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The One-Phase Bifurcation For The p-Laplacian

Alaa Haj Ali, Peiyong Wang

Abstract

A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the p-Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second part, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.

The One-Phase Bifurcation For The p-Laplacian

Abstract

A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the p-Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second part, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.

Paper Structure

This paper contains 4 sections, 7 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. A Third Solution
  3. A Comparison Principle for Evolution
  4. Convergence of Evolution

Key Result

Lemma 2.1

For any $a$ and $b\in\mathbb{R}^n$, it holds where $C(p) > 0$. If $1 < p < 2$, then where $C(p) = p(p-1)$.

Theorems & Definitions (7)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 4.1