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Evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

José M. Mazón, Marcos Solera, Julián Toledo

Abstract

In this paper we study evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in the framework of metric random walk spaces. This covers cases with the $p$-Laplacian operator in weighted discrete graphs and nonlocal operators with nonsingular kernel in $\mathbb{R}^N$.

Evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

Abstract

In this paper we study evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in the framework of metric random walk spaces. This covers cases with the -Laplacian operator in weighted discrete graphs and nonlocal operators with nonsingular kernel in .

Paper Structure

This paper contains 12 sections, 20 theorems, 332 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Metric random walk spaces
  3. Completely accretive operators and semigroup theory
  4. The nonhomogeneous Neumann problem for evolution problems of Leray-Lions type
  5. Nonlocal Leray--Lions operators
  6. Neumann boundary operators
  7. Neumann boundary conditions of Gunzburger--Lehoucq type
  8. Neumann boundary conditions of Dipierro--Ros-Oton--Valdinoci type
  9. Particular cases
  10. The homogeneous Neumann boundary value problem
  11. Nonlocal problems with nonsingular kernels
  12. Weighted graphs

Key Result

Proposition \oldthetheorem

An operator $\mathcal{A} \subset L^1(X,\nu)\times L^1(X,\nu)$ is completely accretive if, for every $(u_i, v_i) \in \mathcal{A}$, $i=1,2$,

Theorems & Definitions (43)

  • Example \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem: BCr2
  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 33 more