Geodesic
On an Extremal Problem Involving Harmonic Functions
Canadian mathematical bulletin, Tome 31 (1988) no. 1, pp. 63-69

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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 P1 by 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.
DOI : 10.4153/CMB-1988-010-6
Mots-clés : harmonic functions, boundary values, boundary control, 31A20, 31B20, 49B22 
Schmidt, E. J. P. Georg. On an Extremal Problem Involving Harmonic Functions. Canadian mathematical bulletin, Tome 31 (1988) no. 1, pp. 63-69. doi: 10.4153/CMB-1988-010-6
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     author = {Schmidt, E. J. P. Georg},
     title = {On an {Extremal} {Problem} {Involving} {Harmonic} {Functions}},
     journal = {Canadian mathematical bulletin},
     pages = {63--69},
     year = {1988},
     volume = {31},
     number = {1},
     doi = {10.4153/CMB-1988-010-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1988-010-6/}
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