# Article

 Title: A global differentiability result for solutions of nonlinear elliptic problems with controlled growths (English) Author: Fattorusso, Luisa Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 58 Issue: 1 Year: 2008 Pages: 113-129 Summary lang: English . Category: math . Summary: Let $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$, $n>2$. In $\Omega$ we deduce the global differentiability result $u \in H^{2}(\Omega , \mathbb{R}^{N})$ for the solutions $u \in H^{1}(\Omega , \mathbb{R}^{n})$ of the Dirichlet problem $u-g \in H^{1}_{0}(\Omega , \mathbb{R}^{N}), -\sum _{i}D_{i}a^{i}(x,u,Du)=B_{0}(x,u,Du)$ with controlled growth and nonlinearity $q=2$. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure. (English) Keyword: global differentiability of weak solutions Keyword: elliptic problems Keyword: controlled growth Keyword: nonlinearity with $q=2$ MSC: 35B65 MSC: 35D10 MSC: 35J60 MSC: 58B10 idZBL: Zbl 1174.35039 idMR: MR2402529 . Date available: 2009-09-24T11:53:55Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/128249 . Reference: [1] S.  Campanato: Equazioni ellittiche del  $II^{\circ }$ ordine e spazi $L^{2,\lambda }$.Ann. Mat. Pura App. 69 (1965), 321–382. (Italian) MR 0192168, 10.1007/BF02414377 Reference: [2] S.  Campanato: Sistemi ellittici in forma di divergenza.Quad. Scuola Normale Superiore Pisa (1980). (English) MR 0668196 Reference: [3] S.  Campanato: Differentiability of the solutions of nonlinear elliptic system with natural growts.Ann. Mat. Pura Appl., 4.  Ser. 131 (1982), 75–106. MR 0681558, 10.1007/BF01765147 Reference: [4] S.  Campanato: A maximum principle for nonlinear elliptic system: Boundary fundamental estimates.Adv. Math. 66 (1987), 291–317. MR 0915857, 10.1016/0001-8708(87)90037-5 Reference: [5] S.  Campanato, P.  Cannarsa: Differentiability and partial Hölder continuity of the solutions of non linear elliptic systems of order  $2m$ with quadratic growth.Ann. Sc. Norm. Super. Pisa Cl. Sci., IV.  Ser. 8 (1981), 285–309. MR 0623938 .

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