Mathematical Sciences: Theory of Capacities

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Work will be done investigating geometric-analytic properties of domains as they relate to solutions of partial differential equations. The work concentrates on the boundaries of domains for it is at the boundary where the nature of solutions tends to be obscure. Among the concepts which have evolved in the measurement of boundary regularity, one of the most important is that of capacity, or more precisely, the capacity of a set. Although there are a number of types of capacity one may define, the source of theory almost surely rests with electrostatics and measurement of how well a conductor can support a charge. This in itself is a question about solutions of a partial differential equations. Several questions in the theory of capacities will be addressed in this work. The first is concerned with nature of weighted capacities arising in the study of degenerate elliptic equations. One wishes to chacterize the boundary regular points or the removable singularities for solutions. The capacities (set functions) are built out of the differential operator. Certain fundamental questions as yet are incompletely answered. They include how weighted capacity depends on the weight at least as far as null sets are concerned. One would also want to be able to classify regular points into equivalence classes according to types of degeneracy. On a larger scale, results on equivalence of capacities and concepts of regularity for nonlinear equations are only just appearing. This research will have application to the theory of partial differential equations and potential theory.