On a Steklov Spectrum in Electromagnetics

In this chapter, we first presented various concepts and results concerning the classical Steklov eigenproblem. Further—out of this problem’s numerous applications in different areas—we discussed the sloshing problem, the problem of electrical prospection, and the problem of vibrating membranes. Next, we gave some details on the formulation of the classical Steklov eigenproblem (i.e., for the scalar Laplacian) and its connections to trace theory. We then presented an electromagnetic analogue of the former; in particular, we considered the time-harmonic Maxwell’s equations in a cavity, and we performed a spectral analysis of the Steklov eigenproblem in an appropriate energy space. To this end, we considered a modified Steklov eigenproblem for the curl curl operator, whose coercivity was obtained by introducing a suitably selected penalty term. Characterizations of the naturally associated trace spaces in terms of the eigenpairs of this problem were then established. Finally, we found explicit formulae for the eigenvalues and eigenfunctions of this problem in the unit ball of R^3, by using classical vector spherical harmonics. Two families of eigenvalues diverging to negative normal infinity−∞ were derived, corresponding, respectively, to divergence-free and nondivergence-free eigenfunctions.