# Tangential Touch Between Free And Fixed Boundaries

Abstract (Summary)

This thesis consists of the following three papers concerning the tangential touch between free and fixed boundaries in elliptic, parabolic and fully non-linear free boundary problems. Paper I. Tangential touch between free and fixed boundaries in a problem from superconductivity In this paper we study regularity properties of the free boundary problem ?u = ?{|?u|?=0} in B+, u = 0 on B ? {x1 = 0}, where B+ = {|x| < 1, x1 > 0} and B = {|x| < 1}. If the origin is a free boundary point, then we show that the free boundary ?? touches the fixed boundary {x1 = 0} tangentially. Paper II. Global solutions and the parabolically tangential touch of the free and fixed boundaries In Q+ = B+×(?1, 0) we consider the following free boundary problem ? ? ? Hu = ?? in Q+ for some open set ? ? Q+, u = |?u| = 0 in Q+ \ ?, u = 0 on {x1 = 0} ? Q, where H = ? ? ?t, B+ is as above and Q = B × (?1, 0). The exact representation of global solutions (i.e., solutions in the entire half-space Rn + × R? ) is established. Using this we obtain that the free boundary touches the fixed one in a parabolically-tangent way. For the problem with u ? 0 the same results were obtained earlier by D.E. Apushkinskaya, H. Shahgholian and N.N. Uraltseva. Paper III. Behavior of the free boundary near contact points with fixed boundary for nonlinear elliptic equations, with P. Markowich The aim of this paper is to study the following free boundary problem ? ? ? F (D2u) = ?? in B+ for some open set ? ? B+, u = |?u| = 0 in B+ \ ?, u = 0 on {x1 = 0} ? B. Here F is a uniformly elliptic fully non-linear operator and the equation is satisfied in the viscosity sense. We show that nonnegative global solutions are one dimensional. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points.
Bibliographical Information:

Advisor:

School:Kungliga Tekniska högskolan

School Location:Sweden

Source Type:Doctoral Dissertation

Keywords:free boundary problems; regularity; contact points

ISBN:91-7283-550-8

Date of Publication:01/01/2003