inegalitatea gradienti in W_0^{1, p}(\Omega)

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Cezar Lupu
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inegalitatea gradienti in W_0^{1, p}(\Omega)

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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

\( \left|\nabla\left(\left(\frac{u^p+v^p}{2}\right)^{1/p}\right)\right| ^p\leq\frac{1}{2}|\nabla u|^p+\frac{1}{2}|\nabla v|^p. \)
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
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