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Logarithmic continuity for the Nonlocal degenerate two-phase Stefan problem

Kyeongbae Kim, Ho-Sik Lee, Harsh Prasad

Abstract

We establish certain oscillation estimates for weak solutions to nonlinear, anomalous phase transitions modeled on the nonlocal two-phase Stefan problem. The problem is singular in time, is scaling deficient and influenced by far-off effects. We study the the problem in a geometry adapted to the solution and obtain oscillation estimates in intrinsically scaled cylinders. Furthermore, via certain uniform estimates, we construct a continuous weak solution to the corresponding initial boundary value problem with a quantitative modulus of continuity.

Logarithmic continuity for the Nonlocal degenerate two-phase Stefan problem

Abstract

We establish certain oscillation estimates for weak solutions to nonlinear, anomalous phase transitions modeled on the nonlocal two-phase Stefan problem. The problem is singular in time, is scaling deficient and influenced by far-off effects. We study the the problem in a geometry adapted to the solution and obtain oscillation estimates in intrinsically scaled cylinders. Furthermore, via certain uniform estimates, we construct a continuous weak solution to the corresponding initial boundary value problem with a quantitative modulus of continuity.

Paper Structure

This paper contains 15 sections, 20 theorems, 337 equations.

Table of Contents

  1. Introduction
  2. History and Motivation
  3. Interior Oscillation Estimate
  4. Oscillation Estimate near the Boundary
  5. Existence of a continuous solution
  6. Technical Tools
  7. Organization of the paper
  8. Preliminaries
  9. Interior Estimates
  10. Oscillation estimate of solutions
  11. Estimates on the Lateral Boundary
  12. Uniform continuity
  13. Estimates on the Initial Boundary
  14. Uniform continuity
  15. Existence of a continuous solution

Key Result

theorem 1

Let $u_\epsilon$ be a bounded weak solution to Let us take Suppose $Q_{\rho_0}^{(\omega_0/4)^{2-p}}(z_0)\subset {Q}_{\mathcal{R}}(z_0)$ for some constant $\rho_0\in(0,\mathcal{R}]$. Then there is a constant $\varsigma=\varsigma(n,s,p,\Lambda)\in(0,1)$ such that for any $r< \rho_0$, for some constant $c=c(n,s,p,\Lambda)$ independent of $\epsilon$.

Theorems & Definitions (41)

  • definition 1
  • theorem 1
  • definition 2
  • theorem 2
  • theorem 3
  • remark 1
  • definition 3
  • theorem 4
  • lemma 1
  • proof
  • ...and 31 more