Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A nonlinear model for long-range segregation

Howen Chuah, Stefania Patrizi, Monica Torres

Abstract

We study a system of fully nonlinear elliptic equations, depending on a small parameter $\eps$, that models long-range segregation of populations. The diffusion is governed by the negative Pucci operator. In the linear case, this system was previously investigated by Caffarelli, the second author, and Quitalo in \cite{CL2} as a model in population dynamics. We establish the existence of solutions and prove convergence as $\eps\to0^+$ to a free boundary problem in which populations remain segregated at a positive distance. In addition, we show that the supports of the limiting functions are sets of finite perimeter and satisfy a semi-convexity property.

A nonlinear model for long-range segregation

Abstract

We study a system of fully nonlinear elliptic equations, depending on a small parameter , that models long-range segregation of populations. The diffusion is governed by the negative Pucci operator. In the linear case, this system was previously investigated by Caffarelli, the second author, and Quitalo in \cite{CL2} as a model in population dynamics. We establish the existence of solutions and prove convergence as to a free boundary problem in which populations remain segregated at a positive distance. In addition, we show that the supports of the limiting functions are sets of finite perimeter and satisfy a semi-convexity property.

Paper Structure

This paper contains 15 sections, 22 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Main results
  4. Organization of the paper
  5. Notations
  6. Preliminaries
  7. The Pucci extremal operators.
  8. Regularity results
  9. The comparison principle
  10. The Schauder fixed-point theorem
  11. Scaling properties of $H_R$.
  12. Existence of Solutions for the Model \ref{['main_problems']}
  13. Uniform Estimates and the Limit Problem
  14. A Semiconvexity Property of the Free Boundary
  15. Appendix: a proof of Theorem \ref{['key']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded Lipschitz domain of $\mathbb{R}^n$. Assume fiassumption holds true. Then for any $\varepsilon > 0$, and $0<R\leq 1$, there exist positive functions $u^{\varepsilon}_1,\ldots, u^{\varepsilon}_K\in C^{\alpha}(\overline{\Omega})\cap C_{loc}^{2,\alpha}(\Omega)$, for some $0<\

Theorems & Definitions (29)

  • Theorem 1.1: Existence of Solutions
  • Theorem 1.2: Limit Problem
  • Theorem 1.3: A Semiconvexity Property of the Free Boundary
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5: Comparison Principle
  • Theorem 2.6: Minimum Principle
  • ...and 19 more