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 konarovskyi@gmail.com, marx@math.uni-leipzig.de, renesse@uni-leipzig.de ^†^†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)
	$\displaystyle Af=\Delta f\mathbb{I}_{\Omega}+(\beta\Delta^{\tau}f-\gamma\frac{\partial f}{\partial\nu})\mathbb{I}_{\partial\Omega}$		(1.1)

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.

Refer to caption — 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)
		$\displaystyle\quad+\int_{\Sigma}\left(H(\frac{\partial f}{\partial\nu})^{2}+\mathrm{II}(\nabla^{\tau}f,\nabla^{\tau}f)+2\Delta^{\tau}f\frac{\partial f}{\partial\nu}\right)dS$		(3.10)

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}

\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 $K_{2}=0$ . ∎

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.

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

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

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

Abstract

1 Introduction and statement of main results

2 An interpolation approach

2.1 Generalized framework

Proposition 2.1.

Proof.

2.2 Continuity of CαC_{\alpha}

Proposition 2.2.

Proof.

Remark 2.3.

3 Examples

3.1 Brownian motion on balls with sticky boundary diffusion

Proposition 3.1.

Proof.

3.2 Smooth manifold with boundary

Proposition 3.2.

Proposition 3.3.

Proof.

Remark 3.4.

Proposition 3.5.

Proof.

Proposition 3.6.

Proof.

Corollary 3.7.

3.3 Brownian motion on balls with partial sticky reflecting boundary diffusion

Proposition 3.8.

Lemma 3.9.

Proof.

Lemma 3.10.

Proof.

Proof of Proposition 3.8.

3.4 Ball with a needle

Proposition 3.11.

Lemma 3.12.

Proof.

Lemma 3.13.

Proof.

Proof of Proposition 3.11.

Remark 3.14.

References

2.2 Continuity of $C_{\alpha}$