Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Free boundary regularity for the inhomogeneous one-phase Stefan problem

Fausto Ferrari, Nicolò Forcillo, Davide Giovagnoli, David Jesus

TL;DR

The paper addresses the problem of free boundary regularity for the inhomogeneous one-phase Stefan problem. It develops a hodograph-transported reformulation to a fixed-boundary quasilinear parabolic equation with an oblique boundary condition, and establishes an inhomogeneous $H$-property together with an improvement-of-flatness scheme to propagate regularity across scales. By comparing the nonlinear transformed problem to its homogeneous limiting problem and proving oscillation decay for the error function, the authors prove that flat free boundaries are $C^{1,\alpha}$. This advances the understanding of parabolic free boundary regularity under inhomogeneous sources and provides a robust framework that can be extended to fully nonlinear parabolic settings.

Abstract

In this paper, we prove that flat free boundaries of solutions to inhomogeneous one-phase Stefan problem are $C^{1,α}$.

Free boundary regularity for the inhomogeneous one-phase Stefan problem

TL;DR

The paper addresses the problem of free boundary regularity for the inhomogeneous one-phase Stefan problem. It develops a hodograph-transported reformulation to a fixed-boundary quasilinear parabolic equation with an oblique boundary condition, and establishes an inhomogeneous -property together with an improvement-of-flatness scheme to propagate regularity across scales. By comparing the nonlinear transformed problem to its homogeneous limiting problem and proving oscillation decay for the error function, the authors prove that flat free boundaries are . This advances the understanding of parabolic free boundary regularity under inhomogeneous sources and provides a robust framework that can be extended to fully nonlinear parabolic settings.

Abstract

In this paper, we prove that flat free boundaries of solutions to inhomogeneous one-phase Stefan problem are .
Paper Structure (12 sections, 12 theorems, 172 equations)

This paper contains 12 sections, 12 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Some remarks about the scalings of the problem
  5. The nonlinear problem arising from the hodograph transform
  6. The inhomogeneous $H$-property
  7. Statement and discussion of our main result
  8. Regularity result for the error function
  9. Regularity results for the limiting profile
  10. Comparison principle
  11. The oscillation decay properties of the error function $w$
  12. Proofs of the main results

Key Result

Theorem 1.1

Let $K>1$. There exists $0 <\lambda \leq 1$ such that if $u$ is a viscosity solution of the one-phase Stefan problem eq:StefanNH in $B_{2\lambda} \times [-2K^{-1}\lambda,0]$, $0\in \mathcal{F}(u),$ and with for some small $\varepsilon$ only depending on $K$ and $n$, then the free boundary $\mathcal{F}(u)$ is a $C^{1,\alpha}$-graph in the $x_n$-direction in $B_\lambda \times[-K^{-1}\lambda ,0]$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Definition 3.5: $H(\sigma)$-property
  • Proposition 3.6: Improvement of flatness
  • Remark 3.7
  • Proposition 4.1: Interior estimates
  • ...and 12 more