# 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

## Office Telephone Number:

202-806-7727
## Fax Number:

## Electronic Mail Address:

jad@scs.howard.edu

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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