Mathematical Sciences: Partial Differential Equations and Applications

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This work focuses on a large class of nonlinear partial differential equations arising from physical and engineering models. A particular set of problems centers on free boundary problems occurring in the study of melting and solidification processes, cavity and jet flows and electrochemical processes from machining or electroplating. In such problems, a part of the boundary of the domain where the solution is to be found is unknown and must be found simultaneously with the solution. Substantial progress has been made on problems of this type but newer and more realistic models lead to complex equations, especially when going from two to three-dimensional descriptions. Work on optimal design involving free boundaries will also be carried out. These can often be formulated as a minimization problem for some function of a control in which the solution may not be differentiable. Problems of this type occur, for example, in trying to minimize the pressure in a dam. The control variables appear either in the data or in the coefficients of the resulting differential equation. The free boundary conditions may occur on a surface which is the control variable. Recent work on construction for an optical lens with certain refraction properties reduces to solving systems of two nonlinear functional differential equations. The construction of a global lens with one axis of symmetry which focuses all rays from two sources into two given collecting points all on the axis has been described although it is not clear what happens when not all points are on the axis of symmetry. Future work is planned on this equation where applications to satellite antenna modeling are expected.