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Young measures generated by sequences in Morrey spaces

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Abstract

Let \({\Omega\subset\mathbb{R}^n}\) be open and bounded. For 1 ≤ p < ∞ and 0 ≤ λ < n, we give a characterization of Young measures generated by sequences of functions \({\{{\bf f}_j\}_{j=1}^\infty}\) uniformly bounded in the Morrey space \({L^{p,\lambda}(\Omega;\mathbb{R}^N)}\) with \({\{\left|{{\bf f}_j}\right|^p\}_{j=1}^\infty}\) equiintegrable. We then treat the case that each f j = u j for some \({{\bf u}_j\in W^{1,p}(\Omega;\mathbb{R}^N)}\). As an application of our results, we consider the functional

$${\bf u} \mapsto \int\limits_{\Omega}f({\bf x}, {\bf u}({\bf x}), {\bf {\nabla}}{\bf u}({\bf x})){\rm d}{\bf x},$$

and provide conditions that guarantee the existence of a minimizing sequence with gradients uniformly bounded in \({L^{p,\lambda}(\Omega;\mathbb{R}^{N\times n})}\).

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Correspondence to Kyle Fey.

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Communicated by J. Ball.

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Fey, K. Young measures generated by sequences in Morrey spaces. Calc. Var. 39, 183–201 (2010). https://doi.org/10.1007/s00526-009-0306-6

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