Splitting-type variational problems with asymmetrical growth conditions

Abstract

Splitting-type variational problems

n i=1 fi(∂iw) dx → min with superlinear growth conditions are studied by assuming hi(t) ≤ f i (t) ≤ Hi(t) (∗) with suitable functions hi , Hi : R → R+, i = 1, …, n, measuring the growth and ellipticity of the energy density. Here, as the main feature, we do not impose a symmetric behaviour like hi(t) ≈ hi(−t) and Hi(t) ≈ Hi(−t) for large |t|. Assuming quite weak hypotheses on the functions appearing in (∗), we establish higher integrability of |∇u| for local minimizers u ∈ L∞() by using a Caccioppoli-type inequality with some power weights of negative exponent.