A Generalization of Separability in Boundary Value Problems

Abstract

We consider solutions Un(X, y) of Laplace's equation which are regular, and of bounded gradient, in the interior and exterior of a smooth closed plane curve c, and coupled across c by the jump conditions [u] 0, [Ou/Ov] 2,,gu, where g is a sufficiently smooth, positive, periodic and prescribed function of the arclength, and u,, 2, are eigenfunctions and eigenvalues to be determined. For large 2, we show that 2, O(n) and that u, is trigonometric asymptotically. Thus, the functions u, generalize the sets of solutions usually employed in solving boundary value problems in "separable" cases, and the results are confirmed in those eases, which arise by special choice of g(s). The method employed is the reduction of the problem to consideration of the eigenfunctions of the homogeneous simple layer potential. We show that iteration leads to an integral equation whose kernel behaves like the Green's function of an ordinary differential equation. Results due to Hammerstein [1] are then applied.

Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime

“…The way to implement this technique is one of the cores of the analysis in this paper. The use of separation of variables to solve boundary value problems for the Laplace equation in an arbitrary domain was considered in the literature and was based on the integral method, see e.g., [14]. The analysis presented here is based on the idea of transformation optics and the reflecting technique.…”