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Existence and regularity in the fully nonlinear one-phase free boundary problem

Matteo Carducci, Bozhidar Velichkov

TL;DR

This work analyzes a fully nonlinear one-phase free boundary problem with a right-hand side and a free boundary condition depending on the normal. It proves existence of viscosity solutions using Perron’s method, defining an admissible supersolution family and a strict minorant to obtain a Perron solution that is Lipschitz and non-degenerate. The paper then develops a quadratic improvement of flatness to show that flat free boundaries are $C^{2,\\alpha}$, and it leverages a hodograph transform to bootstrap to higher regularity, including $C^{k+2,\\beta}$ or analytic regularity under smooth data. The results extend the Alt–Caffarelli framework to general fully nonlinear operators, providing a comprehensive theory for existence and high-order regularity of free boundaries in this nonlinear setting.

Abstract

We consider viscosity solution to one-phase free boundary problems for general fully nonlinear operators and free boundary condition depending on the normal vector. We show existence of viscosity solutions via the Perron's method and we prove $C^{2,α}$ regularity of flat free boundaries via a quadratic improvement of flatness. Finally, we obtain the higher regularity of the free boundary via an hodograph transform.

Existence and regularity in the fully nonlinear one-phase free boundary problem

TL;DR

This work analyzes a fully nonlinear one-phase free boundary problem with a right-hand side and a free boundary condition depending on the normal. It proves existence of viscosity solutions using Perron’s method, defining an admissible supersolution family and a strict minorant to obtain a Perron solution that is Lipschitz and non-degenerate. The paper then develops a quadratic improvement of flatness to show that flat free boundaries are , and it leverages a hodograph transform to bootstrap to higher regularity, including or analytic regularity under smooth data. The results extend the Alt–Caffarelli framework to general fully nonlinear operators, providing a comprehensive theory for existence and high-order regularity of free boundaries in this nonlinear setting.

Abstract

We consider viscosity solution to one-phase free boundary problems for general fully nonlinear operators and free boundary condition depending on the normal vector. We show existence of viscosity solutions via the Perron's method and we prove regularity of flat free boundaries via a quadratic improvement of flatness. Finally, we obtain the higher regularity of the free boundary via an hodograph transform.
Paper Structure (30 sections, 29 theorems, 265 equations)

This paper contains 30 sections, 29 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. State of the art
  3. Main results
  4. Existence of viscosity solutions
  5. Regularity of the free boundary
  6. Sketch of the proof and some remarks.
  7. Existence of viscosity solutions
  8. Regularity of the free boundary
  9. Remarks about the Perron's solutions
  10. Preliminaries
  11. Notations
  12. Viscosity settings
  13. Remarks on the admissible family $\mathcal{A}$ and the strict minorant $\underline u$
  14. Interior Harnack inequality
  15. Construction of barriers
  16. ...and 15 more sections

Key Result

Theorem 1.4

Suppose that $\mathcal{F}\in\mathcal{E}(\lambda,\Lambda)$, $f\in C^0( \overline B_1)$ and $g\in {C^0}(\overline B_1,\mathbb{S}^{d-1})$ satisfies the hypothesis hyp:h1. Let $\phi:\partial B_1\to\mathbb{R}$ be a non-negative continuous function, $\mathcal{A}$ be the set of admissible supersolutions fr is non-empty and the function $u:\overline B_1\to\mathbb{R}$ defined as is a viscosity solution to

Theorems & Definitions (63)

  • Definition 1.1: Viscosity solutions to \ref{['def:def-viscosity-solution']}
  • Definition 1.2: Admissible family of supersolutions $\mathcal{A}$
  • Definition 1.3: Strict minorant
  • Theorem 1.4: Existence of viscosity solutions
  • Theorem 1.5: Flatness implies $C^{2,\alpha}$
  • Corollary 1.6: Higher regularity
  • Definition 2.1: Viscosity solutions of elliptic equations
  • Lemma 2.2
  • proof
  • Lemma 2.3: A strong maximum principle
  • ...and 53 more