Inegalitate de interpolare gradient-laplacian
Posted: Sat Jun 27, 2009 10:57 pm
Fie \( \Omega\subset\mathbb{R}^{n} \) un domeniu cu frontiera neteda si \( u\in C^{2}(\overline{\Omega}) \) astfel incat \( u=0 \) pe \( \partial\Omega \). Demonstrati urmatoarea inegalitate de interpolare:
\( \int_{\Omega}|\nabla u|^{2}dx\leq\epsilon\int_{\Omega}(\Delta u)^{2}dx+\frac{1}{4\epsilon}\int_{\Omega}u^{2}dx, \forall\epsilon>0. \)
\( \int_{\Omega}|\nabla u|^{2}dx\leq\epsilon\int_{\Omega}(\Delta u)^{2}dx+\frac{1}{4\epsilon}\int_{\Omega}u^{2}dx, \forall\epsilon>0. \)