  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.
