Regularity of the Free Boundary in Semilinear Problems

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Ghaffarinia, Omid | 2016

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 49451 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Fotouhi Firouzabad, Morteza
Abstract:
There are some situations where we would like to solve a partial differential equation (PDE) in a domain whose boundary is not known a priori; such a problem is called free boundary problem and the boundary is called free boundary. For this kind of problems, aside from standard boundary conditions, an extra condition is imposed at the free boundary and the goal would be finding the free boundary in addition to finding a solution for PDE. One of the classical examples of free boundary problems is the Stefan problem (melting of ice) in which ice and water temperatures are determined from the heat equation and we would be interested in finding the boundary between ice and water. Another famous free boundary problem is the ”obstacle problem”, which is the main interest of this thesis in which regularity and free boundary of obstacle problem are studied in both one-phase and two-phase cases. It is shown that solutions to both one-phase and two-phase obstacle problems areW2;1 (or C1;1) in space and W1;1 (or C0;1) in time. It is also shown that free boundary in both cases is a C1 graph
Keywords:
Parabolic Equations ; Partial Differential Equations ; Free Boundaries ; Obstacle Problem