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The transmission boundary value problem for a perturbed Dirac operator on arbitrary bounded Lipschitz domains in R3 is formulated and solved in terms of layer potentials of Clifford-Cauchy type. As a byproduct of this analysis, an elliptization procedure for the Maxwell system is devised which allows us to show that the Maxwell and Helmholtz transmission boundary value problems are well-posed as a corollary of the unique solvability of this more general Dirac transmission problem.

Transmission boundary problems for Dirac operators on Lipschitz domains and applications to Maxwell’s and Helmholtz’s equations