inegalitatea gradienti in W_0^{1, p}(\Omega)
Posted: Thu Mar 18, 2010 7:26 pm
Fie \( u, v\in W_{0}^{1, p}(\Omega) \), unde \( \Omega\subset\mathbb{R}^{N} \) este o submultime deschisa, nevida si marginita. Sa se arate ca, daca \( p\geq 2 \), atunci avem
\( \left|\nabla\left(\left(\frac{u^p+v^p}{2}\right)^{1/p}\right)\right| ^p\leq\frac{1}{2}|\nabla u|^p+\frac{1}{2}|\nabla v|^p. \)
\( \left|\nabla\left(\left(\frac{u^p+v^p}{2}\right)^{1/p}\right)\right| ^p\leq\frac{1}{2}|\nabla u|^p+\frac{1}{2}|\nabla v|^p. \)