Caarelli, Luis1752633052008-08-282017-05-112008-08-282017-05-112007http://hdl.handle.net/2152/3374In this dissertation, we study a free boundary problem obtained as a limit as [epsilon omplies 0] to the following regularizing family of semilinear equations [Delta]u = [Beta subscript epsilon](u)F([gradient]u), where [Beta subscript epsilon] approximates the Dirac delta in the origin and F is a Lipschitz function bounded away from 0 and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces that are uniform. This allows to prove that the free boundary of the limit has the "right" weak geometry, in the measure theoretical sense. By the construction of some barriers with curvature, the classification of global profiles for the blow-up analysis is carried out and the limit function is proven to be a viscosity and pointwise solution (a.e) to a free boundary problem. Finally, the free boundary is proven to be a C[superscript 1, alpha] surface around H[superscript n-1] a.e. point.electronicengCopyright © is held by the author. Presentation of this material on the Libraries' web site by University Libraries, The University of Texas at Austin was made possible under a limited license grant from the author who has retained all copyrights in the works.Boundary value problems--Numerical solutionsThesis