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Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems

Joan Domingo-Pasarin, Xavier Ros-Oton

TL;DR

This work establishes that Lipschitz domains yield locally smooth free boundaries for weak solutions to the inhomogeneous one-phase Alt-Caffarelli problem when the data $f$ and $Q$ are smooth and $f\ge0$, $Q>0$. The authors develop a robust weak-solution framework, prove an improvement-of-flatness result, and perform a blow-up analysis to obtain 1-homogeneous limits; a Weiss monotonicity formula then drives the classification of blow-ups and the regularity. Two complementary pathways to regularity are employed: a direct Lipschitz-cone classification with an induction on dimension, and a viscosity-solution approach via positive free-boundary density that leverages De Silva's theory. Consequently, the paper yields an alternative proof of Serrin's overdetermined problem in Lipschitz domains and links domain regularity to Poisson-kernel regularity, enhancing the bridge between free boundary regularity and domain geometry in weak formulations.

Abstract

Motivated by the Serrin problem, we study weak solutions of the generalised Alt-Caffarelli problem $-Δu = f$ in $Ω$, $u = 0$ on $\partialΩ$, $\partial_νu = Q$ on $\partialΩ$. Our main result establishes that if $Ω$ is Lipschitz, then it is actually $C^{\infty}$ (provided that $f$ and $Q$ are smooth). This was known before only for viscosity solutions. As a corollary, we obtain an alternative solution of Serrin's problem in the case of Lipschitz domains. We also discuss the characterisation of the regularity of Lipschitz domains in terms of their Poisson kernel.

Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems

TL;DR

This work establishes that Lipschitz domains yield locally smooth free boundaries for weak solutions to the inhomogeneous one-phase Alt-Caffarelli problem when the data and are smooth and , . The authors develop a robust weak-solution framework, prove an improvement-of-flatness result, and perform a blow-up analysis to obtain 1-homogeneous limits; a Weiss monotonicity formula then drives the classification of blow-ups and the regularity. Two complementary pathways to regularity are employed: a direct Lipschitz-cone classification with an induction on dimension, and a viscosity-solution approach via positive free-boundary density that leverages De Silva's theory. Consequently, the paper yields an alternative proof of Serrin's overdetermined problem in Lipschitz domains and links domain regularity to Poisson-kernel regularity, enhancing the bridge between free boundary regularity and domain geometry in weak formulations.

Abstract

Motivated by the Serrin problem, we study weak solutions of the generalised Alt-Caffarelli problem in , on , on . Our main result establishes that if is Lipschitz, then it is actually (provided that and are smooth). This was known before only for viscosity solutions. As a corollary, we obtain an alternative solution of Serrin's problem in the case of Lipschitz domains. We also discuss the characterisation of the regularity of Lipschitz domains in terms of their Poisson kernel.
Paper Structure (18 sections, 24 theorems, 153 equations)

This paper contains 18 sections, 24 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Poisson kernel
  4. Strategy of the proof
  5. Organization of the paper
  6. Acknowledgements
  7. Weak solutions and basic properties
  8. Improvement of flatness for weak solutions
  9. Blow-up limits of weak solutions
  10. Homogeneity of blow-up limits
  11. Stationary solutions
  12. Weiss monotonicity formula
  13. Proof of Theorem \ref{['thm:1']}
  14. Lipschitz cones
  15. Proof using viscosity solutions
  16. ...and 3 more sections

Key Result

theorem 2

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain such that $0 \in \partial \Omega$ and assume $u$ solves eq:inhom-alt-caff-prob in the following weak sense: $u \in H^1_{\mathrm{loc}}(B_1)$, $u \geq 0$, $u = 0$ on $B_1 \setminus \Omega$ and If $f,Q \in C^{\infty}(B_1)$, $f \geq 0$ and $Q > 0$, then $\partial \Omega \cap B_1$ is locally a smooth surface in $B_1$.

Theorems & Definitions (64)

  • remark 1
  • theorem 2
  • theorem 3
  • theorem 4
  • remark 5
  • remark 6
  • definition 7
  • remark 8
  • remark 9
  • remark 10
  • ...and 54 more