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The evolution problem associated with the fractional first eigenvalue

Begoña Barrios, Leandro M. Del Pezzo, Alexander Quaas, Julio D. Rossi

Abstract

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as $t\to \infty$ with a bound that is given by the first eigenvalue for this problem that we also study.

The evolution problem associated with the fractional first eigenvalue

Abstract

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as with a bound that is given by the first eigenvalue for this problem that we also study.
Paper Structure (4 sections, 14 theorems, 145 equations)

This paper contains 4 sections, 14 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Regularity, up to the boundary, for the elliptic problem
  3. Eigenvalue problem
  4. Evolution problem

Key Result

Theorem 1.1

If $\Omega$ is a bounded strictly convex domain, $u_0{\in\mathcal{C}(\mathbb{R}^N)}$ and $g$ verifying hip, then there exists a unique viscosity solution to Moreover, a comparison principle holds; that is, if $\overline{u}$ is a supersolution and $\underline{u}$ is a subsolution to ev_eq then it holds that for every $(x,t) \in \Omega \times (0,\infty)$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 20 more