Single and double layer potentials can be defined for an elliptic differential operator without ever constructing its fundamental solution, by characterising them as the unique solutions of variational identities supplied by the Babuska-Lax-Milgram theorem: the single layer potential of Neumann data g is the element S g satisfying B(phi, S g) equal to the pairing of g against the trace of phi, and the double layer potential is assembled from the Newton potential, itself defined by an identical variational characterisation. Because the construction needs only a bounded coercive bilinear form on two Hilbert spaces, a bounded trace operator, and restriction maps onto a domain and its complement, it applies to divergence-form operators of any even order 2m on Lipschitz domains, where fundamental solutions are difficult to build. The classical properties survive in this abstract setting: the Green representation formula writing a solution as a double layer potential of its trace plus a single layer potential of its Neumann data, adjoint relations pairing the potentials for a form with those for its transpose, jump and continuity relations across the boundary, and Verchota's equivalence between well-posedness of the Dirichlet and Neumann problems and invertibility of the corresponding boundary operators.
Layer potentials are integral operators built from boundary data that turn a boundary value problem for an elliptic differential operator into an equation posed on the boundary itself. Classically th…