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Elliptic operators with non-local Wentzell-Robin boundary conditions

Markus Kunze, Jonathan Mui, David Ploss

TL;DR

This work analyzes strictly elliptic second-order operators on bounded Lipschitz domains subject to non-local Wentzell–Robin boundary conditions, formalized via integral kernels on the boundary. It proves generation of strongly continuous semigroups on $L^2(\Omega)\times L^2(\Gamma)$ and on spaces of continuous functions, and characterizes when these semigroups are positive, sub-Markovian, or Markovian. A detailed spectral theory explains asymptotic behavior, with explicit results for positive semigroups and dominant eigenvalues; the paper also develops criteria for eventual positivity and provides a concrete 1D example illustrating when positivity or eventual positivity holds under boundary perturbations. The results advance understanding of non-local boundary effects in elliptic problems and have potential probabilistic interpretations and applications in physics and engineering.

Abstract

In this article, we study strictly elliptic, second-order differential operators on a bounded Lipschitz domain in $\mathbb{R}^d$, subject to certain non-local Wentzell-Robin boundary conditions. We prove that such operators generate strongly continuous semigroups on $L^2$-spaces and on spaces of continuous functions. We also provide a characterisation of positivity and (sub-)Markovianity of these semigroups. Moreover, based on spectral analysis of these operators, we discuss further properties of the semigroup such as asymptotic behaviour and, in the case of a non-positive semigroup, the weaker notion of eventual positivity of the semigroup.

Elliptic operators with non-local Wentzell-Robin boundary conditions

TL;DR

This work analyzes strictly elliptic second-order operators on bounded Lipschitz domains subject to non-local Wentzell–Robin boundary conditions, formalized via integral kernels on the boundary. It proves generation of strongly continuous semigroups on and on spaces of continuous functions, and characterizes when these semigroups are positive, sub-Markovian, or Markovian. A detailed spectral theory explains asymptotic behavior, with explicit results for positive semigroups and dominant eigenvalues; the paper also develops criteria for eventual positivity and provides a concrete 1D example illustrating when positivity or eventual positivity holds under boundary perturbations. The results advance understanding of non-local boundary effects in elliptic problems and have potential probabilistic interpretations and applications in physics and engineering.

Abstract

In this article, we study strictly elliptic, second-order differential operators on a bounded Lipschitz domain in , subject to certain non-local Wentzell-Robin boundary conditions. We prove that such operators generate strongly continuous semigroups on -spaces and on spaces of continuous functions. We also provide a characterisation of positivity and (sub-)Markovianity of these semigroups. Moreover, based on spectral analysis of these operators, we discuss further properties of the semigroup such as asymptotic behaviour and, in the case of a non-positive semigroup, the weaker notion of eventual positivity of the semigroup.

Paper Structure

This paper contains 11 sections, 31 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Second order elliptic operators
  3. The sectorial form
  4. Positivity and Markov properties
  5. The semigroup on the space of continuous functions
  6. Spectral theory and asymptotic behavior
  7. The case of positive semigroups
  8. Perturbations of dissipative, self-adjoint operators
  9. Eventual positivity
  10. A 1-dimensional example in detail
  11. Bounded perturbations of weak star semigroups

Key Result

Lemma 2.3

Let $f\in L^2(\Omega)$ and $g\in L^2(\Gamma)$. If $u\in H^1(\Omega)$ satisfies for all $v\in H^1(\Omega)$, then $u\in D_{\mathrm{max}}(L)$, $Lu=f$ and $\partial^L_\nu u = g$. In this case, we say that $u$ is a weak solution of the Neumann boundary value problem

Figures (2)

  • Figure 1: Spectral regions $\mathbb{H}$ and $\mathbb{S}$, and integration path $\gamma$.
  • Figure 2: The function $f(\mu):=\mu\sqrt{2\mu\cot(\mu)+1-\mu^2}$.

Theorems & Definitions (78)

  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Example 3.2
  • Theorem 3.3
  • proof
  • ...and 68 more