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The free boundary for semilinear problems with highly oscillating singular terms

Mark Allen, Dennis Kriventsov, Henrik Shahgholian

TL;DR

The paper develops a theory for free boundaries in one-phase semilinear problems $\Delta u=f(u)$ where $f$ can oscillate and fail homogeneity near $u=0$. By combining a Kelvin-transform-based flatness-to-Lipschitz argument with a hodograph transform that yields a degenerate elliptic PDE, and then applying refined $W^{1,p}$ estimates in a weighted, near-identity setting, the authors prove that flat free boundaries are $C^{\infty}$-smooth. The work introduces a flexible nonlinearity class $\mathcal{F}(M,\gamma_1,\gamma_2)$, proves the existence and growth control of a canonical profile $h$ solving $h''=f(h)$, and develops new Cordes-Nirenberg-type estimates for the degenerate equation to bridge from Lipschitz to $C^{\infty}$ regularity. These results extend classical obstacle and Alt-Phillips-type regularity to highly oscillatory, non-homogeneous nonlinearities with robust analytical techniques.

Abstract

We investigate general semilinear (obstacle-like) problems of the form $Δu = f(u)$, where $f(u)$ has a singularity/jump at $\{u=0\}$ giving rise to a free boundary. Unlike many works on such equations where $f$ is approximately homogeneous near $u = 0$, we work under assumptions allowing for highly oscillatory behavior. We establish the $C^\infty$ regularity of the free boundary $\partial \{u>0\}$ at flat points. Our approach is to first establish that flat free boundaries are Lipschitz, using a comparison argument with the Kelvin transform. For higher regularity, we study the highly degenerate PDE satisfied by ratios of derivatives of $u$, using changes of variable and then the hodograph transform. Along the way, we prove and make use of new Caffarelli-Peral type $W^{1, p}$ estimates for such degenerate equations. Much of our approach appears new even in the case of Alt-Phillips and classical obstacle problems.

The free boundary for semilinear problems with highly oscillating singular terms

TL;DR

The paper develops a theory for free boundaries in one-phase semilinear problems where can oscillate and fail homogeneity near . By combining a Kelvin-transform-based flatness-to-Lipschitz argument with a hodograph transform that yields a degenerate elliptic PDE, and then applying refined estimates in a weighted, near-identity setting, the authors prove that flat free boundaries are -smooth. The work introduces a flexible nonlinearity class , proves the existence and growth control of a canonical profile solving , and develops new Cordes-Nirenberg-type estimates for the degenerate equation to bridge from Lipschitz to regularity. These results extend classical obstacle and Alt-Phillips-type regularity to highly oscillatory, non-homogeneous nonlinearities with robust analytical techniques.

Abstract

We investigate general semilinear (obstacle-like) problems of the form , where has a singularity/jump at giving rise to a free boundary. Unlike many works on such equations where is approximately homogeneous near , we work under assumptions allowing for highly oscillatory behavior. We establish the regularity of the free boundary at flat points. Our approach is to first establish that flat free boundaries are Lipschitz, using a comparison argument with the Kelvin transform. For higher regularity, we study the highly degenerate PDE satisfied by ratios of derivatives of , using changes of variable and then the hodograph transform. Along the way, we prove and make use of new Caffarelli-Peral type estimates for such degenerate equations. Much of our approach appears new even in the case of Alt-Phillips and classical obstacle problems.
Paper Structure (8 sections, 30 theorems, 189 equations)

This paper contains 8 sections, 30 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Assumptions on nonlinearity
  3. Main results
  4. Preliminaries
  5. Flat implies Lipschitz
  6. Consequence of directional monotonicity
  7. Hodograph Transform
  8. $W^{1,p}$ estimates

Key Result

Theorem 1.1

Let $u$ be a $H_{\leq 1}$-minimizer of e:func in $B_1$, and fix $\delta>0$. There exists $\epsilon>0$ depending on $\delta,n,M,\gamma_1,\gamma_2$, such that if $u$ satisfies eq:flatness then $\partial_\nu u \geq 0$ in $B_{1/2}$ whenever $\nu \cdot e_n \geq \delta$. In particular the free boundary $\

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • ...and 46 more