Ngô Quốc Anh

February 22, 2010

The Poincaré inequality: W^{1,p} vs. W_0^{1,p}

Filed under: Giải Tích 6 (MA5205), Giải tích 8 (MA5206), Nghiên Cứu Khoa Học, PDEs — Tags: — Ngô Quốc Anh @ 1:50

In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods of the calculus of variations. A very closely related result is the Friedrichs’ inequality.

This topic will cover two versions of the Poincaré inequality, one is for W^{1,p}(\Omega) spaces and the other is for W_o^{1,p}(\Omega) spaces.

The classical Poincaré inequality for W^{1,p}(\Omega) spaces. Assume that 1\leq p \leq \infty and that \Omega is a bounded open subset of the n–dimensional Euclidean space \mathbb R^n with a Lipschitz boundary (i.e., \Omega is an open, bounded Lipschitz domain). Then there exists a constant C, depending only on \Omega and p, such that for every function u in the Sobolev space W^{1,p}(\Omega),

\displaystyle \| u - u_{\Omega} \|_{L^{p} (\Omega)} \leq C \| \nabla u \|_{L^{p} (\Omega)},

where

\displaystyle u_{\Omega} = \frac{1}{|\Omega|} \int_{\Omega} u(y) \, \mathrm{d} y

is the average value of u over \Omega, with |\Omega| standing for the Lebesgue measure of the domain \Omega.

Proof. We argue by contradiction. Were the stated estimate false, there would exist for each integer k = 1,... a function u_k \in W^{1,p}(\Omega) satisfying

\displaystyle \| u_k - (u_k)_{\Omega} \|_{L^{p} (\Omega)} \geq k \| \nabla u_k \|_{L^{p} (\Omega)}.

We renormalize by defining

\displaystyle {v_k} = \frac{{{u_k} - {{({u_k})}_\Omega }}}{{{{\left\| {{u_k} - {{({u_k})}_\Omega }} \right\|}_{{L^p}(\Omega )}}}}, \quad k \geqslant 1.

Then

\displaystyle {({v_k})_\Omega } = 0, \quad {\left\| {{v_k}} \right\|_{{L^p}(\Omega )}} = 1

and therefore

\displaystyle\| \nabla v_k \|_{L^{p} (\Omega)} \leqslant \frac{1}{k}.

In particular the functions \{v_k\}_{k\geq 1} are bounded in W^{1,p}(\Omega).

By mean of the Rellich-Kondrachov Theorem, there exists a subsequence {\{ {v_{{k_j}}}\} _{j \geqslant 1}} \subset {\{ {v_k}\} _{k \geqslant 1}} and a function v \in L^p(\Omega) such that

\displaystyle v_{k_j} \to v in L^p(\Omega).

Passing to a limit, one easily gets

\displaystyle v_\Omega = 0, \quad {\left\| {{v}} \right\|_{{L^p}(\Omega )}} = 1.

On the other hand, for each i=\overline{1,n} and \varphi \in C_0^\infty(\Omega),

\displaystyle\int_\Omega {v{\varphi _{{x_i}}}dx} = \mathop {\lim }\limits_{{k_j} \to \infty } \int_\Omega {{v_{{k_j}}}{\varphi _{{x_i}}}dx} = - \mathop {\lim }\limits_{{k_j} \to \infty } \int_\Omega {{v_{{k_j},{x_i}}}\varphi dx} = 0.

Consequently, v\in W^{1,p}(\Omega) with \nabla v=0 a.e. Thus v is constant since \Omega is connected. Since v_\Omega=0 then v \equiv 0. This contradicts to \|v\|_{L^p(\Omega)}=1.

The Poincaré inequality for W_0^{1,2}(\Omega) spaces. Assume that \Omega is a bounded open subset of the n-dimensional Euclidean space \mathbb R^n with a Lipschitz boundary (i.e., \Omega is an open, bounded Lipschitz domain). Then there exists a constant C, depending only on \Omega such that for every function u in the Sobolev space W_0^{1,2}(\Omega),

\displaystyle \| u \|_{L^2(\Omega)} \leq C \| \nabla u \|_{L^2(\Omega)}.

Proof. Assume \Omega can be enclosed in a cube

\displaystyle Q=\{ x \in \mathbb R^n: |x_i| \leqslant a, 1\leqslant i \leqslant n\}.

Then for any x \in Q, we have

\displaystyle\begin{gathered} {u^2}(x) = {\left( {\int_{ - a}^{{x_1}} {{u_{{x_1}}}(t,{x_2},...,{x_n})dt} } \right)^2} \hfill \\ \qquad\leqslant ({x_1} + a)\int_{ - a}^{{x_1}} {{{({u_{{x_1}}})}^2}dt} \hfill \\ \qquad\leqslant 2a\int_{ - a}^a {{{({u_{{x_1}}})}^2}dt} . \hfill \\ \end{gathered}.

Thus

\displaystyle\int_{ - a}^a {{u^2}(x)dx} \leqslant 4{a^2}\int_{ - a}^a {{{({u_{{x_1}}})}^2}dt}.

Integration over x_2,...,x_n from -a to a gives the result.

The Poincaré inequality for W_0^{1,p}(\Omega) spaces. Assume that 1\leq p<n and that \Omega is a bounded open subset of the n-dimensional Euclidean space \mathbb R^n with a Lipschitz boundary (i.e., \Omega is an open, bounded Lipschitz domain). Then there exists a constant C, depending only on \Omega and p, such that for every function u in the Sobolev space W_0^{1,p}(\Omega),

\displaystyle \| u \|_{L^{p^\star} (\Omega)} \leq C \| \nabla u \|_{L^{p} (\Omega)},

where p^\star is defined to be \frac{np}{n-p}.

Proof. The proof of this version is exactly the same to the proof of W^{1,p}(\Omega) case.

Remark. The point u =0 on the boundary of \Omega is important. Otherwise, the constant function will not satisfy the Poincaré inequality. In order to avoid this restriction, a weight has been added like the classical Poincaré inequality for W^{1,p}(\Omega) case. Sometimes, the Poincaré inequality for W_0^{1,p}(\Omega) spaces is called the Sobolev inequality.

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