** Mihai Putinar ** University of California at Santa Barbara

*
Carleman factorization of layer potentials on smooth domains
*

Abstract:

One of the classical paths towards solving Dirichlet problem in any dimensions
was to invert, when possible, the double layer potential transform, known today
as the Neumann-Poincaré operator. The study of the resulting integral equation
was one of the main sources and motivations of XX-th Century spectral analysis.
When regarded as a linear bounded transform of Lebesgue space L^{2} of
the respective boundary, the Neumann-Poincaré operator is not self-adjoint,
although it has real spectrum. It is only symmetrizable. Resurrecting some
pertinent observations of Carleman and M. G. Krein, we show that the
Neumann-Poincaré operator can be divided, with a bounded quotient, by the single
layer potential. This grossly overlooked structure allows to develop the
spectral analysis of the Neumann-Poincaré operator to the amenable Lebesgue
space setting, rather than bouncing back and forth the computations between
Sobolev spaces of negative or positive fractional order.
I will present an enhanced, fresh new look at symmetrizable linear transforms
combined with geometric-microlocal analysis techniques. The outcome is manyfold,
complementing recent advances on the theory of layer potentials, in the smooth
boundary setting.