g(x)=\Big{(}\frac{2-\beta}{2}\text{\rm ln}\delta(x)+\frac{1}{2}\text{\rm
ln}\text{\rm ln}\frac{1}{\delta(x)}\Big{)}\Psi(x)$ (5.52)

where $\Psi$ (see (3.8)) is the same as in the proof of Theorem 3.1. Next, we
compute $\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla g(x)\big{)}$on
$\Gamma_{\mu}$. For $\delta(x)<\frac{\nu_{0}}{2}$, $\Psi(x)=1$ hence for
$x\in\Gamma_{\mu}$

$\nabla g(x)=\frac{2-\beta}{2}\delta(x)^{-1}\Big{(}1-\frac{1}{2\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}\nabla\delta(x)$ (5.53)

which together with (5.50) leads to

$\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla
g(x)\big{)}=\Big{(}\frac{2-\beta}{2}\Big{)}^{2}\delta(x)^{\beta-2}\Big{[}\Big{(}1-\frac{1}{2\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+c_{iso}(\nu_{0})\delta^{1-s}(x)\Big{]}.$
(5.54)

Noticing that for $\delta(x)$ sufficiently small

$\Big{(}1-\frac{1}{2\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+c_{iso}(\nu_{0})\delta^{1-s}(x)\leqslant 1$
(5.55)

from this point the proof repeats the proof of Theorem 4.5. Since
$V=q=0,\;\rho=1$, (4.22) gives the condition
$\Big{(}\frac{\beta-1}{\beta-2}\Big{)}^{2}\geqslant 1$ hence
$\beta\geqslant\frac{3}{2}$. ∎

######  Remark 5.9.

A similar argument leads to the fact that if (5.46) holds true then Theorem
4.5 implies Theorem 5.7 i. i.e., $H_{0}$ is essentially self-adjoint for
$\beta>\frac{3}{2}$.

At the level of power behaviour and for ${\mathbb{D}}$ asimptotically
isotropic as $x\rightarrow\partial\Omega$ the problem of dependence of self-
adjointness of $H_{0}$ on the degeneracy of ${\mathbb{D}}$ is completely
solved by Theorem 5.7 and Proposition 5.8. The transition from essential self-
adjoint to not essential self-adjoint takes place at the critical value
$\beta=\frac{3}{2}$. It is an interesting problem to ”zoom” the neighbourhood
of the critical value and to obtain a more precise description of the
essential self-adjoint to not essential self-adjoint transition. In
particular, assuming that as $x\rightarrow\partial\Omega$,
${\mathbb{D}}(x)\sim\delta(x)^{\frac{3}{2}}\big{(}\text{\rm
ln}\frac{1}{\delta(x)}\big{)}$ the problem is to find the critical value,
$\alpha_{c}$, of $\alpha$ where the transition essential self-adjoint to not
essential self-adjoint takes place. Notice that from Proposition 5.8
$\alpha=0$ lies in the essential self-adjoint region.

######  Theorem 5.10.

Let $\Omega$ is bounded, simply connected and with ${\mathcal{C}}^{2}$
boundary. Suppose that
${\mathbb{D}}\in{\mathcal{C}}^{1}(\Omega;{\mathbb{R}}^{d\times d})$ and there
exists $r_{0}>0,\;s<1$ such that on $\Gamma_{r_{0}}$

$\|{\mathbb{D}}(x)\delta(x)^{-\frac{3}{2}}\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\big{)}^{-\alpha}-d(x)\mathds{1}\|\leqslant
c_{r_{0}}\delta(x)^{1-s}$ (5.56)

with $d\in{\mathcal{C}}^{1}(\Omega)$ and $d,\;\frac{1}{d},\;|\nabla d|\in
L^{\infty}\big{(}\Gamma_{r_{0}}\big{)}$. Then $H_{0}$ is essentially self-
adjoint for $\alpha\leqslant\frac{1}{4}$.

###### Proof.

Proceeding as in the proof of Proposition 5.8 we obtain from Theorem 5.3 that
for $\nu$ sufficiently small

$h_{0}\big{[}\varphi,\varphi\big{]}\geqslant-
c_{1}(\nu)\|\chi_{\Omega\setminus\Gamma{\nu}}\varphi\|^{2}+\int_{\Gamma_{\nu}}{\mathcal{H}}_{\alpha}(x)|\varphi(x)|^{2}\rho(x)dx$
(5.57)

with

$\displaystyle{\mathcal{H}}_{\alpha}(x)=$
$\displaystyle\frac{1}{4}d(x)\delta(x)^{-\frac{1}{2}}\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{\alpha}\Big{[}\big{(}\frac{1}{2}-\frac{\alpha}{\text{\rm
ln}\frac{1}{\delta(x)}}\big{)}^{2}+\frac{1-2\alpha-\alpha^{2}}{\big{(}\text{\rm
ln}\frac{1}{\delta(x)}\big{)}^{2}}\Big{]}=$
$\displaystyle\frac{1}{16}d(x)\delta(x)^{-\frac{1}{2}}\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{\alpha}\Big{(}1-\frac{4\alpha}{\text{\rm
ln}\frac{1}{\delta(x)}}+\frac{4(1-2\alpha)}{\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{2}}\Big{)}.$ (5.58)

Choosing $g(x)$ as in Proposition 5.8 one obtains in this case

$\displaystyle\nabla g(x)\cdot\big{(}{\mathbb{D}}(x)\nabla g(x)\big{)}=$
$\displaystyle\frac{1}{16}\delta(x)^{-\frac{1}{2}}\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{\alpha}\Big{[}\Big{(}1-\frac{1}{2\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+c_{iso}(\nu_{0})\delta^{1-s}(x)\Big{]}.$
(5.59)

Then as in Theorem 4.5 the essential self-adjointness of $H_{0}$ for
$\alpha\leqslant\frac{1}{4}$ follows from the fact that for sufficient small
$\delta(x)$

$\frac{1-\frac{4\alpha}{\text{\rm
ln}\frac{1}{\delta(x)}}+\frac{4(1-2\alpha)}{\Big{(}\text{\rm
ln}\frac{1}{\delta(x)}\Big{)}^{2}}\Big{)}}{\Big{(}1-\frac{1}{2\text{\rm
ln}\frac{1}{\delta(x)}}\Big{)}^{2}+c_{iso}(\nu_{0})\delta^{1-s}(x)}\geqslant
1.$ (5.60)

∎

8\. During the last decade U. Boscain and his collaborators made a detailed
study of essential self-adjointness (aka quantum confinement) of Laplace-
Beltrami operator on two-dimensional manifolds endowed with almost Riemannian
structures (2-ARS) i.e. Riemannian structures which become singular on some
one-dimensional submanifolds; we send the reader to the recent paper [5] for a
detailed discussion and further references.

Since, as noticed in [24], the drift-diffussion operator can be viewed (up to
a sign) as Laplace-Beltrami operator on weighted Riemannian manifolds it is a
natural questions whether the results on drift-diffusion operator are related
to the results for Laplace-Beltrami operators 2-ARS. Let us notice first that
the result in [24] as well as those in [27], [28] do not apply to the setting
in [5] since they require ${\mathbb{D}}(x)$ to be in a neighbourhood of the
isotropic case (see Comment 6 above). On the contrary since the results in
this paper do not require ${\mathbb{D}}(x)$ to be almost isotropic Corollary
5.11 below can applied to the crucial step in proving the essential self-
adjointness of Laplace-Beltrami operator on 2-ARS [6], [5], namely the
analysis in a tubular neighbourhood of the singular set.

Let $\Omega$ be the unit disc in ${\mathbb{R}}^{2}$,
$\Omega=\big{\\{}x=(x_{1},x_{2})\big{|}|x|<1\big{\\}}$ and for $0<\nu_{0}<1$,

$\Gamma_{\nu_{0}}=\big{\\{}x\in\Omega\big{|}\big{(}1-|x|\big{)}<\nu_{0}\big{\\}}.$

Consider on $\Omega$ a Riemannian distance such that on $\Gamma_{\nu_{0}}$:

$ds^{2}=dr^{2}+\frac{1}{(1-r)^{2\alpha}e^{2\Phi(r,\theta)}}d\theta^{2},$
(5.61)

where $\alpha\in{\mathbb{R}};\;r,\theta$ are the standard polar coordinates in
$\Omega$ and
$\Phi\in{\mathcal{C}}^{\infty}\big{(}\Gamma_{\nu_{0}};{\mathbb{R}}\big{)}$
with uniformly bounded second order derivatives. On $\Gamma_{\nu_{0}}$ the
Riemannian volume $\omega_{\alpha}$, Laplace-Beltrami operator
$\Delta_{\alpha}$ and the curvature $K_{\alpha}$ have the form:

$\omega_{\alpha}=\frac{1}{f_{\alpha}(r,\theta)}drd\theta,$ (5.62)

$\Delta_{\alpha}=\partial_{r}^{2}+f_{\alpha}^{2}\partial_{\theta}^{2}+\frac{\big{(}\partial_{r}f_{\alpha}\big{)}}{f_{\alpha}}\partial_{r}+f_{\alpha}\big{(}\partial_{r}f_{\alpha}\big{)}\partial_{\theta},$
(5.63)

$K_{\alpha}(r,\theta)=\frac{f_{\alpha}(r,\theta)\big{(}\partial_{r}^{2}f_{\alpha}(r,\theta)\big{)}-4\big{(}\partial_{r}f_{\alpha}(r,\theta)\big{)}^{2}}{f_{\alpha}(r,\theta)^{2}}$
(5.64)

where

$f_{\alpha}(r,\theta)\big{)}=(1-r)^{\alpha}e^{\Phi(r,\theta)}.$ (5.65)

Consider now the operator [5]

$H_{\alpha,c}=-\Big{(}\frac{1}{2}\Delta_{\alpha}+cK_{\alpha}\Big{)};\quad
c\in{\mathbb{R}};\quad{\mathcal{D}}(H_{\alpha,c})={\mathcal{C}}_{0}^{\infty}(\Omega).$
(5.66)

######  Corollary 5.11.

Suppose that

$\Big{(}\frac{1+\alpha}{2}\Big{)}^{2}-2c\alpha(1+\alpha)\geqslant 1.$ (5.67)

Then $H_{\alpha,c}$ is essentially self-adjoint in
$L^{2}\big{(}\Omega;\omega_{\alpha}(x)dx\big{)}$.

###### Proof.

By direct computation, on $\Gamma_{\nu_{0}}$:

$K_{\alpha}(r,\theta)=-\frac{\alpha(\alpha+1)}{(1-r)^{2}}\big{(}1+W_{\alpha}(r,\theta)\big{)}$
(5.68)

with

$\sup_{x\in\Gamma_{\nu_{0}}}\big{|}W_{\alpha}(r,\theta)\big{|}(1-r)=w(\alpha)<\infty.$
(5.69)

On $\Gamma_{\nu_{0}}$, with the identifications:

$\rho(x)=\frac{1}{(1-r)^{\alpha}re^{\Phi(r,\theta)}},$ (5.70)

${\mathbb{D}}(x)=\frac{1}{2}\Big{(}{\mathbb{P}}_{\mathbf{e}_{r}}+(1-r)^{2\alpha}r^{2}e^{2\Phi(r,\theta)}{\mathbb{P}}_{\mathbf{e}_{\theta}}\Big{)}$
(5.71)

where ${\mathbb{P}}_{\mathbf{e}_{r}},\;{\mathbb{P}}_{\mathbf{e}_{\theta}}$ are
the orthogonal projections in ${\mathbb{R}}^{2}$ along
$\mathbf{e}_{r}=(\cos\theta,\sin\theta),\;\mathbf{e}_{\theta}=(-\sin\theta,\cos\theta)$
respectively and

$V(x)=K_{\alpha}(r,\theta),$ (5.72)

$H_{\alpha,c}$ has the form (2.7).

From (5.70)-(5.72) $\rho,\;{\mathbb{D}},\;V$ satisfy Asssumption A and (4.21)
(notice that $\delta(x)=(1-r)$) with
$\beta=0,\;\gamma(x)=-\alpha,\;a(x)=\frac{1}{2},\;v(x)=-c\alpha(\alpha+1),s=0$
and $q(x)=0$. Then Corollary 5.11 follows from Theorem 4.5ii. ∎

Particular cases of Corrolary 5.11 are related to the sufficient conditions
for esential self-adjointnes in [6], [5]; in the case $\alpha=1$,
$\Phi(1,\theta)=0$ it gives another proof of Theorem 1 in [6] while the case
$\Phi(r,\theta)=0$ corresponds to the sufficient part of Proposition 4 in [5].
One may speculate that the results in Section 4 can provide sufficient
conditions ensuring essential self-adjointness of Laplace-Beltrami operator on
larger classes of ARS in $d\geqslant 2$.

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