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

Vitalii Konarovskyi, Victor Marx, Max von Renesse

Abstract

Introducing an interpolation method we derive lower bounds for 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.

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

Abstract

Introducing an interpolation method we derive lower bounds for 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.

Paper Structure

This paper contains 12 sections, 13 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction and statement of main result
  2. Direct Approach
  3. Symmetrizing measure and variational formulation
  4. Estimate from Reilly Formula
  5. Approach via Interpolation
  6. Energy Decomposition
  7. Application to manifold case
  8. Further Examples
  9. Brownian motion on Euclidean balls with sticky boundary diffusion
  10. Brownian motion on balls with partial sticky reflecting boundary diffusion
  11. Stratified spaces -- Ball with a needle
  12. Continuity of C alpha

Key Result

Theorem 1.1

Let $\Omega$ be a smooth compact $d$-dimensional Riemmannian manifold with boundary $\partial \Omega$ such that for some $k_R, k_2 >0$ Then for $\sigma=\sigma_\alpha$ in eq:full_ev_pde with $\alpha \in [0,1]$ given by eq:def_of_alpha it holds that where and

Figures (3)

  • 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 \ref{['example_full_sphere_dim2']}.
  • Figure 2: The above two figures show the upper estimate given by the r.h.s of \ref{['estim_partial']}. In the case $\delta=0.9$ (Figure \ref{['delta2']}), the curve interpolates between the extremal constants $C_\Sigma$ and $C_\Omega$, as opposed to the half-sphere case (Figure \ref{['delta1']}).
  • 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 $\mathcal{L}$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 21 more