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 L² 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.