Mathematical Sciences: Partial Differential Equations and Classical Analysis

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This project will focus primarily on problems arising in the theory of elliptic partial differential equations. They are motivated to some extent by considerations of quasiregular mappings in n-dimensional Euclidean space. These are considered as natural extensions to higher dimension of classical analytic function theory. In addition, solutions of quasilinear elliptic partial differential equations replace harmonic functions. The conformally invariant extension to n-dimensions of the Laplace operator is the so-called N-Laplacian. Work will be done in seeking conditions on bounded weak solutions of quasilinear equations to have gradients of bounded mean oscillation. Even in the linear case this question is still open. Estimates on the oscillation of boundary values of the gradient will be sought. Another line of investigation will focus on the existence of branched quasiregular maps. When the maps are sufficiently smooth, no branching occurs. The minimal smoothness assumptions are not known. Finally, studies will be made of parabolic divergence type equations to establish whether Holder continuity can be expected of bounded weak solutions. This work is related to research in conformal geometry and potential theory.