Abstract of Talk by Dr. James A. Donaldson

Dirichlet-Neumann Operators and Evolution Equations

Dr. James A. Donaldson
Department of Mathematics
Howard University  
Washington, DC 20059

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Let U be harmonic in D, a simply connected region in n-dimensional Euclidean space with smooth boundary C; f be the restriction of U to C; and g be the outward normal derivative of U on C. A Dirichlet-Neumann operator is an operator that maps f to g. Various properties of these operators and their generalizations are considered. In a Hilbert space, an initial-value problem for a second order differential equation involving a special class of these operators is investigated.

These results play a key role in the establishment of the linear ``shallow water'' theory, a theory which provides an important example of the approximation of the solution of an initial-boundary value problem for an elliptic partial differential equation by the solution of an initial-value problem for a hyperbolic partial differential equation.

This is joint work with Daniel A. Williams.

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