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Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions

Marcos Solera, Julián Toledo

Abstract

We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of $p$-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others, weighted discrete graphs and $\mathbb{R}^N$ with a random walk induced by a nonsingular kernel. We also study the case of nonlinear dynamical boundary conditions. The generality of the nonlinearities considered allow us to cover the nonlocal counterparts of a large scope of local diffusion problems: Stefan problems, Hele-Shaw problems, diffusion in porous media problems, obstacle problems, and more. Nonlinear semigroup theory is the basis for this study.

Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions

Abstract

We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of -Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others, weighted discrete graphs and with a random walk induced by a nonsingular kernel. We also study the case of nonlinear dynamical boundary conditions. The generality of the nonlinearities considered allow us to cover the nonlocal counterparts of a large scope of local diffusion problems: Stefan problems, Hele-Shaw problems, diffusion in porous media problems, obstacle problems, and more. Nonlinear semigroup theory is the basis for this study.

Paper Structure

This paper contains 15 sections, 13 theorems, 509 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Metric random walk spaces
  3. Yosida approximation and a Bénilan-Crandall relation
  4. Nonlocal stationary problems
  5. Nonlocal diffusion operators of Leray-Lions type and nonlocal Neumann boundary operators
  6. Existence and uniqueness of solutions of doubly nonlinear stationary problems under nonlinear boundary conditions
  7. Existence of solutions of an approximate problem
  8. Monotonicity of the solutions of the approximate problems
  9. An $L^{p}$-estimate for the solutions of the approximate problems
  10. Existence of solutions of $(GP_\varphi)$
  11. Other boundary conditions
  12. Doubly nonlinear diffusion problems
  13. Nonlinear dynamical boundary conditions
  14. Nonlinear boundary conditions
  15. Poincaré type inequalities

Key Result

lemma 1

Let $\Omega\subset X$ be a $\nu$-finite set and let $\{u_k\}_{k\in\mathbb N}\subset L^p(\Omega,\nu)$ such that $u_k\stackrel{k}{\longrightarrow} u\in L^p(\Omega,\nu)$ in $L^p(\Omega,\nu)$ and pointwise $\nu$-a.e. in $\Omega$. Suppose also that there exists $h\in L^p(\Omega,\nu)$ such that $|u_k|\le and, in particular,

Theorems & Definitions (41)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • lemma 1
  • proof
  • remark 1
  • remark 2
  • definition 5
  • remark 3: Integration by parts formula
  • ...and 31 more