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Interaction between initial behavior of temperature and the mean curvature of the interface in two-phase heat conductors

Shigeru Sakaguchi

TL;DR

The paper addresses how the initial temperature profile in a two-phase heat conductor encodes the mean curvature of the interface. It develops a local blow-up analysis combined with Dong interior estimates to derive a sharp asymptotic expansion: for $x$ on the interface and small $t$, $t^{-1/2}\left(u(x,t) - \frac{\sqrt{\sigma_+}}{\sqrt{\sigma_+}+\sqrt{\sigma_-}}\right) \to \frac{\sqrt{\sigma_+}\sqrt{\sigma_-}}{\sqrt{\pi}\left(\sqrt{\sigma_+}+\sqrt{\sigma_-}\right)}(N-1)H(x)$. This leads to corollaries that stationary isothermic surfaces have constant mean curvature and yields symmetry/rigidity results for overdetermined two-phase problems under weaker regularity assumptions than previously required. The approach combines a one-dimensional barrier, a parabolic blow-up, and interior parabolic estimates to produce a local, geometry-aware understanding of diffusion across interfaces. The results contribute to the broader understanding of how diffusion processes reveal geometric properties of interfaces in heterogeneous media and provide new regularity-relaxed criteria for related rigidity theorems.

Abstract

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media locally with different constant conductivities, where initially one medium has temperature 0 and the other has temperature 1. Under the assumption that a part of the interface between two media with different constant conductivities is of class $C^2$ in a neighborhood of a point $x$ on it, we extract the mean curvature of the interface at $x$ from the initial behavior of temperature at $x$. This result is purely local in space. As a corollary, when the whole Euclidean space consists of two media globally with different constant conductivities, it is shown that if a connected component $Γ$ of the interface is of class $C^2$ and is stationary isothermic, then the mean curvature of $Γ$ must be constant. Moreover, we apply this result to some overdetermined problems for two-phase heat conductors and obtain some symmetry theorems which relax considerably the regularity assumptions of some previous results.

Interaction between initial behavior of temperature and the mean curvature of the interface in two-phase heat conductors

TL;DR

The paper addresses how the initial temperature profile in a two-phase heat conductor encodes the mean curvature of the interface. It develops a local blow-up analysis combined with Dong interior estimates to derive a sharp asymptotic expansion: for on the interface and small , . This leads to corollaries that stationary isothermic surfaces have constant mean curvature and yields symmetry/rigidity results for overdetermined two-phase problems under weaker regularity assumptions than previously required. The approach combines a one-dimensional barrier, a parabolic blow-up, and interior parabolic estimates to produce a local, geometry-aware understanding of diffusion across interfaces. The results contribute to the broader understanding of how diffusion processes reveal geometric properties of interfaces in heterogeneous media and provide new regularity-relaxed criteria for related rigidity theorems.

Abstract

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media locally with different constant conductivities, where initially one medium has temperature 0 and the other has temperature 1. Under the assumption that a part of the interface between two media with different constant conductivities is of class in a neighborhood of a point on it, we extract the mean curvature of the interface at from the initial behavior of temperature at . This result is purely local in space. As a corollary, when the whole Euclidean space consists of two media globally with different constant conductivities, it is shown that if a connected component of the interface is of class and is stationary isothermic, then the mean curvature of must be constant. Moreover, we apply this result to some overdetermined problems for two-phase heat conductors and obtain some symmetry theorems which relax considerably the regularity assumptions of some previous results.
Paper Structure (11 sections, 6 theorems, 82 equations)

This paper contains 11 sections, 6 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Reduction to the case where $\Omega$ is a bounded $C^2$ domain with \ref{['two-phase']}
  3. A key simple estimate and the solution in one dimension
  4. Proof of Theorem \ref{['th: asymptotic formula in time']}
  5. Introducing a principal coordinate system at each point on the interface
  6. Straightening the interface and scaling for blow-up arguments
  7. Utilizing the interior estimates for two-phase problems
  8. Getting the first term in the asymptotic formula
  9. Getting the second term in the asymptotic formula
  10. The uniform convergence on the interface
  11. Proofs of Theorems \ref{['th:hyperplane']} and \ref{['th:sphere theorem']}

Key Result

Theorem 1.1

For every point $x \in \partial\Omega\cap B_{\rho}(p)$, the following formula holds true: where $H(x)$ denotes the mean curvature of $\partial\Omega$ at $x\in\partial\Omega\cap B_\rho(p)$ with respect to the outward normal direction to $\partial\Omega$. The convergence is uniform on $\partial\Omega\cap B_r(p)$ for each $0 < r < \rho$.

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.1
  • Lemma 4.1