The $p$-CurlCurl : Spaces, traces, coercivity and a Helmholtz decomposition in $L^p$

This work provides the foundation for the finite element analysis of an elliptic problem which is the rotational analogue of the $p$-Laplacian and which appears as a model of the magnetic induction in a high-temperature superconductor operating near it's critical current. Whereas the function theory for the $p$-Laplacian requires standard results in $L^p$ Sobolev spaces, this problem requires an extension to $L^p$ spaces of the well-known $L^2$ theory for divergence free vector fields, as used in the finite element method applied to incompressible flows and electromagnetic radiation. Among other things, the analysis requires extensions to $L^p$ of the well-known $H(\operatorname{div}; \Omega)$ and $H(\operatorname{curl};\Omega)$, extensions of traces and Green's theorem, a Helmholtz decomposition and finally a Friedrich's inequality. In this paper, we provide a proof of the existence and uniqueness of weak solutions of our so-called $p$-CurlCurl problem. In a companion paper, the analysis is extended to treat continuous and finite element solutions of the nonlinear parabolic problem whose spatial term is the $p$-CurlCurl operator. Many of the results presented here are either already known, known in slightly different forms or are proven with the help of techniques that are already well-known. The main novelty of this paper appears to be the structured form of this $L^p$ theory and our form of the Helmholtz decomposition and of the Friedrich's inequality. In this respect, we note that some of these results can be found in the works of M. Dauge, M. Mitrea and I. Mitrea.