Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus

Abstract

We show that operators of layer potential type on surfaces that are locally graphs of Lipschitz functions with gradients in vmo are equal, modulo compacts, to pseudodifferential operators (with rough symbols), for which a symbol calculus is available.We build further on the calculus of operators whose symbols have coefficients in L 1 \ vmo, and apply these results to elliptic boundary problems on domains with such boundaries, which in turn we identify with the class of Lipschitz domains with normals in vmo.This work simultaneously extends and refines classical work of Fabes, Jodeit and Rivière, and also work of Lewis, Salvaggi and Sisto, in the context of C 1 surfaces.1. Introduction 116 2. From layer potential operators to pseudodifferential operators 119 2A.General local compactness results 119 2B.The local compactness of the remainder 126 2C.A variable coefficient version of the local compactness theorem 128 3. Symbol calculus 130 3A.Principal symbols 130 3B.Transformations of operators under coordinate changes 133 3C.Admissible coordinate changes on a Lip \ vmo 1 surface 136 3D.Remark on double layer potentials 137 3E.Cauchy integrals and their symbols 138 4. Applications to elliptic boundary problems 140 4A.Single layers and boundary problems for elliptic systems 140 4B.Oblique derivative problems 149 4C.Regular boundary problems for first-order elliptic systems 152 4D.Absolute and relative boundary conditions for the Hodge-Dirac operator 155

Locations

• Analysis & PDE - View - PDF
• Project Euclid (Cornell University) - View - PDF