Thin free boundary problems
Abstract
In this work I consider two free boundary problems of codimension two. The first free boundary problem consists of studying minimizers to the functional [special characters omitted]We prove that there is a separation of the positivity and negativity phases. We also prove results on the regularity of the free boundary. In the second problem we study minimizers of the functional[special characters omitted] This can be considere the two-phase fractional obstacle problem. We prove the optimal regularity of the minimizers. We also show that when a ≥ 0 there is a separation of the positivity and negativity phases which reduces the study of the free boundary to the study of the free boundary in the one-phase fractional obstacle problem. When a < 0 we provide an example showing that the positivity and negativity phases can touch. We show that in such instances the free boundary is a C1,α manifold. When the phases do not touch, the study of the free boundary is again reduced to to the one-phase problem.
Degree
Ph.D.
Advisors
Petrosyan, Purdue University.
Subject Area
Mathematics
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