Maximum principle for second order elliptic equations and applications
Let \(Lu=\sum^{n}_{i,j=1}a_{ij}(x)u_{x_ ix_ j}\) in \(\Omega \subset {\mathbb{R}}^ n\), where \(a_{ij}\in L^{\infty}(\Omega)\), \(a_{ij}=a_{ji}\), \(\sum^{n}_{i=1}a_{ii}=1\), and \(\lambda | \xi |^ 2\leq \sum^{n}_{i,j=1}a_{ij}\xi_ i\xi_ j\) a.e. for \(\lambda >0\) and \(\xi \in {\mathbb{R}}^ n\). For the Dirichlet problem \(Lu=f\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), the author obtains the a priori bound \(\| u\|_{L^{\infty}(\Omega)}\leq k\| Lu\|_{L^ q(\Omega)}\) where \(u\in W^{2,n}(\Omega)\cap C^ 0({\bar \Omega}),\) \(q<n\), and \(k=k(\lambda,\Omega,q)\). A classical result of \textit{C. Pucci} [Ann. Mat. Pura Appl., IV. Ser. 74, 15-30 (1966; Zbl 0144.358)] deduces that \(\| u\|_{L^{\infty}(\Omega)}\leq c\| Lu\|_{L^ n(\Omega)}\) but a physical application is presented here which calls for \(q<n\). The estimate is obtained by using an argument of \textit{E. B. Fabes} and \textit{D. W. Stroock} [Duke Math. J. 51, 997-1016 (1984; Zbl 0567.35003)] and Gehring's theorem [see \textit{G. Moscariello} and the author, Manuscr. Math. 52, 151-170 (1985; Zbl 0576.42022)].
scientific article; zbMATH DE number 4120631