Existence of minimisers of variational problems posed in spaces of mixed smoothness

The present work constitutes a first step towards establishing systematic framework for treating variational problems that depend on given input function through mixture of its derivatives different orders in directions. For fixed vector \(\mathbf {a} := (a_1, \ldots , a_N) \in \mathbb {N}^N\) and \(u :\mathbb {R}^N \supset \varOmega \rightarrow {R}^n\) we denote by \(\nabla _{\mathbf {a}} u (\partial ^{\alpha } u)_{\langle \alpha \mathbf {a}^{-1} \rangle = 1}\) the matrix whose i-th row is composed \(\partial ^\alpha u^i\) component map u, where multi-indices \(\alpha \) satisfy \(\langle \sum _{j=1}^N \frac{\alpha _j}{a_j} 1\). We study functionals form $$\begin{aligned} \mathrm {W}^{\mathbf {a},p}(\varOmega ;\mathbb {R}^n) \ni \mapsto \int _\varOmega F(\nabla u(x)) \, {d} x,\end{aligned}$$where \(\mathrm ; {R}^n)\) an appropriate Sobolev space mixed smoothness F integrand. existence minimisers such under prescribed Dirichlet boundary conditions. characterise coercivity, lower semicontinuity, envelopes relaxation functionals, terms generalisation Morrey’s quasiconvexity.