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 { −div A(x, u(x), Du(x)) = μ in Ω, u = 0 on ∂Ω, where Aα i (x, y, ξ) = N∑ β=1 n∑ j=1 aα,β i,j (x, y) ξβ j 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 aα,β i,j are bounded Carathéodory functions; (ii) ellipticity of the diagonal coefficients aα,α i,j ; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients aα,β i,j (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) aα,β i,j = −aβ,α j,i (skew-symmetry); (b) |aα,β i,j | is small; (c) aα,β i,j 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.