Harmonic Analysis and PDE

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Harmonic Analysis and P D E Abstract of Proposed ResearchSagun Chanillo This grant is to support research on some problems involving partial differential equations and topics in geometry, physics and material science. One class of problems concerns how to design a body of prescribed shape and mass, with certain bounds on the density, that has least first eigenvalue. This is related to certain models of composite materials. In earlier work we have proved certain properties of the solutions of this problem and I will continue studies on the properties of the interfaces between the different materials in the solution. These are free boundary problems and the existing regularity theory for such problems does not extend to these systems. Jointly with Carlos Kenig, we intend to study some examples and analyze these models in considerable detail. A second problem we are studying is the Plateau problem for constant mean curvature surfaces. This is a joint project with A. Malchiodi and continuation of our earlier work where the Dirichlet situation was analysed. Now we are trying to understand the conformal situation.A clear understanding of the composite membrane problem will lead to an understanding and design of useful composite materials with important properties for use in areas such as medicine, aeronautics and wherever light-weight and strong materials are to be used. Our earlier work showed that one sometimes has to build these materials by breaking the natural symmetry of the desired body, for example to build the ubiquitous washer one needs to arrange the composite materials non-symmetrically. The loss of symmetry has a direct bearing on the regularity of the interface of the materials, the free boundary so to say and this will be hopefully revealed in the current research investigation.