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Keywords:
Miranda-Talenti inequality; nonvariational elliptic equations; Hölder regularity
Miranda-Talenti inequality; nonvariational elliptic equations; Hölder regularity
Summary:
Let $\Omega $ be an open bounded set in $\Bbb R^{n}$ $(n\geq 2)$, with $C^2$ boundary, and $N^{p,\lambda}(\Omega )$ ($1 < p < +\infty $, $0\leq \lambda < n$) 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: $$ \cases \sum_{i,j=1}^n a_{ij}(x) \frac{\partial ^2 u}{\partial x_i \partial x_j} = f(x) \in N^{p,\lambda }(\Omega) \quad & \text{ in } \Omega \ u=0 & \text{ on } \partial \Omega \endcases $$ has a unique strong solution in the functional space $$ \left\{ u \in W^{2,p} \cap W^{1,p}_o(\Omega ) : \frac{\partial ^2 u}{\partial x_i \partial x_j} \in N^{p,\lambda}(\Omega ), i,j=1,2,\,\ldots, n\right\}. $$
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