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On elliptic systems with $k$-wise interactions in the strong competition regime: uniform Hölder bounds and properties of the limiting configurations

Lorenzo Giaretto

Abstract

In this paper we investigate a class of variational reaction-diffusion systems with strong competition driven by beyond-pairwise interactions. The model involves $d$ nonnegative components interacting through $k$-wise terms, with $3 \leq k \leq d$, and includes symmetric interaction coefficients accounting for multi-component effects as well as suitable nonlinear terms. We focus on minimal energy solutions, proving uniform-in-$β$ Hölder bounds up to an explicit threshold exponent depending only on the dimension of the space and on the order $k$ of the interaction. As $β\to +\infty$, we show that minimizers converge strongly in $H^1$ and in Hölder spaces to a partially segregated configuration, characterized as minimizer of a natural variational problem under a $k$-segregation constraint. Finally, we prove that every minimizer of the limit problem enjoys the Hölder regularity and we derive some basic extremality conditions.

On elliptic systems with $k$-wise interactions in the strong competition regime: uniform Hölder bounds and properties of the limiting configurations

Abstract

In this paper we investigate a class of variational reaction-diffusion systems with strong competition driven by beyond-pairwise interactions. The model involves nonnegative components interacting through -wise terms, with , and includes symmetric interaction coefficients accounting for multi-component effects as well as suitable nonlinear terms. We focus on minimal energy solutions, proving uniform-in- Hölder bounds up to an explicit threshold exponent depending only on the dimension of the space and on the order of the interaction. As , we show that minimizers converge strongly in and in Hölder spaces to a partially segregated configuration, characterized as minimizer of a natural variational problem under a -segregation constraint. Finally, we prove that every minimizer of the limit problem enjoys the Hölder regularity and we derive some basic extremality conditions.
Paper Structure (15 sections, 33 theorems, 175 equations)

This paper contains 15 sections, 33 theorems, 175 equations.

Table of Contents

  1. Introduction and main results
  2. Main assumptions and results
  3. Previous works on partial segregation phenomena
  4. Further perspectives
  5. Preliminaries and known results
  6. Existence of minimizers with fixed trace
  7. Definition of $\alpha_{\ell,N}$
  8. A decay estimate
  9. A Liouville-type theorem
  10. New Liouville-type theorems
  11. Liouville-type theorems for $k$-partially segregated functions
  12. Liouville theorem for solutions to elliptic systems with $k$-wise interaction
  13. Interior uniform Hölder bounds
  14. Global Hölder bounds
  15. Characterization and properties of the limit

Key Result

Proposition 1

Under the above assumptions, eq:min Jbeta problem is achieved by at least one minimizer.

Theorems & Definitions (62)

  • Proposition 1
  • Remark 1
  • Theorem 1.1: Interior uniform bounds in Hölder spaces
  • Remark 2
  • Remark 3
  • Theorem 1.2: Global uniform Hölder bounds for minimizers with fixed traces
  • Remark 4
  • Theorem 1.3: Characterization of the limit problem
  • Corollary 1
  • Remark 5
  • ...and 52 more