On the validity of the Euler-Lagrange system without growth assumptions

Abstract

The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_\Omega F(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(\Omega , \mathbb R^N )\cap K$, where $K$ is a closed convex subset of the Dirichlet class $\mathrm W^{k,1}_{g}(\Omega , \mathbb R^N ),$ are characterised as the energy solutions to the Euler-Lagrange inequality for $\mathscr F$. We assume that the essentially smooth integrand $F\colon \mathbb R^{N} \otimes \odot^{k}\mathbb R^{n} \to \mathbb R\cup\{+\infty\}$ is convex, lower semi-continuous, proper and at least super-linear at infinity. In the unconstrained case $K=\mathrm W^{k,1}_{g}(\Omega , \mathbb R^N )$, if the integrand $F$ is convex, real-valued, and satisfies a demi-coercivity condition, then $$ \int_{\Omega} \! F^{\prime}(\nabla^{k} u) \cdot \nabla^{k}\phi \, \mathrm d x =0 $$ holds for all $\phi \in \mathrm W_{0}^{k}( \Omega , \mathbb R^{N})$, where $\nabla^{k} u$ is the absolutely continuous part of the vector measure $D^{k}u$.