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Non-Positivity of the heat equation with non-local Robin boundary conditions

Jochen Glück, Jonathan Mui

Abstract

We study heat equations $\partial_t u - \operatorname{div}(A\nabla u) = 0$ on bounded Lipschitz domains $Ω$, where $-\operatorname{div}(A\nabla\,\cdot\,)$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $ν\cdot A\nabla u + Bu=0$, where $B\in\mathcal{L}(L^2(\partialΩ))$ is a general operator. In contrast to large parts of the literature on non-local Robin boundary conditions, we also allow for operators $B$ that destroy the positivity preserving property of the solution semigroup. Nevertheless, we obtain ultracontractivity of the semigroup under quite mild assumptions on $B$. For a certain class of operators $B$ we demonstrate that the semigroup is in fact eventually positive rather than positivity preserving.

Non-Positivity of the heat equation with non-local Robin boundary conditions

Abstract

We study heat equations on bounded Lipschitz domains , where is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by , where is a general operator. In contrast to large parts of the literature on non-local Robin boundary conditions, we also allow for operators that destroy the positivity preserving property of the solution semigroup. Nevertheless, we obtain ultracontractivity of the semigroup under quite mild assumptions on . For a certain class of operators we demonstrate that the semigroup is in fact eventually positive rather than positivity preserving.
Paper Structure (15 sections, 19 theorems, 89 equations, 1 figure)

This paper contains 15 sections, 19 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results and outline of paper
  3. Earlier work on non-local Robin problems
  4. A taste of eventual positivity
  5. Setting the stage
  6. Lipschitz domains
  7. Real and positive functions and operators
  8. Semigroup theory
  9. Non-local Robin boundary conditions via forms
  10. Ultracontractivity
  11. Eventual positivity: the case
  12. Eventual positivity: the case
  13. Invariance and equivariance under group actions
  14. Equivariance of the Robin Laplacian under group actions
  15. Group actions and their invariant functions

Key Result

Theorem 1.1

Let $\Omega\subseteq\mathbb{R}^d$ be a bounded Lipschitz domain, and let $B$ be a bounded and self-adjoint linear operator on $L^2(\partial\Omega)$ that leaves $L^\infty(\partial\Omega)$ invariant and maps real-valued functions to real-valued functions. The semigroup $(e^{-tL_B})_{t\ge 0}$ enjoys th

Figures (1)

  • Figure 1: The graphs of $f_1, f_2$ encode the positive eigenvalues for a given $b>0$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Lemma 3.1
  • proof
  • ...and 31 more