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Geometric Estimates for Solutions of Semilinear Equations with Singular Potentials

Thialita M. Nascimento, Lei Zhang

Abstract

In this work, we study local minimizers of elliptic functionals with strong absorption terms and unbounded, sign-changing sources. These problems naturally interpolate between two classical free boundary problems: Bernoulli-type (cavity) and obstacle-type. While previous studies have focused on bounded and strictly positive sources, we extend sharp regularity and nondegeneracy estimates to the unbounded, sign-changing setting, providing a comprehensive analysis of how the underlying nonlinearity interacts with minimal integrability assumptions on the source.

Geometric Estimates for Solutions of Semilinear Equations with Singular Potentials

Abstract

In this work, we study local minimizers of elliptic functionals with strong absorption terms and unbounded, sign-changing sources. These problems naturally interpolate between two classical free boundary problems: Bernoulli-type (cavity) and obstacle-type. While previous studies have focused on bounded and strictly positive sources, we extend sharp regularity and nondegeneracy estimates to the unbounded, sign-changing setting, providing a comprehensive analysis of how the underlying nonlinearity interacts with minimal integrability assumptions on the source.
Paper Structure (10 sections, 15 theorems, 162 equations)

This paper contains 10 sections, 15 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Assumptions and Main results
  3. Notation and assumptions
  4. Main Results
  5. Auxiliarity results
  6. Existence and boundedness
  7. Universal Hölder estimates
  8. Regularity along the Free Boundary
  9. Nondegeneracy
  10. A radial example

Key Result

Theorem 1

Let $u$ be a local minimizer of main functional. There exist universal constants $0 < \beta \ll 1$ and $C_2 > 0$, such that $u \in C^{0,\beta} (B_{1/2})$ and

Theorems & Definitions (32)

  • Theorem 1: Local Hölder Regularity
  • Theorem 2: Regularity at the Free Boundary
  • Theorem 3: Nondegeneracy
  • Theorem 4
  • Remark 2.1
  • Lemma 1: Weak embedding
  • proof
  • Lemma 2
  • proof
  • Theorem 5: Existence
  • ...and 22 more