Layer Potential Methods for Doubly-Periodic Harmonic Functions

Abstract

We develop and analyze layer potential methods to represent harmonic functions on finitely-connected tori (i.e., doubly-periodic harmonic functions). The layer potentials are expressed in terms of a doubly-periodic and non-harmonic Green's function that can be explicitly written in terms of the Jacobi theta function or a modified Weierstrass sigma function. Extending results for finitely-connected Euclidean domains, we prove that the single- and double-layer potential operators are compact linear operators and derive the relevant limiting properties at the boundary. We show that when the boundary has more than one connected component, the Fredholm operator of the second kind associated with the double-layer potential operator has a non-trivial null space, which can be explicitly constructed. Finally, we apply our developed theory to obtain solutions to the Dirichlet and Neumann boundary value problems, as well as the Steklov eigenvalue problem. We present numerical results using Nystr\"om discretizations and find approximate solutions to these problems in several numerical examples. Our method avoids a lattice sum of the free-space Green's function, is shown to be spectrally convergent, and exhibits a faster convergence rate than the method of particular solutions for problems on tori with irregularly shaped holes.