MiS Preprint Repository

Let $\mathrm{\Omega }\subset R^n$ be a bounded domain and let $f:\mathrm{\Omega }\times R^N\times R^{N\times n}\to R$. Consider the functional
$I(u):={\int }_{\mathrm{\Omega }}f(x,u,Du)dx$
over the class of Sobolev functions $W^{1,q}(\mathrm{\Omega };R^N)$ ($1\le q\le \mathrm{\infty }$) 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 $1\le r\le \mathrm{\infty }$. 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 $1\le r\le \mathrm{\infty }$. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of "directional convergence".