LAUSR

Abstract In a bounded open subset Ω ⊂ R n , we study Dirichlet problems with elliptic systems, involving a finite Radon measure μ on R n with values into R N , defined by { − div A ( x , u ( x ) , D u ( x ) ) = μ  in  Ω , u = 0  on  ∂ Ω , where A i α ( x , y , ξ ) = ∑ β = 1 N ∑ j = 1 n a i , j α , β ( x , y ) ξ j β with α ∈ { 1 , … , N } the equation index. We prove the existence of a (distributional) solution u : Ω → R N , obtained as the limit of approximations, by assuming: (i) that coefficients a i , j α , β are bounded Caratheodory functions; (ii) ellipticity of the diagonal coefficients a i , j α , α ; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients a i , j α , 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) a i , j α , β = − a j , i β , α (skew-symmetry); (b) | a α , β i , j | is small; (c) a i , j α , 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.