Mathematical Sciences: Harmonic Analysis and Elliptic Partial Differential Equations

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This award supports mathematical research focusing on three areas of analysis and partial differential equations. The first concerns the solvability of the Neumann problem for general second order elliptic operators with bounded measurable coefficients. The essential feature here is the lack of smoothness assumed on the coefficients which brings the research closer to representations of problems occurring in the physical world. Successful work has already been done on such questions for the Dirichlet problem. In the present context one is interested in specifying the boundary flux rather than the boundary values of the solution. The second area will concentrate on the solvability of boundary value problems for higher order elliptic equations in domains with less than smooth boundaries. That is, boundaries with corners and nondifferentiable edges. This problem also relates to problems arising in more concrete applications. Progress has been made on the Dirichlet problem with constant coefficients and on finding sharp conditions on the domain which allow for solvability of the Neumann problem. Work remains to be done in classifying solutions of homogeneous problems with boundary data in the Besov or Sobolev spaces. The third problem concerns the existence of a convex lower bound for the first eigenvalue of the Laplace Beltrami operator on the sphere minus a polar cap. Numerical estimates have been obtained on the value of the eigenvalue as a function of the size of the cap. Sharper, analytic estimates will provide estimates for the growth of subharmonic functions as well as have important consequences for free boundary problems.