Access Type

Dissertation/Thesis

Date of Award

January 2019

Degree Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

First Advisor

Pei-Yong Wang

Abstract

In the first part of this thesis, 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 the first chapter. 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 chapter and third chapter, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.

In the second part of this thesis, we study a singularly perturbed free boundary problem arising from a real problem associated with a Radiographic Integrated Test Stand and concerning a solution of the equation $\Delta u = f(u)$ in a domain $\Omega$ subject to constant boundary data, where the function $f$ in general is not monotone. In chapter 4, we let the domain $\Omega$ be a perfect ring and we incorporate a new idea of radial correction into the classical moving plane method to prove the radial symmetry of a solution. In chapter 5, we let the domain $\Omega$ be slightly shifted from a ring and we establish the stability of the solution by showing the approximate radial symmetry of the free boundary and the solution. For this purpose, we complete the proof via an evolutionary point of view, as an elliptic comparison principle is false, nevertheless a parabolic one holds.

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