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$\Omega \in \mathbb{R}^n$ is open, bounded, not empty and $u \in L^\infty(\Omega)$. For $1\leq p<\infty$ we have

$$ \Phi_p (u) = \frac{1}{|\Omega|} \left(\int_ \Omega |u|^p dx \right)^{\frac{1}{p}} $$

with $|\Omega| = \int_\Omega 1 \, dx$. I wanna show that $p \rightarrow \Phi_p(u)$ is monotonically increasing and $\lim_{n\rightarrow \infty} \Phi_p(u) = \|u\|_{L^\infty(\Omega)}$

For the monotony i think Hölder is the key. I cant get it done though. Especially the second task confuses me. Anyone can help?

$\left\{x\in \Omega \mid u(x) \geq \|u\|_{L^\infty} - \varepsilon \right\}$ Should help as a tip but i dont see it being useful

Michael Hardy
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SquareJoe
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1 Answers1

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It is the quantity $\displaystyle \left( \frac 1{|\Omega|} \int_\Omega |u|^p \, dx \right)^{1/p}$ that is nondecreasing.

You should probably use Jensen's inequality, but if you want to use Hölder's inequality you could assume $p < q$ and let $r = \frac qp$ with Hölder conjugate $r' = \frac q{q-p}$ so that $$ \int_\Omega |u|^p \, dx \le \left( \int_\Omega \, dx \right)^{1/r'} \left( \int_\Omega |u|^{pr} \, dx \right)^{1/r} = |\Omega|^{1 - q/p} \left( \int_\Omega |u|^q \right)^{p/q}$$ which simplifies to $$ \frac 1{|\Omega|} \int_\Omega |u|^p \, dx \le \left( \frac 1{|\Omega|} \int_\Omega |u|^q \right)^{p/q}.$$ Now take the $1/p$ power on both terms.

If $u$ is essentially bounded it is evident that $\displaystyle \left( \frac 1{|\Omega|} \int_\Omega |u|^p \, dx \right)^{1/p} \le \|u\|_\infty$. Now let $0 < \epsilon < \|u\|_\infty$ and define $A_\epsilon = \{x \in \Omega \mid |u(x)| > \|u\|_\infty - \epsilon\}$. Then $|A_\epsilon| > 0$ and $$ (\|u\|_\infty - \epsilon)^p |A_\epsilon| \le \int_{A_\epsilon} |u|^p \, dx \le \int_\Omega |u|^p \, dx$$ so that $$ \|u\|_\infty - \epsilon \le \left( \frac{|\Omega|}{|A_\epsilon|} \right)^{1/p} \left( \frac 1{|\Omega|} \int_\Omega |u|^p \, dx \right)^{1/p}.$$As $p \to \infty$ you have $$\left( \frac{|\Omega|}{|A_\epsilon|} \right)^{1/p} \to 1$$ so that $$\|u\|_\infty \le \lim_{p \to \infty} \left( \frac 1{|\Omega|} \int_\Omega |u|^p \, dx \right)^{1/p} + \epsilon.$$

Now let $\epsilon \to 0^+$.

Umberto P.
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