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

Marmolejo-Olea, Emilio; Mitrea, Irina; Mitrea, Marius; Shi, Qiang

doi:10.1090/S0002-9947-2012-05606-6

Transmission boundary problems for Dirac operators on Lipschitz domains and applications to Maxwell’s and Helmholtz’s equations
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by Emilio Marmolejo-Olea, Irina Mitrea, Marius Mitrea and Qiang Shi PDF

Trans. Amer. Math. Soc. 364 (2012), 4369-4424 Request permission

Abstract:

The transmission boundary value problem for a perturbed Dirac operator on arbitrary bounded Lipschitz domains in $\mathbb {R}^3$ 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.

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