Lipschitz estimates for systems with ellipticity conditions at infinity

Abstract

In the general vector-valued case \(N \ge 1\), we prove the Lipschitz continuity of local minimizers to some integrals of the calculus of variations of the form \( \int _{\Omega } g(x, |Du|)\,\mathrm{d}x\), with p, q-growth conditions only for \(|Du| \rightarrow +\infty \) and without further structure conditions on the integrand \(g=g(x,|Du|)\). We apply the regularity results to weak solutions to nonlinear elliptic systems of the form \(\sum _{i=1}^{n}\frac{\partial }{\partial x_{i}}a_{i}^{\alpha }\left( x,Du\right) =0\), \(\alpha =1,2,\ldots ,N\).