On the solvability of the zaremba problem in infinite sectors and the invertibility of associated singular integral operators

Authors

Hussein Awala, College of Science and Technology
Irina Mitrea, College of Science and Technology
Katharine A. Ott, Bates CollegeFollow

Publication Title

Applied and Numerical Harmonic Analysis

Document Type

Book Chapter

Department or Program

Mathematics

Publication Date

2017

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 ℝ².

Comments

Original version is available from the publisher at: https://doi.org/10.1007/978-3-319-55556-0_10

Recommended Citation

Awala, H., Mitrea, I., Ott, K. A. (2017). On the Solvability of the Zaremba Problem in Infinite Sectors and the Invertibility of Associated Singular Integral Operators. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis Series. Birkhäuser, Cham. (Springer International Publishing)