Parabolic problems with thin free boundaries

Wenhui Shi, Purdue University

Abstract

This thesis contains two papers on parabolic thin free boundary problems. The first paper is based on a joint work with A. Petrosyan on the parabolic boundary Harnack principle in domains with thin Lipschitz complement. We establish the forward and backward parabolic boundary Harnack principle in such domains by carefully defining the corkscrew points associated to the boundary points on the thin complement. This result has an application in the one-phase parabolic Signorini problem which was recently studied in [1]. The second paper is based on a joint work with M. Allen on a two-phase parabolic Signorini problem [2]. This problem models the heat control on certain portion of the boundary of a bounded domain. In the paper, we first show that the weak solutions (in the sense of variational inequalities) are Lipschitz up to the boundary. Then we study the "blow-up" profiles and conclude that the two free boundaries (corresponding to the positive phase and negative phase) cannot touch. Hence one can locally reduce the study of this two-phase problem to the above one-phase parabolic Signorini problem, for which the optimal regularity of the solutions as well as the structure of the free boundaries are known [1, 3].

Degree

Ph.D.

Advisors

Petrosyan, Purdue University.

Subject Area

Mathematics

COinS