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

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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 \(u :\mathbb {R}^N \supset \varOmega \rightarrow \mathbb {R}^n\) 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

$$\begin{aligned} \mathrm {W}^{\mathbf {a},p}(\varOmega ;\mathbb {R}^n) \ni u \mapsto \int _\varOmega F(\nabla _{\mathbf {a}} u(x)) \, \mathrm {d} x,\end{aligned}$$

where \(\mathrm {W}^{\mathbf {a},p}(\varOmega ; \mathbb {R}^n)\) 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 semicontinuity, and envelopes of relaxation of such functionals, in terms of an appropriate generalisation of Morrey’s quasiconvexity.

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Acknowledgements

This work is based on the author’s doctoral thesis [72] prepared under the supervision of Prof. Jan Kristensen, whose support and guidance have been of immense help. The author would also like to thank Prof. Sir John Ball FRS, Prof. Gregory Seregin, Prof. Luc Nguyen, Prof. Gui-Qiang Chen, and Prof. Kewei Zhang, who have all acted as referees for the thesis at various stages of its completion. The generous financial support of Oxford Engineering and Physical Sciences Research Council Centre for Doctoral Training in Partial Differential Equations, the Clarendon Fund, and St John’s College Oxford is gratefully acknowledged.

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Prosinski, A. Existence of minimisers of variational problems posed in spaces of mixed smoothness. Calc. Var. 62, 6 (2023). https://doi.org/10.1007/s00526-022-02342-0

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