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

Comment.Math.Univ.Carolinae 43,1 (2002) 43-59.

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

Keywords: Miranda-Talenti inequality, nonvariational elliptic equations, H\"older regularity
AMS Subject Classification: 35B45, 35B65, 35J25, 35J60, 35R05