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

Abstract

The constrained minimisers of convex integral functionals of the form $\mathcal{F}(v)={\int }_{\Omega }F({\nabla }^{k}v(x))\mathrm{d}x$ defined on Sobolev mappings $v\in {\mathrm{W}}_g^{k,1}(\Omega ,{\mathbb{R}}^N)\cap K$, where $K$ is a closed convex subset of the Dirichlet class ${\mathrm{W}}_g^{k,1}(\Omega ,{\mathbb{R}}^N),$ are characterised as the energy solutions to the Euler-Lagrange inequality for $\mathcal{F}$. We assume that the essentially smooth integrand $F:{\mathbb{R}}^N\otimes {\odot }^k{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ is convex, lower semi-continuous, proper and at least super-linear at infinity. In the unconstrained case $K={\mathrm{W}}_g^{k,1}(\Omega ,{\mathbb{R}}^N)$, if the integrand $F$ is convex, real-valued, and satisfies a demi-coercivity condition, then $${\int }_{\Omega }{F}^{\prime }({\nabla }^{k}u)\cdot {\nabla }^k\varphi \mathrm{d}x=0$$ holds for all $\varphi \in {\mathrm{W}}_0^k(\Omega ,{\mathbb{R}}^N)$, where ${\nabla }^{k}u$ is the absolutely continuous part of the vector measure $D^{k}u$.