Morrey regularity for almost minimizers of asymptotically convex functionals with nonstandard growth (Q2855504)

The paper deals with the regularity for almost minimizers of integral functionals of the form: NEWLINE\[NEWLINE F(u)= \int _{\Omega} f(x,u,D u) \, dx NEWLINE\]NEWLINE where \(u: \Omega \rightarrow \mathbb{R}^N\) and the integrand \(f\) behaves asymptotically like the function \(h(|D u)^{\alpha(x)}\) where \(h\) is an \(N\)-function such that \(th''\) is comparable to \(h'\). Under a suitable continuity assumption on the function \(\alpha=\alpha(x)\), the authors prove that if \(u\) is a minimizer then \(h(|D u|)^{\alpha(x)}\) belongs to the Morrey space \(L^{1,\lambda} (\Omega)\). Moreover, by using a version of Ekeland's variational principle, they prove the existence of a Morrey regular minimizing sequence. An interesting application to a class of PDEs is also given.