Green Identities and Layer Potentials

This chapter is devoted to the Green Identities and to the layer potentials corresponding to the fundamental solution of the Laplace operator, cf. ( 3.2 ). We first consider the Green Identities for bounded domains and then the case of exterior domains for functions which are harmonic at infinity. Next, we introduce the single and double layer potentials and the corresponding mapping properties, especially in spaces of Hölder continuous functions and in Schauder spaces. Although such properties can be found in the classic monographs of Günter (Potential theory and its applications to basic problems of mathematical physics. Translated from the Russian by John R. Schulenberger. Frederick Ungar Publishing, New York, 1967) and Kupradze et al. (Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, volume 25 of North-Holland Series in Applied Mathematics and Mechanics. North-Holland Publishing, Amsterdam-New York, Russian edition, 1979. Edited by V. D. Kupradze.), here we prove the corresponding statements with optimal Hölder exponents. To do so, we exploit a result of Miranda (Atti Accad Naz Lincei Mem Cl Sci Fis Mat Natur Sez I (8), 7:303–336, 1965) on the Hölder continuity of singular integrals of convolution type, for which we provide a complete proof.