 
Summary: ON POINCARŽE'S VARIATIONAL PROBLEM IN
POTENTIAL THEORY
DMITRY KHAVINSON, MIHAI PUTINAR, AND HAROLD S. SHAPIRO
2000 Mathematics Subject Classification. 31B15, 31B20, 30C40, 47A75.
Key words and phrases. Newtonian potential, NeumannPoincarŽe operator, sym
metrizable operator, PoincarŽe's variational problem, Schiffer's operator, Hilbert
Beurling transform, Fredholm spectrum.
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Abstract. One of the earliest attempts to rigorously prove the
solvability of Dirichlet's boundary value problem was based on
seeking the solution in the form of a "potential of double layer", and
this leads to an integral equation whose kernel is (in general) both
singular and nonsymmetric. C. Neumann succeeded with this ap
proach for smoothly bounded convex domains, and H. PoincarŽe,
by a tremendous tour de force, showed how to push through the
analysis for domains with sufficiently smooth boundaries but no
hypothesis of convexity. In this work he was (according to his own
account) guided by consideration of a variational problem involving
the partition of energy of an electrostatic field induced by charges
