Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients

Leonardi, S.

Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 43-59.

Résumé

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 $$ \left\{ u \in W^{2,p} \cap W^{1,p}_o(\Omega ) : \frac{\partial ^2 u}{\partial x_i \partial x_j} \in N^{p,\lambda}(\Omega ), i,j=1,2,\,\ldots, n\right\}. $$

Détail

Classification : 35B45, 35B65, 35J25, 35J60, 35R05
Keywords: Miranda-Talenti inequality; nonvariational elliptic equations; Hölder regularity