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We prove a maximum principle for linear second order elliptic systems in divergence form with discontinuous coefficients under a suitable condition on the dispersion of the eigenvalues of the coefficients matrix.

Let $\Omega$ be an open bounded set in ${ℝ}^n$ $(n\ge 2)$, with $C^2$ boundary, and $N^{p,\lambda }\left(\Omega \right)$ ($1<p<+\infty$, $0\le \lambda <n$) be a weighted Morrey space. In this note we prove a weighted version of the Miranda-Talenti inequality and we exploit it to show that, under a suitable condition of Cordes type, the Dirichlet problem: $$\left\{\begin{array}{ccc}\sum _{i,j=1}^n{a}_{ij}\left(x\right)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}=f\left(x\right)\in N^{p,\lambda }\left(\Omega \right)\hfill & \text{in}\Omega u=0\hfill & \text{on}\partial \Omega \end{array}\right.$$ has a unique strong solution in the functional space $$\left\{u\in W^{2,p}\cap W_o^{1,p}\left(\Omega \right):\frac{{\partial }^{2}u}{\partial x_i\partial x_j}\in N^{p,\lambda }\left(\Omega \right),i,j=1,2,...,n\right\}.$$

Using a Hardy-type inequality, the authors weaken certain assumptions from the paper [1] and derive existence results for equations with a stronger degeneration.

We give an example of a bounded discontinuous divergence-free solution of a linear elliptic system with measurable bounded coefficients in ${ℝ}^3$ and a corresponding example for a .