A-priori gradient bound for elliptic systems under either slow or fast growth conditions

We obtain an a-priori Wloc1,∞(Ω;Rm)-bound for weak solutions to the elliptic system divA(x,Du)=∑i=1n∂∂xiaiα(x,Du)=0,α=1,2,…,m,where Ω is an open set of Rn, n≥ 2 , u is a vector-valued map u: Ω ⊂ Rn→ Rm. The vector field A(x, ξ) has a variational nature in the sense that A(x, ξ) = Dξf(x, ξ) , where f= f(x, ξ) is a convex function with respect to ξ∈ Rm×n. In this context of vector-valued maps and systems, a classical assumption finalized to the everywhere regularity is a modulus-dependence in the energy integrand; i.e., we require that f(x, ξ) = g(x, | ξ|) , where g(x, t) is convex and increasing with respect to the gradient variable t∈ [0 , ∞). We allow x-dependence, which turns out to be a relevant difference with respect to the autonomous case and not only a technical perturbation. Our assumptions allow us to consider both fast and slow growth. We consider fast growth even of exponential type; and slow growth, for instance of Orlicz-type with energy-integrands such as g(x, | Du|) = a(x) | Du| p(x)log (1 + | Du|) or, when n= 2 , 3 , even asymptotic linear growth with energy integrals of the type ∫Ωg(x,|Du|)dx=∫Ω{|Du|-a(x)|Du|}dx.