Variational integrals of nearly linear growth

Abstract

We study variational integrals and related equations whose integrand grows almost linearly with respect to the gradient. A prototype of such functionals is $$ I[u] =\int |\nabla u|\,A\big(|\nabla u|\big)\,dx, $$ where $A$ is slowly increasing to $\infty $. For instance, $A(t)=\log^\alpha(1+t)$, $\alpha >0$, or $A(t)=\log\log (e+t)$, etc. We show that the minimizer $u$ subject to the Dirichlet data $v$ satisfies the estimate $$ \int |\nabla u|\,A^{1\pm\epsilon}\big(|\nabla u|\big)\,dx\le C\int|\nabla v|\, A^{1\pm\epsilon}\big(|\nabla v|\big)\,dx $$ at least for some small $\epsilon >0$. This extends previous results [12], [16] on integrals with power growth.