The aim of this proposal is to investigate several free boundary problems that arise naturally while studying a number of physical phenomena in elasticity, in phase change of materials, in the flow of two liquids in models of jets and cavities as well as in shape optimization. The project consists of four main topics: (1) Stefan-type free boundary problems describing solid-fluid phase transition. The objective is to prove existence and regularity results for those moving-boundary problems in which the moving interface has an active role in the overall process. (2) Thin obstacle type problems describing the behavior of semi-permeable membranes. Mathematically the questions to be investigated include regularity properties of free boundaries and of solutions of systems of equations that model phenomena in elasticity such as the vectorial Signorini problem. (3) Free boundary regularity problems for elliptic measures appearing in optimal design and optimal control problems. The proposed research includes the study of the singular set of the free boundary as well as questions of how the geometry of a domain can be recovered from the regularity of its elliptic measure by applying Geometric Measure Theory techniques. (4) Problems appearing in image processing and optimization problems involving domains with cracks and their origins are present in applications from fracture mechanics. These questions involve the study of the regularity for minimizers for Mumford-Shah type functionals, stability questions of Dirichlet eigenvalues and nonlinear Neumann problems in Reifenberg flat domains.