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On an Extremal Problem Involving Harmonic Functions

Published online by Cambridge University Press:  20 November 2018

E. J. P. Georg Schmidt
E. J. P. Georg Schmidt*
Affiliation:
Department of Mathematics and McGill University, 805 Sherbrooke Street West Montreal, P.Q. CanadaCanada H3A 2K6
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Abstract

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Given a domain D in R” and two specified points P0 and P1 in D we consider the problem of minimizing u(p1) over all functions harmonic in D with values between 0 and 1 normalised by the requirement u(P0) = 1/2. We show that when D is suitably regular the problem has a unique solution u* which necessarily takes on boundary values 0 or 1 almost everywhere on the boundary. In the process we prove that it is possible to separate P0 and P1by a harmonic function whose boundary value is supported in an arbitrary set of positive measure. These results depend on the fact that (under suitable regularity conditions) a harmonic function which vanishes on an open subset of the boundary has a normal derivative which is almost everywhere non-vanishing in that set.

Keywords

harmonic functionsboundary valuesboundary control31A2031B2049B22
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 1 , 01 March 1988 , pp. 63 - 69
Copyright
Copyright © Canadian Mathematical Society 1988

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