Smallness and cancellation in some elliptic systems with measure data

Abstract

In a bounded open subset Ω⊂Rn, we study Dirichlet problems with elliptic systems, involving a finite Radon measure μ on Rn with values into RN, defined by{−divA(x,u(x),Du(x))=μ in Ω,u=0 on ∂Ω, where Aiα(x,y,ξ)=∑β=1N∑j=1nai,jα,β(x,y)ξjβ with α∈{1,…,N} the equation index. We prove the existence of a (distributional) solution u:Ω→RN, obtained as the limit of approximations, by assuming: (i) that coefficients ai,jα,β are bounded Carathéodory functions; (ii) ellipticity of the diagonal coefficients ai,jα,α; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients ai,jα,β (i.e. α≠β) verifying a r-staircase support condition with r>0. Such a smallness condition is satisfied, for instance, in each one of these cases: (a) ai,jα,β=−aj,iβ,α (skew-symmetry); (b) |aα,βi,j| is small; (c) ai,jα,β may be decomposed into two parts, the first enjoying skew-symmetry and the second being small in absolute value. We give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works. A Brezis's type nonexistence result is also given for general (smooth) elliptic-hyperbolic systems.