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Lower semicontinuity of \(L^\infty\) functionals. (English) Zbl 1034.49008

The authors study the lower semicontinuity and existence of minimizers for the so-called supremal functionals, i.e., functionals of the form \[ \operatorname{ess}\sup_{x \in \Omega} f (x,u,Du). \] The ambient space is \(W^{1,\infty} (\Omega, \mathbb R^m)\). The scalar case was previously settled by the first two authors and W. Liu [“Hopf-Lax-type formula for \(u_ t+ H(u, Du)=0\)”, J. Differ. Equations 126, 48–61 (1996; Zbl 0857.35023)] and the fundamental notion there was the quasiconvexity also known as levelconvexity (i.e., convexity of level sets). In this paper 4 notions are introduced (weak Morrey quasiconvexity, strong Morrey quasiconvexity, polyquasiconvexity and rank-one quasiconvexity) all stemming from the notion of levelconvexity.
The paper is mainly focused on the notion of strong Morrey quasiconvexity. This last notion is shown to be necessary and sufficient for the lower semicontinuity (up to some minor technical assumption). The relationship between the different notion of quasiconvexity are then investigated in the case of lower semicontinuous \(f\) which depends only on the gradient variable. Finally, in the last section the so-called Aronsson-Euler equation is derived (formally). This paper has a companion paper [the same authors, “The Euler equation and absolute minimizers of \(L^\infty\) functionals”, Arch. Ration. Mech. Anal. 157, 255–283 (2001; Zbl 0979.49003)].

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
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References:

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