ABSTRACT
This chapter is devoted to the introduction of single and double layer potentials for some simple elliptic operators. Potential theory is a very powerful tool to prove important results on existence of solutions to boundary value problems. We will present this theory, first variationally and then in ‘integral form’ for the Yukawa operators u ↦ − Δ u + c 2 u https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math14_1.jpg"/> (with c > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math14_2.jpg"/> ), since this avoids bringing in new Sobolev spaces. The goal of these sections will be to understand the variational properties of the layer potentials and their associated boundary integral operators and use them to provide equivalent formulations for homogeneous boundary value problems in the exterior of a Lipschitz domain. We then generalize the results for the Laplacian in three dimensions, which we will use as an excuse to introduce weighted Sobolev spaces. Finally, we will give a taste of the coupling of boundary integral formulations in an exterior domain with variational formulations inside the domain.
