A note on Meyers’ Theorem in 𝑊^{𝑘,1}

Fonseca, Irene; Leoni, Giovanni; Malý, Jan; Paroni, Roberto

doi:10.1090/S0002-9947-02-03027-1

A note on Meyers’ Theorem in $W^{k,1}$
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by Irene Fonseca, Giovanni Leoni, Jan Malý and Roberto Paroni PDF

Trans. Amer. Math. Soc. 354 (2002), 3723-3741 Request permission

Abstract:

Lower semicontinuity properties of multiple integrals \[ u\in W^{k,1}(\Omega ;\mathbb {R}^{d})\mapsto \int _{\Omega }f(x,u(x), \cdots ,\nabla ^{k}u(x)) dx\] are studied when $f$ may grow linearly with respect to the highest-order derivative, $\nabla ^{k}u,$ and admissible $W^{k,1}(\Omega ;\mathbb {R}^{d})$ sequences converge strongly in $W^{k-1,1}(\Omega ;\mathbb {R}^{d}).$ It is shown that under certain continuity assumptions on $f,$ convexity, $1$-quasiconvexity or $k$-polyconvexity of \[ \xi \mapsto f(x_{0},u(x_{0}),\cdots ,\nabla ^{k-1}u(x_{0}),\xi )\] ensures lower semicontinuity. The case where $f(x_{0},u(x_{0}),\cdots ,\nabla ^{k-1}u(x_{0}),\cdot )$ is $k$-quasiconvex remains open except in some very particular cases, such as when $f(x,u(x),\cdots ,\nabla ^{k}u(x))=h(x)g(\nabla ^{k}u(x)).$

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