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On the Geometry of Solutions of the Fully Nonlinear Inhomogeneous One-Phase Stefan Problem

Fausto Ferrari, Davide Giovagnoli, David Jesus

TL;DR

This work addresses the geometry and regularity of the free boundary for the one-phase fully nonlinear Stefan problem with an inhomogeneous source. The authors develop a framework linking free-boundary flatness to solution flatness, using a parabolic Hopf-Oleinik lemma and a careful nondegeneracy condition that accounts for negative source terms. They prove a flatness-improvement result (Theorem \ref{Theorem:Flatsoltoflatfree}) showing that, upon rescaling, the solution near the free boundary is trapped between two close affine profiles, enabling the use of linear theory to deduce higher regularity; they also establish the equivalence between integral and pointwise nondegeneracy under Lipschitz control. The results extend prior regularity theories for homogeneous or linear cases to the fully nonlinear, inhomogeneous setting and clarify how flat free boundaries become smooth under minimal assumptions, with explicit dependence on the negative part of the source $f^-$.

Abstract

In this paper, we characterize the geometry of solutions to one-phase inhomogeneous fully nonlinear Stefan problem with flat free boundaries under a new nondegeneracy assumption. This continues the study of regularity of flat free boundaries for the linear inhomogeneous Stefan problem started in [9], as well as justifies the definition of flatness assumed in [15].

On the Geometry of Solutions of the Fully Nonlinear Inhomogeneous One-Phase Stefan Problem

TL;DR

This work addresses the geometry and regularity of the free boundary for the one-phase fully nonlinear Stefan problem with an inhomogeneous source. The authors develop a framework linking free-boundary flatness to solution flatness, using a parabolic Hopf-Oleinik lemma and a careful nondegeneracy condition that accounts for negative source terms. They prove a flatness-improvement result (Theorem \ref{Theorem:Flatsoltoflatfree}) showing that, upon rescaling, the solution near the free boundary is trapped between two close affine profiles, enabling the use of linear theory to deduce higher regularity; they also establish the equivalence between integral and pointwise nondegeneracy under Lipschitz control. The results extend prior regularity theories for homogeneous or linear cases to the fully nonlinear, inhomogeneous setting and clarify how flat free boundaries become smooth under minimal assumptions, with explicit dependence on the negative part of the source .

Abstract

In this paper, we characterize the geometry of solutions to one-phase inhomogeneous fully nonlinear Stefan problem with flat free boundaries under a new nondegeneracy assumption. This continues the study of regularity of flat free boundaries for the linear inhomogeneous Stefan problem started in [9], as well as justifies the definition of flatness assumed in [15].

Paper Structure

This paper contains 6 sections, 6 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Parabolic Hopf-Oleinik lemma
  5. From flat free boundaries to flat solutions
  6. The equivalence of the integral and pointwise nondegeneracy condition

Key Result

Theorem 1.1

Let $K>1$. There exists $0 <\lambda \leq 1$ such that if $u$ is a viscosity solution of which satisfies and for all $t_0 \ \in (-(K\lambda)^{-1},0)$ there exists $x_0 \in \partial B_{\frac{3}{4}}$ Then for all $\eta >0$ we have where the coefficients satisfy and the constants $p_0, \,\beta\in(0,1)$ and $c,C>0$ depend only on $K$ and $n$, provided that $\varepsilon_0$ is sufficiently small, dep

Figures (1)

  • Figure 1: The flatness condition expressed in Theorem \ref{['Theorem:Flatsoltoflatfree']}

Theorems & Definitions (11)

  • Theorem 1.1
  • Proposition 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Parabolic Weak Harnack inequality
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • ...and 1 more