# Spectral gap estimates for Brownian motion on domains with sticky-reflecting
boundary diffusion

Vitalii Konarovskyi∗†, Victor Marx∗, and Max von Renesse∗

###### Abstract

Introducing an interpolation method we estimate the spectral gap for Brownian
motion on general domains with sticky-reflecting boundary diffusion associated
to the first nontrivial eigenvalue for the Laplace operator with corresponding
Wentzell-type boundary condition. In the manifold case our proofs involve
novel applications of the celebrated Reilly formula.

††$*$ Universität Leipzig, Fakultät für Mathematik und Informatik,
Augustusplatz 10, 04109 Leipzig, Germany; $\dagger$ Universität Hamburg,
Bundesstraße 55, 20146 Hamburg, Germany; Institute of Mathematics of NAS of
Ukraine, Tereschenkivska st. 3, 01024 Kiev, Ukraine<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>†† _Mathematics Subject
Classification (2020):_ Primary 26D10, 35A23, 34K08 ; Secondary 46E35, 53B25,
60J60, 47D07.

##  1 Introduction and statement of main results

Brownian motion on smooth domains with sticky-reflecting diffusion along the
boundary has a long history, dating back at least to Wentzell [34]. As a
prototype consider a diffusion on the closure $\overline{\Omega}$ of a smooth
domain $\Omega$ with Feller generator $(\mathcal{D}(A),A)$

$\displaystyle\mathcal{D}(A)=\\{f\in C_{0}(\overline{\Omega})\,|\,Af\in
C_{0}(\overline{\Omega})\\}$ (1.1) $\displaystyle Af=\Delta
f\mathbb{I}_{\Omega}+(\beta\Delta^{\tau}f-\gamma\frac{\partial
f}{\partial\nu})\mathbb{I}_{\partial\Omega}$

where $\frac{\partial}{\partial\nu}$ is the outer normal derivative,
$\Delta^{\tau}$ is the Laplace-Beltrami operator on the boundary
$\partial\Omega$ and $\beta>0,\gamma\in\mathbb{R}$. The case of pure sticky
reflection but no diffusion along the boundary corresponds to the regime
$\beta=0$; models with $\beta>0$ have appeared recently in interacting
particle systems with singular boundary or zero-range pair interaction [1, 7,
13, 19, 27]. The first rigorous process constructions on special domains
$\Omega$ were given in [16, 33, 37] and were later extended to jump-diffusion
processes on general domains [6] cf. [32]. An efficient construction in
symmetric cases was given by Grothaus and Voßhall via Dirichlet forms in [15].
Qualitative regularity properties of the associated semigroups were studied
e.g. in [14]. In this note we address the problem of estimating the spectral
gap for such processes, which is a natural question also in algorithmic
applications. To our knowledge this question has been considered only for
$\beta=0$ by Kennedy [17] and Shouman [30]. However, for $\beta>0$ the
properties of the process change significantly, which is indicated by the fact
that the energy form of $A$ now also contains a boundary part and which also
constitutes the main difference to the closely related work [18].

In the sequel we treat the case when $\gamma>0$ which corresponds to an inward
sticky reflection at $\partial\Omega$. Our ansatz to estimate the spectral gap
is based on a simple interpolation idea. To this aim assume that $\Omega$ and
$\partial\Omega$ have finite (Hausdorff) measure so that we may choose
$\alpha\in(0,1)$ for which

$\frac{\alpha}{1-\alpha}\frac{|\partial\Omega|}{|\Omega|}=\gamma.$

Introducing $\lambda_{\Omega}$ and $\lambda_{\partial}$ as normalized volume
and Hausdorff measures on $\Omega$ and $\partial\Omega$ and setting

$\lambda_{\alpha}=\alpha\lambda_{\Omega}+(1-\alpha)\lambda_{\partial},$

we find that $-A$ is $\lambda_{\alpha}$-symmetric with first nonzero
eigenvalue/spectral gap characterized by the Rayleigh quotient

$\sigma_{\alpha,\beta}=\inf_{\begin{subarray}{c}f\in
C^{1}(\overline{\Omega})\\\
\operatorname{Var}_{\lambda_{\alpha}}(f)>0\end{subarray}}\frac{\mathcal{E}_{\alpha,\beta}(f)}{\operatorname{Var}_{\lambda_{\alpha}}f},$

where

$\operatorname{Var}_{\lambda_{\alpha}}f=\int_{\Omega}f^{2}d\lambda_{\alpha}-\left(\int_{\Omega}fd\lambda_{\alpha}\right)^{2}$

and

$\mathcal{E}_{\alpha,\beta}(f)=\alpha\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}+(1-\alpha)\int_{\partial\Omega}\beta\|\nabla^{\tau}f\|^{2}d\lambda_{\partial},$

and $\nabla^{\tau}$ denotes the tangential derivative operator on
$\partial\Omega$.

This representation of $\sigma_{\alpha,\beta}$ formally interpolates between
the two extremal cases of the spectral gap for reflecting Brownian motion on
$\Omega$ when $\alpha=1$ and for Brownian motion on the surface
$\partial\Omega$ when $\alpha=0$. As our main result, in Proposition 2.1 we
propose a simple method to estimate $\sigma_{\alpha,\beta}$ from below using
only $\sigma_{0}$ and $\sigma_{1}$ and estimates for certain bulk-boundary
interaction terms which are independent of $\alpha$. The method can lead to
quite good results which is illustrated by the example when
$\Omega=B_{1}\subset\mathbb{R}^{d}$ is a $d$-dimensional unit ball. When $d=2$
and $\beta=1$, for instance, it yields the estimate

$\displaystyle\sigma_{\alpha}\geq\frac{8(1+\alpha)\sigma_{\Omega}}{8(1-\alpha)\sigma_{\Omega}+16\alpha+3\alpha(1-\alpha)\sigma_{\Omega}}\mbox{
with }\alpha=\frac{\gamma}{2+\gamma},$

where $\sigma_{0}\approx 3.39$ is the spectral gap for the Neumann Laplacian
on the 2-dimensional unit ball, c.f. Section 3.1. – In case when $\Omega$ is a
$d$-dimensional manifold with Ricci curvature bounded from below by $k_{R}>0$
and with boundary $\partial\Omega$ whose second fundamental form $\mathop{\rm
II}_{\partial\Omega}$ is bounded from below by $k_{2}>0$ we obtain (again with
$\beta=1$, for simplicity) that

$\displaystyle\sigma_{\alpha}\geq\min\left(\frac{dk_{R}}{C_{\Omega}dk_{R}+(1-\alpha)(d-1)},\frac{dk_{R}}{C_{\partial\Omega}}\frac{2(1-\alpha)+\alpha
k_{2}C_{\partial\Omega}}{2(1-\alpha)dk_{R}+\alpha
dk_{2}k_{R}C_{\Omega}+\alpha(1-\alpha)(d-1)k_{2}}\right),$

where $C_{\Omega}$ and $C_{\partial\Omega}$ are the usual (Neumann) Poincaré
constants of $\Omega$ and $\partial\Omega$ respectively. To derive this result
we combine Escobar’s lower bound [9] on the first Steklov eigenvalue [12, 20]
of $\Omega$ with a novel estimate on the optimal zero mean trace Poincaré
constant of $\Omega$ [22, 26], for which we obtain that

$\int_{\Omega}f^{2}dx\leq\frac{d-1}{dk_{R}}\int_{\Omega}|\nabla f|^{2},$

for all $f\in C^{1}(\Omega)$ with $\int_{\partial\Omega}fdS=0$, and which is
of independent interest. The proof is based on a novel application of Reilly’s
formula [28] which is also used for a complementary lower bound of $\sigma$
independent of the interpolation approach stating that

$\sigma_{\alpha}\geq\min\left(\frac{dk_{2}}{3d-1}\frac{\alpha}{1-\alpha}\frac{|\partial\Omega|}{|\Omega|},\frac{d}{d-1}k_{R}\right),$

but which is generally weaker for small values of $\alpha$, c.f. Section 3.2.

The interpolation approach also yields a sufficient condition for the
continuity of $\sigma_{\alpha}$ at $\alpha\in\\{0,1\\}$, which in general may
fail. In Section 2.2 we present sufficient conditions for continuity and
discontinuity of $\sigma_{\alpha}$ at $\\{0,1\\}$ which hints towards a phase
transition in the associated family of variational problems.

We conclude with the discussion of two applications of the method in non-
standard or singular situations, c.f. Sections 3.3 and 3.4.

##  2 An interpolation approach

###  2.1 Generalized framework

It will be convenient to work with a slight generalisation of the setup above.
To this aim let $\Omega$ be an open domain in $\mathbb{R}^{d}$ or a Riemannian
manifold with a piecewise smooth boundary $\partial\Omega$. Let $\Sigma$ be a
smooth compact and connected subset of $\partial\Omega$. We denote by
$\partial\Sigma$ the boundary of $\Sigma$ in the space $\partial\Omega$, i.e.
$\partial\Sigma=\Sigma\cap\overline{\partial\Omega\backslash\Sigma}$. We
consider two probability measures $\lambda_{\Omega}$ and $\lambda_{\Sigma}$
with support $\Omega$ and $\Sigma$, which are absolutely continuous with
respect to the Lebesgue and the Hausdorff measure on $\Omega$ and $\Sigma$,
respectively.

Let $\mathop{D}:C^{1}(\Omega)\mapsto\Gamma^{0}(\Omega)$ and
${\mathop{D}}^{\tau}:C^{1}(\partial\Omega)\mapsto\Gamma^{0}(\partial\Omega)$
denote given first order gradient operators mapping differentiable functions
into (tangential) vector fields on $\Omega$ and on $\partial\Omega$,
respectively, and for $\alpha\in[0,1]$ let

$\displaystyle\lambda_{\alpha}$
$\displaystyle:=\alpha\lambda_{\Omega}+(1-\alpha)\lambda_{\Sigma},$
$\displaystyle\mathcal{E}_{\alpha}(f)$
$\displaystyle:=\alpha\int_{\Omega}\|\mathop{D}f\|^{2}d\lambda_{\Omega}+(1-\alpha)\int_{\Sigma}\|{\mathop{D}}^{\tau}f\|^{2}d\lambda_{\Sigma},\quad
f\in\mathcal{D}_{0},$

where $\mathcal{D}_{0}\subset\mathcal{C}^{1}(\overline{\Omega})$ is dense in
$C_{0}(\Omega)$. We assume that for $\alpha\in[0,1]$ the quadratic form
$(\mathcal{E}_{\alpha},\mathcal{D}_{0})$ is a pre-Dirichlet form on
$L^{2}(\overline{\Omega},\lambda_{\alpha})$ whose closure we shall denote by
$(\mathcal{E}_{\alpha},\mathcal{D})$, c.f. [15] for details. We wish to
estimate from above $\sigma_{\alpha}^{-1}=C_{\alpha}$, where $C_{\alpha}$ is
the optimal Poincaré constant given by

$C_{\alpha}:=\sup_{\begin{subarray}{c}f\in\mathcal{D}_{0}\\\
\mathcal{E}_{\alpha}(f)>0\end{subarray}}\frac{\operatorname{Var}_{\lambda_{\alpha}}f}{\mathcal{E}_{\alpha}(f)}.$
(2.1)

In the interpolation method presented below it is assumed that $C_{\alpha}$
are known or can be estimated at the two extremals $\alpha\in\\{0,1\\}$. For
instance, when $\mathop{D}=\nabla$, ${\mathop{D}}^{\tau}=\nabla^{\tau}$ are
the standard gradient resp. tangential gradient operators and
$\lambda_{\Omega}$ and $\lambda_{\Sigma}$ are normalized Lebesgue resp.
Hausdorff measures on $\Omega$ and $\Sigma\subset\partial\Omega$,
$C_{\Omega}:=C_{1}$ is the optimal Poincaré constant associated to the Laplace
operator on $\Omega$ with Neumann boundary conditions, whereas
$C_{\Sigma}:=C_{0}$ is the optimal Poincaré constant associated to the
Laplace-Beltrami operator on $\Sigma$ with Neumann boundary conditions on
$\partial\Sigma$.

The following proposition establishes an estimate of $C_{\alpha}$ in terms of
$C_{\Omega}$ and $C_{\Sigma}$.

######  Proposition 2.1.

Assume there exists constants $K_{\Sigma,\Omega}$, $K_{1},K_{2}$ such that for
any $f\in\mathcal{D}_{0}$

$\displaystyle\operatorname{Var}_{\lambda_{\Sigma}}f\leq
K_{\Sigma,\Omega}\int_{\Omega}\|\mathop{D}f\|^{2}d\lambda_{\Omega},$ (2.2)

and

$\displaystyle\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}\leq
K_{1}\int_{\Omega}\|\mathop{D}f\|^{2}d\lambda_{\Omega}+K_{2}\int_{\Sigma}\|{\mathop{D}}^{\tau}f\|^{2}d\lambda_{\Sigma},$
(2.3)

then it holds for any $\alpha\in(0,1)$,

$\displaystyle C_{\alpha}\leq\max\left(C_{\Omega}+(1-\alpha)K_{1},\alpha
K_{2},\frac{(1-\alpha)K_{\Sigma,\Omega}C_{\Sigma}+\alpha
C_{\Omega}C_{\Sigma}+\alpha(1-\alpha)(K_{\Sigma,\Omega}K_{2}+C_{\Sigma}K_{1})}{(1-\alpha)K_{\Sigma,\Omega}+\alpha
C_{\Sigma}}\right).$ (2.4)

###### Proof.

By definition of $C_{\Sigma}$ and by (2.2), for any $f\in\mathcal{D}_{0}$

$\displaystyle\operatorname{Var}_{\lambda_{\Sigma}}f\leq
tK_{\Sigma,\Omega}\int_{\Omega}\|\mathop{D}f\|^{2}d\lambda_{\Omega}+(1-t)C_{\Sigma}\int_{\Sigma}\|{\mathop{D}}^{\tau}f\|^{2}d\lambda_{\Sigma},$

for any $t\in[0,1]$. Let $\alpha\in(0,1)$. For any $f\in\mathcal{D}_{0}$ and
any $t\in[0,1]$

$\displaystyle\operatorname{Var}_{\lambda_{\alpha}}f$
$\displaystyle=\alpha\operatorname{Var}_{\lambda_{\Omega}}f+(1-\alpha)\operatorname{Var}_{\lambda_{\Sigma}}f+\alpha(1-\alpha)\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}$
$\displaystyle\leq\left(C_{\Omega}+\frac{(1-\alpha)t}{\alpha}K_{\Sigma,\Omega}+(1-\alpha)K_{1}\right)\alpha\int_{\Omega}\|\mathop{D}f\|^{2}d\lambda_{\Omega}$
$\displaystyle\quad+\left((1-t)C_{\Sigma}+\alpha
K_{2}\right)(1-\alpha)\int_{\Sigma}\|{\mathop{D}}^{\tau}f\|^{2}d\lambda_{\Sigma}.$

Therefore,

$\displaystyle
C_{\alpha}\leq\inf_{t\in[0,1]}\max\left(C_{\Omega}+\frac{(1-\alpha)t}{\alpha}K_{\Sigma,\Omega}+(1-\alpha)K_{1},(1-t)C_{\Sigma}+\alpha
K_{2}\right).$

For any positive constants $a,b,c,d$, we have

$\displaystyle\inf_{t\in[0,1]}\max\left(a+bt,c-dt\right)=\begin{cases}a&\text{if
}c-a<0,\\\ c-d&\text{if }c-a>b+d,\\\ \frac{bc+ad}{b+d}&\text{if }0\leq c-a\leq
b+d.\end{cases}$

Therefore

$\displaystyle C_{\alpha}\leq\begin{cases}C_{\Omega}+(1-\alpha)K_{1}&\text{\ \
if }\alpha K_{2}-(1-\alpha)K_{1}+C_{\Sigma}-C_{\Omega}<0,\\\ \alpha
K_{2}&\text{\ \ if }\alpha
K_{2}-(1-\alpha)K_{1}-C_{\Omega}>\frac{1-\alpha}{\alpha}K_{\Sigma,\Omega},\\\
\frac{(1-\alpha)K_{\Sigma,\Omega}C_{\Sigma}+\alpha
C_{\Omega}C_{\Sigma}+\alpha(1-\alpha)(K_{\Sigma,\Omega}K_{2}+C_{\Sigma}K_{1})}{(1-\alpha)K_{\Sigma,\Omega}+\alpha
C_{\Sigma}}&\begin{array}[]{r}\text{if }0\leq\alpha
K_{2}-(1-\alpha)K_{1}+C_{\Sigma}-C_{\Omega}\\\ \leq
C_{\Sigma}+\frac{1-\alpha}{\alpha}K_{\Sigma,\Omega}.\end{array}\end{cases}$

The last term is equivalent to the announced result. ∎

###  2.2 Continuity of $C_{\alpha}$

In general, the function $\alpha\mapsto C_{\alpha}$ might have discontinuities
at $\alpha\in\\{0,1\\}$ in which cases an upper bound for $C_{\alpha}$ which
interpolates continuously between $C_{0}$ and $C_{1}$ cannot exist. For
example, when $\Omega=(0,b)\times(0,1)\subset\mathbb{R}^{2}$ and
$\Sigma=[0,b]\times\\{0\\}$, straightforward computations yield

$\lim_{\alpha\to
0}C_{\alpha}=\max\left\\{C_{\Sigma},\frac{4}{\pi^{2}}\right\\},$

where $C_{\Sigma}=\frac{b^{2}}{\pi^{2}}$. Hence $\alpha\mapsto C_{\alpha}$ is
discontinuous at $\alpha=0$ if and only if $b<2$. – To generalize this to the
framework of Section 2.1 let
$\mathcal{C}^{1}_{0}(\overline{\Omega})=\\{f\in\mathcal{C}^{1}(\overline{\Omega}):f=0\mbox{
on }\Sigma\\}$ and

$\tilde{C}_{0}:=\sup_{\begin{subarray}{c}f\in\mathcal{C}^{1}_{0}(\overline{\Omega})\\\
f\text{ non
constant}\end{subarray}}\frac{\int_{\Omega}f^{2}d\lambda_{\Omega}}{\int_{\Omega}\|\mathop{D}f\|^{2}d\lambda_{\Omega}}.$

(If $\mathop{D}=\nabla$, $\tilde{C}_{0}$ is the inverse of the spectral gap
for Brownian motion on $\Omega$ with killing on $\Sigma$ and normal reflection
at $\partial\Omega\setminus\Sigma$. ) We can then record the following
statement as a partial corollary to Proposition 2.1.

######  Proposition 2.2.

In the setting of proposition 2.1 it holds that

$\varliminf_{\alpha\to 0}C_{\alpha}\geq\tilde{C}_{0}.$

In particular, if $C_{\Sigma}<\tilde{C}_{0}$, then $\alpha\mapsto C_{\alpha}$
is discontinuous at $\alpha=0$. Conversely, if $C_{\Sigma}\geq
C_{\Omega}+K_{1}$ then $\alpha\mapsto C_{\alpha}$ is continuous at 0. If
$C_{\Omega}\geq K_{2}$ continuity at 1 holds.

###### Proof.

To prove the second statement, take a non constant function
$g\in\mathcal{C}^{1}_{0}(\overline{\Omega})$ and estimate

$\displaystyle\varliminf_{\alpha\to 0}C_{\alpha}$
$\displaystyle=\varliminf_{\alpha\to
0}\sup_{\begin{subarray}{c}f\in\mathcal{C}^{1}(\overline{\Omega})\\\ f\text{
non
constant}\end{subarray}}\frac{\operatorname{Var}_{\lambda_{\alpha}}f}{\mathcal{E}_{\alpha}(f)}\geq\varliminf_{\alpha\to
0}\frac{\operatorname{Var}_{\lambda_{\alpha}}g}{\mathcal{E}_{\alpha}(g)}$
$\displaystyle=\varliminf_{\alpha\to
0}\frac{\alpha\operatorname{Var}_{\lambda_{\Omega}}g+(1-\alpha)\operatorname{Var}_{\lambda_{\Sigma}}g+\alpha(1-\alpha)\left(\int_{\Omega}gd\lambda_{\Omega}-\int_{\Sigma}gd\lambda_{\Sigma}\right)^{2}}{\alpha\int_{\Omega}\|\mathop{D}g\|^{2}d\lambda_{\Omega}+(1-\alpha)\int_{\Sigma}\|{\mathop{D}}^{\tau}g\|^{2}d\lambda_{\Sigma}}.$

Since $g=0$ on $\Sigma$, we obtain

$\displaystyle\varliminf_{\alpha\to 0}C_{\alpha}$
$\displaystyle\geq\varliminf_{\alpha\to
0}\frac{\alpha\operatorname{Var}_{\lambda_{\Omega}}g+\alpha(1-\alpha)\left(\int_{\Omega}gd\lambda_{\Omega}\right)^{2}}{\alpha\int_{\Omega}\|\mathop{D}g\|^{2}d\lambda_{\Omega}}=\frac{\int_{\Omega}g^{2}d\lambda_{\Omega}}{\int_{\Omega}\|\mathop{D}g\|^{2}d\lambda_{\Omega}}.$

Taking the supremum over $g\in\mathcal{C}^{1}_{0}(\overline{\Omega})$ yields
the first statement.

To prove the second assertion note that $\alpha\mapsto C_{\alpha}$ is the
pointwise supremum of a family of continuous functions and therefore lower
semi continuous. Thus $C_{\Sigma}=C_{0}\leq\varliminf_{\alpha\to
0}C_{\alpha}$. If $C_{\Sigma}\geq C_{\Omega}+K_{1}$, the r.h.s. of inequality
(2.4) converges to $C_{\Sigma}$ as $\alpha$ goes to 0, which implies that
$\varlimsup_{\alpha\to 0}C_{\alpha}\leq C_{\Sigma}$. Similarly, if
$C_{\Omega}\geq K_{2}$, the r.h.s. of (2.4) converges to $C_{\Omega}$ as
$\alpha$ goes ∎

######  Remark 2.3.

For smooth enough boundary the constant $K_{2}$ can always be taken equal to
zero, hence by proposition 2.2 continuity at $\alpha=1$ holds. An example
where a phase transition appears at $\alpha=0$ is given in section 3.3. In
section 3.4 we present an example where $C_{\Omega}<K_{2}$ but continuity of
at $\alpha=1$ can be established via Mosco-convergence [23] of the associated
Dirichlet forms, see also [24].

##  3 Examples

###  3.1 Brownian motion on balls with sticky boundary diffusion

As our first example let $\Omega:=B_{1}$ be the unit ball in $\mathbb{R}^{d}$,
$\Sigma=\partial\Omega$ and $\mathop{D}=\nabla$ and
${\mathop{D}}^{\tau}=\sqrt{\beta}\,\nabla^{\tau}$ with
$\mathcal{D}_{0}=C^{1}(\overline{\Omega})$.

######  Proposition 3.1.

In the case when $\Omega=B_{1}\subset\mathbb{R}^{d}$ the optimal Poincaré
constant of the generator (1.1) is bounded from above by

$\displaystyle
C_{\alpha}\leq\max\left(C_{\Omega}+(1-\alpha)\frac{d+1}{4d^{2}},\frac{4(1-\alpha)d+4\alpha
d^{2}C_{\Omega}+\alpha(1-\alpha)(d+1)}{4d(\alpha
d+(1-\alpha)\beta(d-1))}\right),$ (3.1)

where $\alpha=\frac{\gamma}{d+\gamma}$ and $C_{\Omega}$ is the optimal
Poincaré constant for reflecting Brownian motion on
$B_{1}\subset\mathbb{R}^{d}$.

###### Proof.

In order to apply Proposition 2.1, it is sufficient to compute the constants
$C_{\Sigma}$, $K_{\Sigma,\Omega}$, $K_{1}$ and $K_{2}$. We claim that
inequalities (2.2) and (2.3) holds with

$C_{\Sigma}=\frac{1}{\beta(d-1)},\quad K_{\Sigma,\Omega}=\frac{1}{d},\quad
K_{1}=\frac{d+1}{4d^{2}},\quad K_{2}=0.$

First, according to [31, Theorem 22.1], the first eigenvalue of the Laplace-
Beltrami operator on the unit sphere of dimension $d-1$ is equal to $d-1$,
thus $C_{\Sigma}=\frac{1}{\beta(d-1)}$.

Moreover, according to [3, Theorem 4], for every
$f\in\mathcal{C}^{1}(\partial\Omega)$ one has

$\left(\int_{\partial\Omega}|f|^{q}d\lambda_{\Sigma}\right)^{\frac{2}{q}}\leq\frac{q-2}{d}\int_{\Omega}\|\nabla
u\|^{2}d\lambda_{\Omega}+\int_{\partial\Omega}f^{2}d\lambda_{\Sigma},$

for $2\leq q<\infty$ if $d=2$ and $2\leq q<\frac{2d-2}{d-2}$ if $d\geq 3$,
where $u$ is the harmonic extension of $f$ to the unit ball $\Omega$. It
implies the logarithmic Sobolev inequality
$\operatorname{Ent}_{\lambda_{\Sigma}}(f^{2})\leq\frac{2}{d}\int_{\Omega}\|\nabla
u\|^{2}d\lambda_{\Omega}$. Repeating the proof of Proposition 5.1.3 in [2], we
get
$\operatorname{Var}_{\lambda_{\Sigma}}f\leq\frac{1}{d}\int_{\Omega}\|\nabla
u\|^{2}d\lambda_{\Omega}$. Moreover, since the harmonic extension of $f$ is
minimizing the energy functional $\mathcal{E}_{1}$ under any function with
boundary condition $f$, the last inequality implies for any
$f\in\mathcal{C}^{1}(\overline{\Omega})$

$\displaystyle\operatorname{Var}_{\lambda_{\Sigma}}f\leq\frac{1}{d}\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega},$ (3.2)

which implies $K_{\Sigma,\Omega}=\frac{1}{d}$.

Furthermore, note that
$\int_{\partial\Omega}f(y)\lambda_{\Sigma}(dy)=\int_{\Omega}f(\pi_{x})\lambda_{\Omega}(dx)$,
where $\pi_{x}=\frac{x}{\|x\|}$, $x\not=0$. Hence, using Jensen’s inequality
and polar coordinates

$\displaystyle\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\partial\Omega}fd\lambda_{\Sigma}\right)^{2}$
$\displaystyle\leq\int_{\Omega}(f(x)-f(\pi_{x}))^{2}\lambda_{\Omega}(dx)$
$\displaystyle=\frac{1}{|\Omega|}\int_{\partial\Omega}\int_{0}^{1}\left(f(ry)-f(y)\right)^{2}r^{d-1}drdy$
$\displaystyle=\frac{1}{|\Omega|}\int_{\partial\Omega}\int_{0}^{1}\left(\int_{r}^{1}\frac{d}{ds}f(sy)ds\right)^{2}r^{d-1}drdy$
$\displaystyle\leq\frac{1}{|\Omega|}\int_{\partial\Omega}\int_{0}^{1}(1-r)\left(\int_{r}^{1}\left(\frac{d}{ds}f(sy)\right)^{2}ds\right)r^{d-1}drdy$
$\displaystyle=\frac{1}{|\Omega|}\int_{\partial\Omega}\int_{0}^{1}\left[\int_{0}^{s}(1-r)r^{d-1}dr\right]\left(\frac{d}{ds}f(sy)\right)^{2}dsdy.$

We separately estimate

$\displaystyle\int_{0}^{s}(1-r)r^{d-1}dr=\left(\frac{s}{d}-\frac{s^{2}}{d+1}\right)s^{d-1}\leq\frac{d+1}{4d^{2}}s^{d-1}.$

for any $s\in[0,1]$. Hence,

$\displaystyle\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\partial\Omega}fd\lambda_{\Sigma}\right)^{2}$
$\displaystyle\leq\frac{d+1}{4d^{2}|\Omega|}\int_{\partial\Omega}\int_{0}^{1}\left(\nabla
f(sy)\cdot y\right)^{2}s^{d-1}ds$
$\displaystyle=\frac{d+1}{4d^{2}|\Omega|}\int_{\partial\Omega}\int_{0}^{1}\left\|\nabla
f(sy)\right\|^{2}s^{d-1}dsdy$
$\displaystyle=\frac{d+1}{4d^{2}}\int_{\Omega}\|\nabla
f(x)\|^{2}\lambda_{\Omega}(dx).$ (3.3)

which implies $K_{1}=\frac{d+1}{4d^{2}}$ and $K_{2}=0$. ∎

For illustration, in $d=2,$ we compare the bound from Proposition 3.1 for
$\beta=1,\gamma>0$ to the optimal constant $C_{\alpha}$ which will be computed
numerically. To evaluate the bound (3.1), note that in this case

$C_{\Omega}=\frac{1}{\sigma_{\Omega}}\approx\frac{1}{3.39},$ (3.4)

where $\sigma_{\Omega}$ is the smallest positive eigenvalue of the Laplace
operator with Neumann boundary condition on the circle. It is given as the
minimal positive solution to the equation $J_{m}^{\prime}(\sqrt{\gamma})=0$,
$m\in\mathbb{N}_{0}$, where $J_{m}$ is the Bessel function of the first kind
of parameter $m$, defined by
$J_{m}(x)=\frac{1}{\pi}\int_{0}^{\pi}\cos(mt-x\sin t)dt$, $x\geq 0$. As a
consequence, inequality (3.1) becomes

$\displaystyle
C_{\alpha}\leq\frac{8(1-\alpha)\sigma_{\Omega}+16\alpha+3\alpha(1-\alpha)\sigma_{\Omega}}{8(1+\alpha)\sigma_{\Omega}}.$
(3.5)

For the numerical computation of $C_{\alpha}$ one notes that the generator
$A_{\alpha}$ associated with $\mathcal{E}_{\alpha}$ is defined on
$D(A_{\alpha})\subset\mathcal{C}^{2}(\overline{\Omega})$ as

$A_{\alpha}f=\mathbb{I}_{\Omega}\Delta
f+\mathbb{I}_{\partial\Omega}\left(\Delta^{\tau}f-\frac{2\alpha}{1-\alpha}\frac{\partial
f}{\partial\nu}\right),$

where $\Delta^{\tau}$ and $\frac{\partial}{\partial\nu}$ denote the Laplace-
Beltrami operator and the outer normal derivative on the circle
$\partial\Omega$. Hence, an eigenvector of $-A_{\alpha}$ for eigenvalue
$\lambda\geq 0$ is a function $f\in D(A_{\alpha})$ such that

$A_{\alpha}f=-\lambda f\quad\mbox{in}\ \ \Omega.$

This equation is equivalent to the system of partial differential equations

$\begin{cases}\Delta f=-\lambda f&\mbox{in}\ \ \Omega,\\\
\Delta^{\tau}f-\frac{2\alpha}{1-\alpha}\frac{\partial f}{\partial\nu}=-\lambda
f&\mbox{on}\ \ \partial\Omega,\end{cases}$

which by the continuity of $f$ can be rewritten as

$\begin{cases}\Delta f=-\lambda f&\mbox{in}\ \ \Omega,\\\ \Delta
f=\Delta^{\tau}f-\frac{2\alpha}{1-\alpha}\frac{\partial
f}{\partial\nu}&\mbox{on}\ \ \partial\Omega.\end{cases}$

Passing to polar coordinates
$(x_{1},x_{2})=(r\cos\theta,r\sin\theta)\in\Omega$ in $d=2$ and separating
variables, we obtain the set of eigenfunctions
$\\{f_{m,l}^{c},f_{m,l}^{s}\\}_{m,l\in\mathbb{N}_{0}}$,

$f_{m,l}^{c}(x_{1},x_{2})=J_{m}(\sqrt{\lambda_{m,l}}r)\cos(m\theta),\quad
m,l\in\mathbb{N}_{0},$
$f_{m,l}^{s}(x_{1},x_{2})=J_{m}(\sqrt{\lambda_{m,l}}r)\sin(m\theta),\quad
m\in\mathbb{N},\ \ l\in\mathbb{N}_{0},$

where $\lambda_{m,l}$, $l\in\mathbb{N}_{0}$, are countable family of positive
solutions to the equation

$\sqrt{\lambda}J_{m}^{\prime\prime}(\sqrt{\lambda})+\frac{1+\alpha}{1-\alpha}J_{m}^{\prime}(\sqrt{\lambda})=0$
(3.6)

for every $m\in\mathbb{N}_{0}$. Since the family $\\{f_{m,l}^{c},\
m,l\in\mathbb{N}_{0}\\}\cup\\{f_{m,l}^{s},\ m\in\mathbb{N}_{0},\
l\in\mathbb{N}_{0}\\}$ is dense in $L_{2}(\Omega,\lambda_{\alpha})$ and the
operator $A_{\alpha}$ is symmetric, the standard argument implies

$C_{\alpha}=\frac{1}{\lambda_{\alpha,\star}},$ (3.7)

where
$\lambda_{\alpha,\star}=\min\limits_{m,l\in\mathbb{N}_{0}}\lambda_{m,l}$. The
resulting curves are plotted in Figure 1.

Figure 1: The blue curve represents $\alpha\mapsto C_{\alpha}$ the optimal
Poincaré constant when $\Omega$ is the unit ball of $\mathbb{R}^{2}$ with full
boundary diffusion. The red curve is the upper estimate given by (3.5).

###  3.2 Smooth manifold with boundary

Let $\Omega$ be a smooth compact Riemannian manifold of dimension $d$ with
piecewise smooth boundary $\partial\Omega$. We denote by $\operatorname{Ric}$
the Ricci curvature of $\Omega$ and by $\mathrm{II}$ the second fundamental
form on the boundary $\partial\Omega$. Assume in this section that:

$\displaystyle\text{Assumption (M)}:\quad\quad\quad\quad\quad\exists
k_{r}>0,k_{2}>0,\quad\operatorname{Ric}|_{\Omega}\geq
k_{R}\operatorname{id}\quad\text{and}\quad\mathrm{II}|_{\partial\Omega}\geq
k_{2}\operatorname{id}.$

As before we consider $\Sigma=\partial\Omega$, $\mathop{D}=\nabla$ and
${\mathop{D}}^{\tau}=\nabla^{\tau}$ with
$\mathcal{D}_{0}=C^{1}(\overline{\Omega})$.

######  Proposition 3.2.

Under assumption (M), it holds that

$\displaystyle
C_{\alpha}\leq\max\left(C_{\Omega}+\frac{(1-\alpha)(d-1)}{dk_{R}},\frac{C_{\Sigma}}{dk_{R}}\cdot\frac{2(1-\alpha)dk_{R}+\alpha
dk_{2}k_{R}C_{\Omega}+\alpha(1-\alpha)(d-1)k_{2}}{2(1-\alpha)+\alpha
k_{2}C_{\Sigma}}\right)=:M_{1}.$ (3.8)

This statement is obtained via Proposition 2.1 and the two statements below.

######  Proposition 3.3.

Under assumption (M), inequality (2.3) is satisfied with $K_{2}=0$ and

$\displaystyle K_{1}=\frac{d-1}{dk_{R}}.$

###### Proof.

Our goal is to obtain an lower bound of

$\displaystyle\inf_{\begin{subarray}{c}f\in
C^{1}(\overline{\Omega})\end{subarray}}\frac{\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}}{\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}},$

where we recall that $\Sigma=\partial\Omega$. We note that

$\displaystyle\inf_{\begin{subarray}{c}f\in
C^{1}(\overline{\Omega})\end{subarray}}\frac{\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}}{\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}}=\inf_{\begin{subarray}{c}f\in
C^{1}(\overline{\Omega})\\\
\int_{\Sigma}fd\lambda_{\Sigma}=0\end{subarray}}\frac{\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}}{\left(\int_{\Omega}fd\lambda_{\Omega}\right)^{2}}\geq\inf_{\begin{subarray}{c}f\in
C^{1}(\overline{\Omega})\\\
\int_{\Sigma}fd\lambda_{\Sigma}=0\end{subarray}}\frac{\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}}{\int_{\Omega}f^{2}d\lambda_{\Omega}}=:\sigma.$

Let $f\in C^{1}(\overline{\Omega})$ be a minimizer for $\sigma$. Then
$\int_{\Sigma}fd\lambda_{\Sigma}=0$ and

$\displaystyle\int_{\Omega}\nabla f\cdot\nabla\xi
d\lambda_{\Omega}=\sigma\int_{\Omega}f\xi d\lambda_{\Omega}$

for each $\xi\in C^{1}(\overline{\Omega})$ with $\int_{\Sigma}\xi
d\lambda_{\Sigma}=0$. By integration by parts, the latter equality is
equivalent to

$\displaystyle-\int_{\Omega}\Delta f\xi
d\lambda_{\Omega}+\frac{|\Sigma|}{|\Omega|}\int_{\Sigma}\frac{\partial
f}{\partial\nu}\xi d\lambda_{\Sigma}=\sigma\int_{\Omega}f\xi
d\lambda_{\Omega}$

for each $\xi\in C^{1}(\overline{\Omega})$ satisfying $\int_{\Sigma}\xi
d\lambda_{\Sigma}=0$. In particular, choosing $\xi\in C^{\infty}_{0}(\Omega)$
(which obviously satisfies $\int_{\Sigma}\xi d\lambda_{\Sigma}=0$), we get
that $f$ should satisfy $-\Delta f=\sigma f$ in $\Omega$. Hence
$\int_{\Sigma}\frac{\partial f}{\partial\nu}\xi d\lambda_{\Sigma}=0$ for each
$\xi$ with zero mean, so it follows that $\int_{\Sigma}\frac{\partial
f}{\partial\nu}\left(\xi-\int_{\Sigma}\xi
d\lambda_{\Sigma}\right)d\lambda_{\Sigma}=0$ for every $\xi\in
C^{1}(\overline{\Omega})$, which is equivalent to

$\displaystyle\int_{\Sigma}\left(\frac{\partial
f}{\partial\nu}-\int_{\Sigma}\frac{\partial
f}{\partial\nu}d\lambda_{\Sigma}\right)\xi d\lambda_{\Sigma}=0$

for every $\xi\in C^{1}(\overline{\Omega})$. It follows that $\frac{\partial
f}{\partial\nu}$ is constant on $\Sigma$. Therefore, $f$ satisfies

$\begin{cases}\Delta f=-\sigma f&\mbox{in}\ \ \Omega,\\\ \frac{\partial
f}{\partial\nu}\equiv c&\mbox{on}\ \ \partial\Omega,\\\
\int_{\Sigma}fd\lambda_{\Sigma}=0,\end{cases}$ (3.9)

for some constant $c$.

Moreover, recall Reilly’s formula (see [28])

$\displaystyle\int_{\Omega}\left((\Delta f)^{2}-\|\nabla^{2}f\|^{2}\right)dx$
$\displaystyle=\int_{\Omega}\operatorname{Ric}(\nabla f,\nabla f)dx$ (3.10)
$\displaystyle\quad+\int_{\Sigma}\left(H(\frac{\partial
f}{\partial\nu})^{2}+\mathrm{II}(\nabla^{\tau}f,\nabla^{\tau}f)+2\Delta^{\tau}f\frac{\partial
f}{\partial\nu}\right)dS$

where $dx$ and $dS$ denote the Riemannian volume resp. surface measure on
$\Omega$ and $\partial\Omega$, $\nabla^{2}f$ is the Hessian of $f$ and $H$ is
the mean curvature of $\Sigma$ (i.e. the trace of $\mathrm{II}$). Since $f$
satisfies (3.9),

$\displaystyle\int_{\Omega}(\Delta f)^{2}dx=-\sigma\int_{\Omega}f\Delta fdx$
$\displaystyle=\sigma\int_{\Omega}\|\nabla
f\|^{2}dx-\sigma\int_{\Sigma}\frac{\partial f}{\partial\nu}fdS$
$\displaystyle=\sigma\int_{\Omega}\|\nabla f\|^{2}dx-\sigma
c\int_{\Sigma}fdS=\sigma\int_{\Omega}\|\nabla f\|^{2}dx,$

because $\int_{\Sigma}fdS=|\Sigma|\int_{\Sigma}fd\lambda_{\Sigma}=0$.
Furthermore, note that
$\|\nabla^{2}f\|^{2}=\sum_{i,j}(\partial_{ij}^{2}f)^{2}\geq\sum_{i=1}^{d}(\partial_{ii}^{2}f)^{2}\geq\frac{1}{d}(\sum_{i=1}^{d}\partial_{ii}^{2}f)^{2}=\frac{1}{d}(\Delta
f)^{2}$. Therefore, the l.h.s. of (3.10) is bounded by

$\displaystyle\int_{\Omega}\left((\Delta
f)^{2}-\|\nabla^{2}f\|^{2}\right)dx\leq\frac{d-1}{d}\int_{\Omega}(\Delta
f)^{2}dx\leq\frac{d-1}{d}\sigma\int_{\Omega}\|\nabla f\|^{2}dx.$

On the other hand, by assumption (M), $H\geq 0$,
$\mathrm{II}(\nabla^{\tau}f,\nabla^{\tau}f)\geq 0$ and

$\displaystyle\int_{\Omega}\operatorname{Ric}(\nabla f,\nabla f)dx\geq
k_{R}\int_{\Omega}\|\nabla f\|^{2}dx.$

Since

$\displaystyle\int_{\Sigma}\Delta^{\tau}f\frac{\partial
f}{\partial\nu}dS=c\int_{\Sigma}\Delta^{\tau}fdS=0$

the r.h.s. of (3.10) is bounded from below by $k_{R}\int_{\Omega}\|\nabla
f\|^{2}dx$. It turns out that

$\displaystyle\frac{d-1}{d}\sigma\int_{\Omega}\|\nabla f\|^{2}dx\geq
k_{R}\int_{\Omega}\|\nabla f\|^{2}dx,$

which implies that $\sigma\geq\frac{d}{d-1}k_{R}$. It follows that inequality
(2.3) holds with $K_{1}=\frac{d-1}{dk_{R}}$. ∎

######  Remark 3.4.

Instead of using $K_{1}$ from Proposition 3.3 another admissible choice is

$K_{1}^{\prime}=\frac{|\Omega|}{|\partial\Omega|}B^{2}(1+C_{\Omega})<\infty,$

where $B$ is the optimal Sobolev trace constant of $\Omega$, i.e. the norm of
the embedding $H^{1,2}(\Omega)\hookrightarrow L^{2}(\partial\Omega)$. $B^{-2}$
is the first nontrivial eigenvalue of a Steklov-type eigenvalue problem

$\begin{cases}-\Delta f+f=0&\mbox{ in }\Omega\\\ \frac{\partial
f}{\partial\nu}=\sigma f&\mbox{ on }\partial\Omega,\end{cases}$

for which however explicit lower bounds in terms of the geometry of $\Omega$
seem yet unknown [4, 5, 11, 21, 29].

######  Proposition 3.5.

Under assumption (M), inequality (2.2) holds with
$K_{\Sigma,\Omega}=\frac{2}{k_{2}}$.

###### Proof.

The optimal choice for $K_{\Sigma,\Omega}$ is $\sigma^{-1}$, where $\sigma$
given by

$\sigma=\inf_{\begin{subarray}{c}f\in C^{1}(\overline{\Omega})\\\
\int_{\Sigma}fd\lambda_{\Sigma}=0\end{subarray}}\frac{\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}}{\left(\int_{\Sigma}f^{2}d\lambda_{\Sigma}\right)^{2}}$

is the first nontrivial eigenvalue of the Steklov-problem c.f. [12]

$\begin{cases}\Delta f=0&\mbox{ in }\Omega,\\\ \frac{\partial
f}{\partial\nu}=\sigma f&\mbox{ on }\partial\Omega.\end{cases}$

Escobar [9] showed $\sigma\geq\frac{k_{2}}{2}$ in this case. ∎

Alternatively, we obtain another upper bound for $C_{\alpha}$ by a direct
application of Reilly’s formula.

######  Proposition 3.6.

Under assumption (M) it holds that

$\displaystyle C_{\alpha}\leq\max\left(\frac{(3d-1)(1-\alpha)}{d\alpha
k_{2}}\frac{|\Omega|}{|\partial\Omega|},\frac{d-1}{dk_{R}}\right)=:M_{2}.$
(3.11)

###### Proof.

We estimate equivalently from below the first nontrivial eigenvalue
$\sigma=C_{\alpha}^{-1}$ for the problem

$\begin{cases}\Delta f+\sigma f=0&\mbox{ in }\Omega\\\
\Delta^{\tau}f-\gamma\frac{\partial f}{\partial\nu}+\sigma f=0&\mbox{ on
}\partial\Omega,\end{cases}$

where $\gamma=\frac{\alpha}{1-\alpha}\frac{|\partial\Omega|}{|\Omega|}$. As in
the proof of Proposition 3.3 we apply Reilly’s formula (3.10) to the
corresponding eigenfunction $f$. In this case, for the l.h.s. we estimae

$\displaystyle\int_{\Omega}\left((\Delta f)^{2}-\|\nabla^{2}f\|^{2}\right)dx$
$\displaystyle\leq\frac{d-1}{d}\int_{\Omega}(\Delta
f)^{2}dx=-\frac{d-1}{d}\sigma\int_{\Omega}f\Delta fdx$
$\displaystyle=\frac{d-1}{d}\sigma\int_{\Omega}\|\nabla
f\|^{2}dx-\frac{d-1}{d}\sigma\int_{\Sigma}\frac{\partial f}{\partial\nu}fdS$
$\displaystyle=\frac{d-1}{d}\sigma\int_{\Omega}\|\nabla
f\|^{2}dx-\frac{d-1}{d}\frac{\sigma}{\gamma}\int_{\Sigma}(\Delta^{\tau}f+{\sigma}f)fdS$
$\displaystyle=\frac{d-1}{d}\sigma\int_{\Omega}\|\nabla
f\|^{2}dx+\frac{d-1}{d}\frac{\sigma}{\gamma}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS-\frac{d-1}{d}\frac{\sigma^{2}}{\gamma}\int_{\Sigma}f^{2}dS$
$\displaystyle\leq\frac{d-1}{d}\sigma\int_{\Omega}\|\nabla
f\|^{2}dx+\frac{d-1}{d}\frac{\sigma}{\gamma}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS.$

Since

$\displaystyle\int_{\Sigma}\frac{\partial f}{\partial\nu}\Delta^{\tau}fdS$
$\displaystyle=\frac{1}{\gamma}\int_{\Sigma}(\Delta^{\tau}f+\sigma
f)\Delta^{\tau}fdS$
$\displaystyle=\frac{1}{\gamma}\int_{\Sigma}(\Delta^{\tau}f)^{2}dS-\frac{\sigma}{\gamma}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS\geq-\frac{\sigma}{\gamma}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS$

the r.h.s. of (3.10) is bounded from below by

$\displaystyle k_{R}\int_{\Omega}\|\nabla f\|^{2}dx$
$\displaystyle-\frac{2\sigma}{\gamma}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS+\int_{\Sigma}h|\frac{\partial
f}{\partial\nu}|^{2}dS+k_{2}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS$
$\displaystyle\geq k_{R}\int_{\Omega}\|\nabla
f\|^{2}dx-\frac{2\sigma}{\gamma}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS+k_{2}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS.$

Combining the two bounds for (3.10) yields

$\left(\frac{d-1}{d}\sigma-k_{R}\right)\int_{\Omega}\|\nabla
f\|^{2}dx\geq\left(k_{2}-\frac{3d-1}{d}\frac{\sigma}{\gamma}\right)\int_{\Sigma}\|\nabla^{\tau}f\|^{2}dS,$

which implies that either

$k_{2}-\frac{3d-1}{d}\frac{\sigma}{\gamma}\leq 0,\quad\mbox{ i.e.
}\quad\sigma\geq\frac{dk_{2}\gamma}{3d-1}$

or

$\frac{d-1}{d}\sigma-k_{R}\geq 0,\quad\mbox{ i.e.
}\quad\sigma\geq\frac{d}{d-1}k_{r}.$

Consequently,

$\sigma\geq\min\left(\frac{dk_{2}\gamma}{3d-1},\frac{d}{d-1}k_{R}\right).$

∎

######  Corollary 3.7.

Under assumption (M), it holds that

$\displaystyle C_{\alpha}\leq\min(M_{1},M_{2}),$

where $M_{1}=M_{1}(\alpha)$ and $M_{2}=M_{2}(\alpha)$ are defined by (3.8) and
(3.11), respectively.

When $\alpha$ goes to 0, $M_{1}$ tends to
$\max(C_{\Omega},\frac{d-1}{dk_{R}},C_{\Sigma})$ and $M_{2}$ tends to
$+\infty$, so the estimation via the interpolation method is always stronger.
When $\alpha$ goes to $1$, $M_{1}$ tends to $C_{\Omega}$ and $M_{2}$ tends to
$\frac{d-1}{dk_{R}}$, so the relative strength of each method depends on the
values of $C_{\Omega}$, $d$ and $k_{R}$.

###  3.3 Brownian motion on balls with partial sticky reflecting boundary
diffusion

As in Section 3.1, let $\Omega:=B_{1}$ be the unit ball of $\mathbb{R}^{2}$.
Now, define for a fixed $\delta\in(0,1)$

$\Sigma=\\{(\cos\theta,\sin\theta)\in\partial\Omega:-\delta\pi\leq\theta\leq\delta\pi\\},\quad\quad\Sigma_{\operatorname{N}}:=\partial\Omega\backslash\Sigma.$

######  Proposition 3.8.

It holds that

$\displaystyle
C_{\alpha}\leq\max\left(C_{\Omega}+(1-\alpha)K_{1}(\delta),\frac{4(1-\alpha)\delta^{2}+8\alpha\delta^{3}C_{\Omega}+8\alpha(1-\alpha)\delta^{3}K_{1}(\delta)}{(1-\alpha)+8\alpha\delta^{3}}\right),$
(3.12)

where $C_{\Omega}=\frac{1}{\sigma_{\Omega}}\approx\frac{1}{3.39}$ and
$K_{1}(\delta)=\left(\sqrt{1-\delta}\pi+\frac{1}{4}\sqrt{\frac{3}{\delta}}\right)^{2}$.

As previously, we will start by computing the needed constants $C_{\Omega}$,
$C_{\Sigma}$, $K_{\Sigma,\Omega}$, $K_{1}$ and $K_{2}$. The first constant,
$C_{\Omega}=\frac{1}{\sigma_{\Omega}}\approx\frac{1}{3.39}$, remains
unchanged.

######  Lemma 3.9.

The following inequalities hold true

$\displaystyle\operatorname{Var}_{\lambda_{\Sigma}}f$ $\displaystyle\leq
C_{\Sigma}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}d\lambda_{\Sigma},$ (3.13)
$\displaystyle\operatorname{Var}_{\lambda_{\Sigma}}f$ $\displaystyle\leq
K_{\Sigma,\Omega}\int_{\Omega}\|\nabla f\|^{2}d\lambda_{\Omega},$ (3.14)

where $C_{\Sigma}=4\delta^{2}$ and $K_{\Sigma,\Omega}=\frac{1}{2\delta}$.

###### Proof.

Inequality (3.13) corresponds to the Poincaré inequality of the Laplacian on
the one-dimensional interval $[-\delta\pi,\delta\pi]$ with Neumann boundary
conditions. It is well known (see [2, Prop. 4.5.5]) that the optimal Poincaré
constant is given by $C_{\Sigma}=4\delta^{2}$.

Moreover, let us decompose the normalized Hausdorff measure
$\lambda_{\partial}$ on the sphere $\partial\Omega$ into the normalized
Hausdorff measure $\lambda_{\Sigma}$ on $\Sigma$ and the normalized Hausdorff
measure $\lambda_{\operatorname{N}}$ on $\Sigma_{\operatorname{N}}$:
$\lambda_{\partial}=\delta\lambda_{\Sigma}+(1-\delta)\lambda_{\operatorname{N}}$.
Therefore

$\displaystyle\operatorname{Var}_{\lambda_{\partial}}f=\delta\operatorname{Var}_{\lambda_{\Sigma}}f+(1-\delta)\operatorname{Var}_{\lambda_{\operatorname{N}}}f+\delta(1-\delta)\left(\int_{\Sigma}fd\lambda_{\Sigma}-\int_{\Sigma_{\operatorname{N}}}fd\lambda_{\operatorname{N}}\right)^{2}\geq\delta\operatorname{Var}_{\lambda_{\Sigma}}f,$

Furthermore, recall that by inequality (3.2), for any
$f\in\mathcal{C}^{1}(\overline{\Omega})$,
$\operatorname{Var}_{\lambda_{\partial}}f\leq\frac{1}{2}\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}$. It implies (3.14). ∎

######  Lemma 3.10.

It holds that

$\displaystyle\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}\leq
K_{1}(\delta)\int_{\Omega}\|\nabla f\|^{2}d\lambda_{\Omega}$

with
$K_{1}(\delta)=\left(\sqrt{1-\delta}\pi+\frac{1}{4}\sqrt{\frac{3}{\delta}}\right)^{2}$.

###### Proof.

For every $x\in\Omega\backslash\\{0\\}$ with polar coordinates $(r,\theta)$,
$r\in(0,1)$, $\theta\in(-\pi,\pi]$, denote by $p_{x}$ the point of coordinates
$(1,\delta\theta)$ on $\Sigma$. Obviously,
$\int_{\Sigma}f(y)\lambda_{\Sigma}(dy)=\int_{\Omega}f(p_{x})\lambda_{\Omega}(dx)$
and by Jensen’s inequality

$\displaystyle
I:=\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}\leq\int_{\Omega}\left(f(x)-f(p_{x})\right)^{2}\lambda_{\Omega}(dx).$

Define $g(r,\theta):=f(r\cos(\theta),r\sin(\theta))$. Then

$\displaystyle I$
$\displaystyle\leq\frac{1}{\pi}\int_{0}^{1}\int_{-\pi}^{\pi}(g(r,\theta)-g(1,\delta\theta))^{2}rdrd\theta\leq(\sqrt{J_{1}}+\sqrt{J_{2}})^{2},$
(3.15)

where
$J_{1}=\frac{1}{\pi}\int_{0}^{1}\int_{-\pi}^{\pi}(g(r,\theta)-g(r,\delta\theta))^{2}rdrd\theta$
and
$J_{2}=\frac{1}{\pi}\int_{0}^{1}\int_{-\pi}^{\pi}(g(r,\delta\theta)-g(1,\delta\theta))^{2}rdrd\theta$.
On the one hand

$\displaystyle J_{1}$
$\displaystyle=\frac{1}{\pi}\int_{0}^{1}\int_{-\pi}^{\pi}\left(\int_{\delta\theta}^{\theta}\frac{\partial
g}{\partial\theta}(r,u)du\right)^{2}rdrd\theta\leq\frac{1-\delta}{\pi}\int_{0}^{1}\int_{-\pi}^{\pi}|\theta|\int_{-\pi}^{\pi}\left(\frac{\partial
g}{\partial\theta}\right)^{2}(r,u)du\;rdrd\theta$
$\displaystyle\leq(1-\delta)\pi^{2}\frac{1}{\pi}\int_{0}^{1}\int_{-\pi}^{\pi}\left(\frac{1}{r}\frac{\partial
g}{\partial\theta}\right)^{2}(r,u)du\;rdr\leq(1-\delta)\pi^{2}\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}.$ (3.16)

On the other hand

$\displaystyle J_{2}$
$\displaystyle\leq\frac{1}{\pi}\int_{0}^{1}\int_{-\pi}^{\pi}(1-r)\int_{r}^{1}\left(\frac{\partial
g}{\partial
r}\right)^{2}(s,\delta\theta)ds\;rdrd\theta\leq\frac{1}{\pi}\int_{0}^{1}\int_{-\pi}^{\pi}\left(\frac{\partial
g}{\partial r}\right)^{2}(s,\delta\theta)\int_{0}^{s}(1-r)rdrdsd\theta.$

For every $s\in[0,1]$,
$\int_{0}^{s}(1-r)rdr=\frac{s^{2}}{2}-\frac{s^{3}}{3}\leq\frac{3s}{16}$, thus

$\displaystyle J_{2}$
$\displaystyle\leq\frac{3}{16\delta\pi}\int_{0}^{1}\int_{-\delta\pi}^{\delta\pi}\left(\frac{\partial
g}{\partial r}\right)^{2}(s,u)sdsdu\leq\frac{3}{16\delta}\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}.$ (3.17)

The proof of the lemma is completed by putting together (3.15), (3.16) and
(3.17). ∎

###### Proof of Proposition 3.8.

We apply Proposition 2.1 with $C_{\Omega}=\frac{1}{\sigma_{\Omega}}$,
$C_{\Sigma}=4\delta^{2}$, $K_{\Sigma,\Omega}=\frac{1}{2\delta}$,
$K_{1}(\delta)=\left(\sqrt{1-\delta}\pi+\frac{1}{4}\sqrt{\frac{3}{\delta}}\right)^{2}$
and $K_{2}=0$. ∎

(a) $\delta=0.5$

(b) $\delta=0.9$

Figure 2: The above two figures show the upper estimate given by the r.h.s of
(3.12). In the case $\delta=0.9$ (Figure 2b), the curve interpolates between
the extremal constants $C_{\Sigma}$ and $C_{\Omega}$, as opposed to the half-
sphere case (Figure 2a).

For $\delta$ sufficiently large, the map $\alpha\mapsto C_{\alpha}$ is
continuous at $\alpha=0$. Indeed, by Proposition 2.2, a sufficient condition
is $C_{\Sigma}(\delta)>C_{\Omega}+K_{1}(\delta)$, that is

$4\delta^{2}>\frac{1}{\sigma_{\Omega}}+\left(\sqrt{1-\delta}\pi+\frac{1}{4}\sqrt{\frac{3}{\delta}}\right)^{2},$

which is satisfied for any $\delta\geq 0.862$.

###  3.4 Ball with a needle

Our final example is the unit ball $\Omega=B_{1}$ of $\mathbb{R}^{2}$ with a
needle $\mathcal{L}$ of length $L$ attached to one point of the boundary, i.e.
$\mathcal{L}:=\\{(x,0):1\leq x\leq L+1\\}$, see Figure 3. The attachment point
and the endpoint of the needle are denoted by $x_{0}:=(1,0)$ and
$x_{L}=(L+1,0)$, respectively.

Figure 3: The ball (in green) is denoted by $\Omega$, the boundary of the ball
is denoted by $\partial\Omega$ and the needle (in blue) is denoted by
$\mathcal{L}$.

In that setting, we define
$\overline{\Omega}=\overline{B_{1}}\cup\mathcal{L}$, $\Sigma=\partial
B_{1}\cup\mathcal{L}$ and

$\lambda_{\alpha}=\alpha\lambda_{\Omega}+(1-\alpha)\lambda_{\Sigma},$

where $\lambda_{\Omega}$ is as previously the normalized Lebesgue measure on
$\Omega$ and
$\lambda_{\Sigma}=\frac{2\pi}{2\pi+L}\lambda_{\partial}+\frac{L}{2\pi+L}\lambda_{\mathcal{L}}$,
with $\lambda_{\partial}$ and $\lambda_{\mathcal{L}}$ being the normalized
Hausdorff measures on $\partial\Omega$ and $\mathcal{L}$, respectively. We
choose

$\mathcal{D}_{0}=\left\\{f\in C_{0}(\overline{\Omega}))\cap
C^{1}(\overline{\Omega}\setminus\\{x_{0}\\})\,|\,\frac{\partial f}{\partial
e_{1}}+\frac{\partial f}{\partial e_{2}}+\frac{\partial f}{\partial
e_{3}}=0\mbox{ at }x_{0}\right\\},$

where $e_{1}=(0,1)$, $e_{2}=(0,-1)$ and $e_{3}=(1,0)$ are the three ”tangent”
vectors to $\Sigma$ at point $x_{0}$, and $\mathop{D}:=\nabla$,
${\mathop{D}}^{\tau}:=\sqrt{\beta}\nabla^{\tau}$, which is well defined in
$\Sigma\setminus\\{x_{0}\\}$. With this choice, for $\alpha\in[0,1]$
$(\mathcal{E}_{\alpha},\mathcal{D}_{0})$ is a pre-Dirichlet form on
$L^{2}(\overline{\Omega},\lambda_{\alpha})$, whose closure generates Brownian
motion on $\Omega$ with sticky boundary diffusion on $\Sigma$, i.e. whose
generator is given by

$A_{\alpha}(f)=\Delta
f\mathbb{I}_{\Omega}+\beta\Delta_{\Sigma}f\mathbb{I}_{\Sigma}-\frac{\alpha}{1-\alpha}\frac{2\pi+L}{\pi}\frac{\partial
f}{\partial\nu}\mathbb{I}_{\partial\Omega},$

with $\Delta_{\Sigma}$ being the generator of the canonical diffusion on
$\Sigma$ with reflecting boundary condition at $x_{L}$. As before, the optimal
Poincaré constant $C_{\alpha}$ for $A_{\alpha}$ is given by

$\displaystyle C_{\alpha}:=\sup_{\begin{subarray}{c}f\in\mathcal{D}_{0}\\\
\mathcal{E}_{\alpha}(f)>0\end{subarray}}\frac{\operatorname{Var}_{\lambda_{\alpha}}f}{\mathcal{E}_{\alpha}(f)},$

and let $C_{\Omega}:=C_{1}$ and $C_{\Sigma}:=C_{0}$. In this case the
following estimate is obtained.

######  Proposition 3.11.

$\displaystyle
C_{\alpha}\leq\max\left(\frac{1}{\sigma_{\Omega}}+\frac{3}{8}(1-\alpha),\frac{1}{\beta\gamma_{L}}+\alpha\frac{L^{2}(\pi+L)}{\beta(2\pi+L)}\right),$

where $\gamma_{L}>0$ is the smallest positive solution to

$\displaystyle
2\cos(\sqrt{\gamma}L)(1-\cos(\sqrt{\gamma}2\pi))+\sin(\sqrt{\gamma}L)\sin(\sqrt{\gamma}2\pi)=0.$
(3.18)

Note that $\gamma_{L}\leq 1$ for any $L>0$ and if $L=2\pi$,
$\gamma_{2\pi}=\left(\frac{\arccos(-1/3)}{2\pi}\right)^{2}\approx 0.0925$.

Let us compute the constants needed to apply Proposition 2.1. As we do not
expect an inequality of type (2.2) to hold in that case, we set
$K_{\Sigma,\Omega}:=+\infty$. Moreover, $C_{\Sigma}$ can be computed exactly
as follows.

######  Lemma 3.12.

In this case, $C_{\Sigma}=\frac{1}{\beta\gamma_{L}}$.

###### Proof.

The constant $\frac{1}{C_{\Sigma}}$ is the smallest non-zero eigenvalue
$\gamma$ of the following problem:

$\displaystyle\begin{cases}\beta\Delta^{\tau}f=-\gamma f&\mbox{on}\ \
\Sigma\backslash\\{x_{0}\\},\\\ \frac{\partial f}{\partial\nu}=0&\mbox{at
point }x_{L},\\\ \frac{\partial f}{\partial e_{1}}+\frac{\partial f}{\partial
e_{2}}+\frac{\partial f}{\partial e_{3}}=0&\mbox{at point }x_{0},\end{cases}$

where $\Delta^{\tau}$ is the Laplace-Beltrami operator on $\partial\Omega$ and
$\mathcal{L}$. A general solution to that boundary value problem is given by

$\displaystyle
f(x)=\begin{cases}A\cos(\sqrt{\frac{\gamma}{\beta}}y)+B\sin(\sqrt{\frac{\gamma}{\beta}}y)&\mbox{if
}x=(y,0)\in\mathcal{L},\\\
C\cos(\sqrt{\frac{\gamma}{\beta}}\theta)+D\sin(\sqrt{\frac{\gamma}{\beta}}\theta)&\mbox{if
}x=(\cos\theta,\sin\theta)\in\partial\Omega,\end{cases}$

where $A$, $B$, $C$ and $D$ have to satisfy the continuity assumption of $f$
at point $x_{0}$ and both boundary conditions, that is:

$\displaystyle\left\\{\begin{aligned}
A&=C=C\cos(\sqrt{\frac{\gamma}{\beta}}2\pi)+D\sin(\sqrt{\frac{\gamma}{\beta}}2\pi),\\\
0&=-A\sin(\sqrt{\frac{\gamma}{\beta}}L)+B\cos(\sqrt{\frac{\gamma}{\beta}}L),\\\
0&=B+D+C\sin(\sqrt{\frac{\gamma}{\beta}}2\pi)-D\cos(\sqrt{\frac{\gamma}{\beta}}2\pi).\end{aligned}\right.$

A short computation shows that this system has a non-trivial solution if and
only if $\frac{\gamma}{\beta}$ solves (3.18). Therefore,
$\frac{1}{C_{\Sigma}}=\beta\gamma_{L}$. Obviously, $\gamma=1$ is a solution to
(3.18), thus $\gamma_{L}\leq 1$. ∎

Next, we look for the constants $K_{1}$ and $K_{2}$.

######  Lemma 3.13.

Inequality (2.3) holds with $K_{1}=\frac{3}{8}$ and
$K_{2}=\frac{L^{2}(\pi+L)}{\beta(2\pi+L)}$.

###### Proof.

Recall that $\Sigma=\partial\Omega\cup\mathcal{L}$. Let us insert the average
of $f$ over $\partial\Omega$ as follows:

$\displaystyle\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}$
$\displaystyle\leq
2\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\partial\Omega}fd\lambda_{\partial}\right)^{2}+2\left(\int_{\partial\Omega}fd\lambda_{\partial}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}$
$\displaystyle\leq\frac{3}{8}\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}+2\left(\int_{\partial\Omega}fd\lambda_{\partial}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2},$

where the second inequality follows directly from (3.3). Moreover, recalling
that
$\lambda_{\Sigma}=\frac{2\pi}{2\pi+L}\lambda_{\partial}+\frac{L}{2\pi+L}\lambda_{\mathcal{L}}$

$\displaystyle\left(\int_{\partial\Omega}fd\lambda_{\partial}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}$
$\displaystyle=\frac{L^{2}}{(2\pi+L)^{2}}\left(\int_{\partial\Omega}fd\lambda_{\partial}-\int_{\mathcal{L}}fd\lambda_{\mathcal{L}}\right)^{2}.$

For every $x=(\cos\theta,\sin\theta)\in\partial\Omega$, with
$\theta\in(-\pi,\pi]$, we denote by $p_{x}$ the point of $\mathcal{L}$ with
coordinates $(1+L-\frac{|\theta|L}{\pi},0)$. It follows that

$\displaystyle\left(\int_{\partial\Omega}fd\lambda_{\partial}-\int_{\mathcal{L}}fd\lambda_{\mathcal{L}}\right)^{2}=\left(\int_{\partial\Omega}(f(x)-f(p_{x}))d\lambda_{\partial}\right)^{2}\leq\int_{\partial\Omega}(f(x)-f(p_{x}))^{2}d\lambda_{\partial}.$

Denoting by $\lambda_{\partial}^{+}$ and $\lambda_{\partial}^{-}$ the
normalized Hausdorff measures on
$\partial\Omega^{+}:=\\{(x,y)\in\partial\Omega:y>0\\}$ and
$\partial\Omega^{-}:=\\{(x,y)\in\partial\Omega:y<0\\}$, respectively,

$\displaystyle\int_{\partial\Omega}(f(x)-f(p_{x}))^{2}d\lambda_{\partial}=\frac{1}{2}\int_{\partial\Omega^{+}}(f(x)-f(p_{x}))^{2}d\lambda^{+}_{\partial}+\frac{1}{2}\int_{\partial\Omega^{-}}(f(x)-f(p_{x}))^{2}d\lambda^{-}_{\partial}.$

Moreover, for any $\mathcal{C}^{1}$-function $g:[-\pi,L]\to\mathbb{R}$,

$\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\left|g(-\theta)-g(L-\textstyle\frac{\theta
L}{\pi})\right|^{2}d\theta\leq\frac{\pi+L}{2}\int_{-\pi}^{L}|g^{\prime}(t)|^{2}dt,$

so we deduce, identifying $\partial\Omega^{+}$ with $[-\pi,0]$ and
$\mathcal{L}$ with $[0,L]$, that

$\displaystyle\int_{\partial\Omega^{+}}(f(x)-f(p_{x}))^{2}d\lambda^{+}_{\partial}\leq\frac{\pi+L}{2}\left(\pi\int_{\partial\Omega^{+}}\|\nabla^{\tau}f\|^{2}d\lambda^{+}_{\partial}+L\int_{\mathcal{L}}\|\nabla^{\tau}f\|^{2}d\lambda_{\mathcal{L}}\right)$

and using symmetry to deal with $\partial\Omega^{-}$, we obtain

$\displaystyle\int_{\partial\Omega}(f(x)-f(p_{x}))^{2}d\lambda_{\partial}$
$\displaystyle\leq\frac{\pi+L}{4}\left(\pi\int_{\partial\Omega^{+}}\|\nabla^{\tau}f\|^{2}d\lambda^{+}_{\partial}+\pi\int_{\partial\Omega^{-}}\|\nabla^{\tau}f\|^{2}d\lambda^{-}_{\partial}+2L\int_{\mathcal{L}}\|\nabla^{\tau}f\|^{2}d\lambda_{\mathcal{L}}\right)$
$\displaystyle\leq\frac{(\pi+L)(2\pi+L)}{2}\int_{\Sigma}\|\nabla^{\tau}f\|^{2}d\lambda_{\Sigma}.$

Putting together the above inequalities, we get

$\displaystyle\left(\int_{\Omega}fd\lambda_{\Omega}-\int_{\Sigma}fd\lambda_{\Sigma}\right)^{2}$
$\displaystyle\leq\frac{3}{8}\int_{\Omega}\|\nabla
f\|^{2}d\lambda_{\Omega}+2\frac{L^{2}}{(2\pi+L)^{2}}\frac{(\pi+L)(2\pi+L)}{2\beta}\int_{\Sigma}\beta\|\nabla^{\tau}f\|^{2}d\lambda_{\Sigma}$

which leads to inequality (2.3) with $K_{1}=\frac{3}{8}$ and
$K_{2}=\frac{L^{2}(\pi+L)}{\beta(2\pi+L)}$. ∎

###### Proof of Proposition 3.11.

Since $K_{\Sigma,\Omega}=\infty$, we immediately get from Proposition 2.1 that

$\displaystyle C_{\alpha}\leq\max\left(C_{\Omega}+(1-\alpha)K_{1},\alpha
K_{2},C_{\Sigma}+\alpha
K_{2}\right)=\max\left(C_{\Omega}+(1-\alpha)K_{1},C_{\Sigma}+\alpha
K_{2}\right).$

Therefore,

$\displaystyle
C_{\alpha}\leq\max\left(\frac{1}{\sigma_{\Omega}}+\frac{3}{8}(1-\alpha),\frac{1}{\beta\gamma_{L}}+\alpha\frac{L^{2}(\pi+L)}{\beta(2\pi+L)}\right),$
(3.19)

where $\sigma_{\Omega}\approx 3.39$. ∎

######  Remark 3.14.

If $\beta$ is large enough, that is if the diffusion velocity is larger on
$\Sigma$ than on $\Omega$, then the first term in (3.19) dominates. Precisely,
if
$\beta\geq\sigma_{\Omega}\left(\frac{1}{\gamma_{L}}+\frac{L^{2}(\pi+L)}{2\pi+L}\right)$,
then (3.19) rewrites for any $\alpha$

$C_{\alpha}\leq\frac{1}{\sigma_{\Omega}}+\frac{3}{8}(1-\alpha).$

Conversely, if
$\beta\leq\frac{1}{\gamma_{L}}\left(\frac{1}{\sigma_{\Omega}}+\frac{3}{8}\right)^{-1}$,
then (3.19) rewrites for any $\alpha$

$C_{\alpha}\leq\frac{1}{\beta\gamma_{L}}+\alpha\frac{L^{2}(\pi+L)}{\beta(2\pi+L)}.$

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