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A note on Bernoulli type free boundary problem on collapsed RCD(K,N)-spaces

Sitan Lin

TL;DR

This work studies a Bernoulli-type free boundary problem on collapsed $RCD(K,N)$ spaces by minimizing the energy $E(v)=\int_{\Omega}|\nabla v|^2+\lambda_{+}\chi_{\{v>0\}}+\lambda_{-}\chi_{\{v<0\}}\,dm$ over an admissible class $\mathscr{A}_g$. It proves the existence of minimizers, and establishes local Hölder regularity and, under a linear growth condition, Lipschitz continuity; it also shows the free boundaries have locally finite perimeter and derives non-degeneracy. The analysis relies on $RCD(K,N)$-space calculus, stability under $p$-mGH convergence, and BV/perimeter theory, while avoiding the non-collapsed assumption. A key highlight is that the results apply to collapsed spaces, and the paper discusses how collapsed examples can preclude a manifold structure for the free boundary, underscoring the role of ambient geometry in free boundary regularity.

Abstract

In this paper, we investigate Bernoulli type free boundary problem on collapsed RCD(K,N)-spaces. We prove the existence of minimizers and prove the local Lipschitz continuity of minimizers provided that the negative part is locally Lipschitz continuous. In particular, we prove the local Lipschitz continuity of minimizers for the one-phase problem (i.e. when the solution is non-negative). And then we prove that the free boundaries of minimizers have locally finite perimeter. We emphasize that the proof in this paper applies to collapsed RCD(K,N)-spaces and does not rely on the non-collapsed condition.

A note on Bernoulli type free boundary problem on collapsed RCD(K,N)-spaces

TL;DR

This work studies a Bernoulli-type free boundary problem on collapsed spaces by minimizing the energy over an admissible class . It proves the existence of minimizers, and establishes local Hölder regularity and, under a linear growth condition, Lipschitz continuity; it also shows the free boundaries have locally finite perimeter and derives non-degeneracy. The analysis relies on -space calculus, stability under -mGH convergence, and BV/perimeter theory, while avoiding the non-collapsed assumption. A key highlight is that the results apply to collapsed spaces, and the paper discusses how collapsed examples can preclude a manifold structure for the free boundary, underscoring the role of ambient geometry in free boundary regularity.

Abstract

In this paper, we investigate Bernoulli type free boundary problem on collapsed RCD(K,N)-spaces. We prove the existence of minimizers and prove the local Lipschitz continuity of minimizers provided that the negative part is locally Lipschitz continuous. In particular, we prove the local Lipschitz continuity of minimizers for the one-phase problem (i.e. when the solution is non-negative). And then we prove that the free boundaries of minimizers have locally finite perimeter. We emphasize that the proof in this paper applies to collapsed RCD(K,N)-spaces and does not rely on the non-collapsed condition.
Paper Structure (9 sections, 22 theorems, 118 equations)

This paper contains 9 sections, 22 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Calculus tools
  4. Pointed measured Gromov-Hausdorff convergence and stability results
  5. Existence
  6. Hölder continuity
  7. Lipschitz continuity
  8. Non-degeneracy
  9. Local finiteness of perimeter

Key Result

Theorem 1.1

Let $(X,d,m)$ be an $RCD(K,N)$-space with $K\in\mathbb{R}$ and $N\in(1,\infty)$ and let $\Omega\subset X$ be a bounded open domain. Given $g\in W^{1,2}(\Omega)$, we define $\mathscr{A}_{g}:=\{v\in W^{1,2}(\Omega):v-g\in W^{1,2}_{0}(\Omega)\}$ and consider the following minimization problem: where $E(v)=\int_{\Omega}\lvert\nabla v\rvert^{2}+\lambda_{+}\chi_{\{v>0\}}+\lambda_{-}\chi_{\{v<0\}}\mathr

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • ...and 40 more