Regularity Theory In Some Free Boundary Problems

In this doctoral thesis, we focus on the regularity theory for a large category of the free boundary problems. We provide a rather complete description of the regu-larity theory for a family of heterogeneous/homogeneous, one/two-phase variational free boundary problems, governed by nonlinear, degenerate/non-degenerate, elliptic operators. Included in such family are heterogeneous/homogeneous, one/two-phase obstacle type problems, jets and cavities problems of Prandtl-Batchelor type, singu-lar degenerate/non-degenerate equations which model the density of certain chemical specie in reaction with a porous catalyst pellet.One of the main work involved in this thesis is to establish the regularity of min-imizers in heterogeneous/homogeneous, two-phase obstacle type problems, jets and cavities problems, and chemical reaction problems in both Orlicz-Sobolev spaces and Sobolev spaces with variable exponents. The other task performed here is to establish the finite Hausdorff measure/Cloc1,α-regularity for the free boundary in the homoge-neous, one/two-phase obstacle type problems in Sobolev spaces and two-phase jets and cavity problems in Orlicz-Sobolev spaces, respectively.This thesis contributes the following to the free boundary problems’research field: by using Energy method, we solve a growth-problem which has appeared for many years. More precisely, we obtain the optimal growth rate for minimizers near the free boundary in the two-phase obstacle problem governed by p-Laplace operator. We show that the growth rate is of order p/p-1and then establish finite Hausdorff measure for the free boundary. Our method can be used in general equations, including free boundary problems and elliptic equations, to obtain the optimal regularity for solutions.