Geodesic
Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 43-59 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $$ \left\{ u \in W^{2,p} \cap W^{1,p}_o(\Omega ) : \frac{\partial ^2 u}{\partial x_i \partial x_j} \in N^{p,\lambda}(\Omega ), i,j=1,2,\,\ldots, n\right\}. $$
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 $$ \left\{ u \in W^{2,p} \cap W^{1,p}_o(\Omega ) : \frac{\partial ^2 u}{\partial x_i \partial x_j} \in N^{p,\lambda}(\Omega ), i,j=1,2,\,\ldots, n\right\}. $$
Classification : 35B45, 35B65, 35J25, 35J60, 35R05
Keywords: Miranda-Talenti inequality; nonvariational elliptic equations; Hölder regularity 
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     author = {Leonardi, S.},
     title = {Weighted {Miranda-Talenti} inequality and applications to equations with discontinuous coefficients},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {43--59},
     year = {2002},
     volume = {43},
     number = {1},
     mrnumber = {1903306},
     zbl = {1090.35045},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2002_43_1_a4/}
}
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Leonardi, S. Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 43-59. http://geodesic.mathdoc.fr/item/CMUC_2002_43_1_a4/