Abstract
We consider eigenvalue problems for a general p-div-curl system. In particular, we consider a nonlinear eigenvalue problem similar to the standard Dirichlet eigenvalue problem for the p-Laplacian, but due to the presence of the curl, we see that eigenfunctions of the general p-div-curl operator do not enjoy nice regularity as in the p-Laplacian case. We first show that weak solutions of a particular nonlinear p-div-curl eigenvalue problem split into a good and bad part, where the good part is not even divergence free. We then show that the general p-div-curl system admits a Ljusternik-Schnirelman sequence of eigenvalues. Finally, we show a general chain rule for the curl and divergence.
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