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A parabolic free transmission problem: flat free boundaries are smooth

Dennis Kriventsov, María Soria-Carro

Abstract

We study a two-phase parabolic free boundary problem motivated by the jump of conductivity in composite materials that undergo a phase transition. Each phase is governed by a heat equation with distinct thermal conductivity, and a transmission-type condition is imposed on the free interface. We establish strong regularity properties of the free boundary: first, we prove that flat free boundaries are $C^{1,α}$ by means of a linearization technique and compactness arguments. Then we use the Hodograph transform to achieve higher regularity. To this end, we prove a new Harnack-type inequality and develop the Schauder theory for parabolic linear transmission problems.

A parabolic free transmission problem: flat free boundaries are smooth

Abstract

We study a two-phase parabolic free boundary problem motivated by the jump of conductivity in composite materials that undergo a phase transition. Each phase is governed by a heat equation with distinct thermal conductivity, and a transmission-type condition is imposed on the free interface. We establish strong regularity properties of the free boundary: first, we prove that flat free boundaries are by means of a linearization technique and compactness arguments. Then we use the Hodograph transform to achieve higher regularity. To this end, we prove a new Harnack-type inequality and develop the Schauder theory for parabolic linear transmission problems.
Paper Structure (14 sections, 25 theorems, 219 equations, 2 figures)

This paper contains 14 sections, 25 theorems, 219 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Function spaces
  5. Viscosity solutions
  6. Harnack Inequality
  7. Linear transmission problems
  8. Equations with constant coefficients and right-hand side
  9. Equations with variable coefficients and right-hand side
  10. Improvement of flatness
  11. $C^{1,\alpha}$ regularity of the free boundary
  12. Higher regularity of the free boundary
  13. A counterexample to Lipschitz regularity
  14. Appendix

Key Result

Theorem 1.1

Let $u \in C(Q_1)$ be a viscosity solution of FBP. There exists $\bar{\delta} >0$, depending on $n$, $a_+$, and $a_-$, such that if the graph of $u$ is $\delta$-flat in the $e_n$-direction in $Q_1=B_1\times (-1,0]$, i.e., then $F(u)$ is locally smooth in space and time.

Figures (2)

  • Figure 1: Illustration of the domain $D$ and the function $s(t)\phi_2(x_n + \tfrac{5}{8}t)$.
  • Figure 2: Graphs of $M$ (blue) and $U$ (red) intersecting at the unique zero $s_\varepsilon$.

Theorems & Definitions (52)

  • Theorem 1.1: Smoothness
  • Theorem 1.2: $C^{1,\alpha}$ regularity
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 3.2: Hopf Lemma
  • proof : Proof of Lemma \ref{['lem:harnack']}
  • Theorem 3.3: Harnack inequality
  • proof
  • Corollary 3.4
  • ...and 42 more