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

Abstract

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