  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.
