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arXiv · 2604.03084

Embedding transmission problems for Maxwell's equations into elliptic theory

Yuri A. Godin·Boris Vainberg
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Abstract

We embed general boundary value problems for the time-harmonic Maxwell equations into the elliptic boundary value theory. This is achieved by introducing two new scalar functions to the electromagnetic field and imposing additional boundary conditions, after which the problem becomes elliptic. The results are applied to general problems for Maxwell's equations in bounded and unbounded domains, as well as to the transmission problem with inhomogeneities on the right-hand side of the equations and at all boundaries. Relations between the inhomogeneities of the elliptic problem are established that provide a one-to-one correspondence between the solutions of Maxwell's problem and the solutions of the elliptic boundary value problem. As an application, the existence of an orthogonal basis from the eigenfunctions of the general Maxwell's problem is justified.

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Yuri A. Godin, Boris Vainberg. 2026-08-05. Embedding transmission problems for Maxwell's equations into elliptic theory. https://doi.org/10.1016/j.jmaa.2026.131000

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