On the Solvability of the Zaremba Problem in Infinite Sectors and the Invertibility of Associated Singular Integral Operators

Abstract

This paper is concerned with the study of invertibility properties of a singular integral operator naturally associated with the Zaremba problem for the Laplacian in infinite sectors in two dimensions, when considering its action on an appropriate Lebesgue scale of p integrable functions, for 1 < p < ∞. Concretely we consider the case in which a Dirichlet boundary condition is imposed on one ray of the sector, and a Neumann boundary condition is imposed on the other ray. In this geometric context, using Mellin transform techniques, we identify the set of critical integrability indexes p for which the invertibility fails, and we establish an explicit characterization of the L p spectrum of this operator for each p ∈ (1, ∞). This analysis, along with a divergence theorem with non-tangential trace, are then used to establish well-posedness of the Zaremba problem with L p data in infinite sectors in \(\mathbb{R}^{2}\).