Maximum principles for some quasilinear elliptic systems

Abstract

We give maximum principles for solutions u:Ω→RN to a class of quasilinear elliptic systems whose prototype is −∑i=1n∂∂xi∑β=1N∑j=1nai,jα,βx,u(x)∂uβ∂xj(x)=0,x∈Ω,where α∈{1,…,N} is the equation index and Ω is an open, bounded subset of Rn. We assume that coefficients ai,jα,β(x,y) are measurable with respect to x, continuous with respect to y∈RN, bounded and elliptic. In vectorial problems, when trying to bound the solution by means of the boundary data, we need to bypass De Giorgi’s counterexample by means of some additional structure assumptions on the coefficients ai,jα,β(x,y). In this paper, we assume that off-diagonal coefficients ai,jα,β, α≠β, have support in some staircase set along the diagonal in the yα,yβ plane.