Smallness and cancellation in some elliptic systems with measure data

In a bounded open subset $\Omega\subset\mathbb{R}^{n}$, we study Dirichlet problems with elliptic systems, involving a finite Radon measure $\mu$ on $\mathbb{R}^{n}$ with values into $\mathbb{R}^{N}$, defined by $$\left\{\begin{array}{ll} -\mbox{div}\, A(x,u(x),Du(x)) =\mu & \mbox{ in }\Omega,\\ u=0 & \mbox{ on }\partial\Omega, \end{array}\right.$$ where $A_{i}^{\alpha}(x,y,\xi)=\sum\limits_{\beta=1}^{N}\sum\limits_{j=1}^{n}a_{i,j}^{\alpha,\beta}\left( x,y\right) \xi_{j}^{\beta}$ with $\alpha\in\{1,\dots,N\}$ the equation index. We prove the existence of a (distributional) solution $u:\Omega\rightarrow\mathbb{R}^{N}$, obtained as the limit of approximations, by assuming: (i) that coefficients $a_{i,j}^{\alpha,\beta}$ are bounded Carath\'eodory functions; (ii) ellipticity of the diagonal coefficients $a_{i,j}^{\alpha,\alpha}$; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients $a^{\alpha,\beta}_{i,j}$ (i.e. $\alpha\neq\beta$) verifying a $r$-staircase support condition with $r>0$. Such a smallness condition is satisfied, for instance, in each one of these cases: (a) $a^{\alpha,\beta}_{i,j}=-a^{\beta,\alpha}_{j,i}$ (skew-symmetry); (b) $|a^{\alpha,\beta}_{i,j}|$ is small; (c) $a^{\alpha,\beta}_{i,j}$ 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