Remarks on quasiconvexity and stability of equilibria for variational integrals

Zhang, Kewei

doi:10.1090/S0002-9939-1992-1037211-6

Remarks on quasiconvexity and stability of equilibria for variational integrals
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by Kewei Zhang PDF

Proc. Amer. Math. Soc. 114 (1992), 927-930 Request permission

Abstract:

Let $F:{{\mathbf {R}}^{nN}} \to {\mathbf {R}}$ be a uniformly strictly quasiconvex function (see [3, 4]) of class ${C^{2 + \alpha }},(0 < \alpha < 1)$, and be of polynomial growth. Then every smooth solution of the Euler-Lagrangian equation of the multiple integral $I\left ( {u;\Omega } \right ) = {\smallint _\Omega }F(Du(x))dx$ is a minimum of $I$ for variations of sufficiently small supports contained in $\Omega$.

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