Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients
Leonardi, Salvatore
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002), p. 43-59 / Harvested from 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 $$ \left\{ u \in W^{2,p} \cap W^{1,p}_o(\Omega ) : \frac{\partial ^2 u}{\partial x_i \partial x_j} \in N^{p,\lambda}(\Omega ), i,j=1,2,\,\ldots, n\right\}. $$

Publié le : 2002-01-01
Classification:  35B45,  35B65,  35J25,  35J60,  35R05 
@article{119299,
     author = {Salvatore Leonardi},
     title = {Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {43},
     year = {2002},
     pages = {43-59},
     zbl = {1090.35045},
     mrnumber = {1903306},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119299}
}

Leonardi, Salvatore. Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 43-59. http://gdmltest.u-ga.fr/item/119299/

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