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MiS Preprint
86/2000
Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations
Ali Taheri
Abstract
Let $\Omega \subset R^n$ be a bounded domain and let $f:\Omega \times R^N \times R^{N \times n} \to R$. Consider the functional $I(u) :=\int_\Omega f(x,u,Du) dx$ over the class of Sobolev functions $W^{1,q} (\Omega ; R^N) $ ($1 \leq q \leq \infty$) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function $u_0$ and f to ensure that $u_0$ provides an $L^r$ local minimizer for I where $1 \leq r \leq \infty$. The case $r = infty$ is somewhat known and there is a considerable literature on the subject treating the case $\min (n,N)=1$, mostly based on the field theory of the calculus of variations. The main contribution here is to present a set of sufficient conditions for the case $1 \leq r \leq \infty$. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of "directional convergence".
