Convex configurations in free boundary problems

by Petrosyan, Arsak

The thesis consists of the following three papers on free boundary problems for parabolic and elliptic partial differential equations with certain convexity assumptions on the initial data. Convexity and uniqueness in a free boundary problem arising in combustion theory We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain ? ? QT = Rn × (0, T) for some T > 0 and a function u > 0 in ? such that ? ? ?u ? ut = 0 in ? (P) u = 0, |?u| = 1 on ?: = ?? ? QT ? u(·, 0) = u0 on ?0, where ?0 is a given domain in Rn and u0 is a positive and continuous function in ?0, vanishing on ??0. If ?0 is convex and u0 is concave in ?0, then we show that (u, ?) is unique and the time sections ?t are convex for every t ? (0, T), provided the free boundary ? is locally the graph of a Lipschitz function and the fixed gradient condition is understood in the classical sense. On existence and uniqueness in a free boundary problem from combustion with L. Caffarelli We continue the study of problem (P) above under certain geometric assumptions on the initial data. The problem arises in the limit as ? ? 0 of a singular perturbation problem (P?) { ?u ? ? u ? t = ??(u?) in QT , u?(·, 0) = u? 0 on Rn, where ??(s) = (1/?)?(s/?) is a nonnegative Lipschitz function, supp?? = [0, ?] and ? ? 0 ??(s)ds = 1/2. Generally, no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. In the case when ?0 is convex and u0 is log-concave and satisfies the condition ?M ? ?u0 ? 0, we prove that the minimal supersolution is a classical solution of the free boundary problem for a short time interval. A free boundary problem for ?-Laplace equation with J. Manfredi and H. Shahgholian We consider a free boundary problem for the p-Laplacian ?pu = div(|?u|p?2?u), describing the nonlinear potential flow past convex profile K with prescribed pressure gradient |?u(x)| = a(x) on the free stream line. The main purpose of this paper is to study the limit as p ? ? of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding problem for the ?-Laplacian ??u = ?2u?u · ?u, in a certain weak sense, strong however, to guarantee the uniqueness. We show also that in the special case a(x) ? a0 > 0 the limit coincides with an explicit solution, given by a distance function. 2000 Mathematics Subject Classification: Primary 35R35, 35K05, 35J60 Key Words: Free boundary problems, convexity, classical solutions, the heat equation, p-Laplacian, ?-Laplacian, the propagation of flames, nonlinear potential flow.