Grand p-harmonic energy

To every nonlinear differential expression there corresponds the so-called natural domain of definition. Usually, such a domain consists of Sobolev functions, sometimes with additional geometric constraints. There are, however, special nonlinear differential expressions (Jacobian determinants, div-curl products, etc.) whose special properties (higher integrability, weak-continuity, etc.) cannot be detected within their natural domain. We must consider them in a slightly larger class of functions. The grand Lebesgue space, denoted by \(\mathscr {L}^{p})(\mathbb X)\), and the corresponding grand Sobolev space \(\mathscr {W}^{1,p})(\mathbb X)\), turn out to be most effective. They were studied by many authors, largely in analogy with the questions concerning \(\mathscr {L}^p (\mathbb X)\) and \(\mathscr {W}^{1,p}(\mathbb X)\) spaces. The present paper is a continuation of these studies. We take on stage the grand p-harmonic energy integrals. These variational functionals involve both one-parameter family of integral averages and supremum with respect to the parameter. It is for this reason that the existence and uniqueness of the grand p-harmonic minimal mappings becomes a new (rather challenging) problem.