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Equivalence of definitions of fractional caloric functions

Artur Rutkowski

Abstract

We prove equivalence between nonnegative distributional solutions of the fractional heat equation and caloric functions, i.e., functions satisfying the mean value property with respect to the space-time isotropic $α$-stable process. We also provide sufficient conditions for the boundary and exterior data under which the solutions are classical and we give off-diagonal estimates for the derivatives of the Dirichlet heat kernel and the lateral Poisson kernel, which might be of their own interest.

Equivalence of definitions of fractional caloric functions

Abstract

We prove equivalence between nonnegative distributional solutions of the fractional heat equation and caloric functions, i.e., functions satisfying the mean value property with respect to the space-time isotropic -stable process. We also provide sufficient conditions for the boundary and exterior data under which the solutions are classical and we give off-diagonal estimates for the derivatives of the Dirichlet heat kernel and the lateral Poisson kernel, which might be of their own interest.

Paper Structure

This paper contains 7 sections, 15 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Auxiliary estimates on the kernels
  4. Caloric functions are distributional solutions
  5. Caloric functions are classical solutions
  6. Distributional and classical solutions are caloric
  7. Continuity of $\partial_t u$

Key Result

Theorem 1.3

Let $D$ be bounded and Lipschitz and assume that $u\geq 0$ is caloric in $[0,T)\times D$ with the representation eq:urepr. If $g\in C^{\rm Dini}((0,T),L^1(1\wedge \nu))$, and $\mu\in C^{\rm Dini}((0,T),\mathcal{M}(\partial D))$, then $u$ is a classical solution to eq:FHE.

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 26 more