Poincare Inequalities for Any W Continuously Differentiable on 01proof

Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains

Mikhail Borsuk , Vladimir Kondratiev , in North-Holland Mathematical Library, 2006

2.2 The Poincaré inequality

Theorem 2.9

The Poincaré inequality for the domain in ℝ ^N (see e.g. (7.45) [129]).

Let u ∈ W ¹(G) and G is bounded convex domain in ℝ ⁿ . Then $||u-\overline{u}||{}_{2;G}\le c\left(N\right)\frac{{\left(\mathit{diam}G\right)}^N}{{\left(\mathit{meas}S\right)}^{1-1/N}}||\nabla u||{}_{2;G\text{,}}$ (PI 1)

where

$\overline{u}=\frac{1}{\mathit{meas}S}{\displaystyle \underset{S}{\int }u\left(x\right)\mathit{dx}\text{,}}$

and S is any measurable subset of G.

Theorem 2.10

The Poincaré inequality for the domain on the sphere (see e.g. Theorem 3.21 [145]).

Let u ∈ W ¹(Ω) and Ω is convex domain on the unit sphere S ^N-1 . Then||u  − u _Ω||_{2;   Ω}  ≤ c(N,   Ω)   ||   ∇ _ω u||_2;Ω, (PI 2)

where

$u_{\mathrm{\Omega }}=\frac{1}{\mathit{meas}\mathrm{\Omega }}{\displaystyle \underset{\mathrm{\Omega }}{\int }u\left(x\right)d\mathrm{\Omega }.}$

Also the following lemma is true.

Lemma 2.11

(see e.g. Lemma 7.16 [129]). Let u ∈ W ^1,1(G) and G is bounded convex domain in ℝ ^N . Then

(2.2.1) $|u-\overline{u}|\le \frac{{\left(\mathit{diam}G\right)}^N}{N\cdot \mathit{meas}S}{\displaystyle \underset{G}{\int }|x-y|{}^{1-N}|\nabla u\left(y\right)|\mathit{dy}}a.e.\mathit{in}G\text{,}$

where

$\overline{u}=\frac{1}{\mathit{meas}S}{\displaystyle \underset{S}{\int }u\left(x\right)\mathit{dx}\text{,}}$

and S is any measurable subset of G.

Now we shall prove the version of the Poincaré inequality.

Theorem 2.12

Let G ₀ ^d be convex domain, G ₀ ^d   ⊂ G, G is bounded domain in ℝ ^N . Let u ∈ L ²(G) and ${\displaystyle \underset{G_0^d}{\int }{r}^{\alpha -2}|\nabla u|{}^2\mathit{dx}<\infty }$ , α  ≤   4 – N. Then

(2.2.2) ${\displaystyle \underset{G_0^{\mathrm{\varrho }}}{\int }{r}^{\alpha -4}|u-\tilde{u}|{}^2\mathit{dx}\le c}{\displaystyle \underset{G_0^{\mathrm{\varrho }}}{\int }{r}^{\alpha -2}|\nabla u|{}^2\mathit{dx},\forall \mathrm{\varrho }\in \left(0,d\right)}\text{,}$

where

(2.2.3) $\tilde{u}=\frac{1}{\mathit{meas}{G}_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }u\left(y\right)\mathit{dy}}$

and c  >   0 depend only on N, d, measΩ.

Proof. Since α  ≤   4 – N then from our assumption we have u ∈ W ¹ (G). By density of C ^∞(G) ∩ W ¹(G) in W ¹(G) we can consider u ∈ C ¹(G). We use Lemma 2.11, applying it for the domains ${G^{\mathrm{\varrho }}}_{\mathrm{\varrho }/2}$ and $S={G^{\mathrm{\varrho }}}_{\mathrm{\varrho }/2}$ . By this Lemma and the Hölder inequality, we have

(2.2.4) $\begin{array}{l}|u\left(x\right)-\tilde{u}|{}^2\le c{\left({\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x-y|{}^{1-N}|\nabla u\left(y\right)|\mathit{dy}}\right)}^2\le \\ \le c{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x-y|{}^{1-N}|\nabla u\left(y\right)|{}^2\mathit{dy}{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x-y|{}^{1-N}\mathit{dy}}}=\\ =\frac{c}{2}\mathrm{\varrho }\cdot \mathit{meas}\mathrm{\Omega }{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x-y|{}^{1-N}|\nabla u\left(y\right)|{}^2\mathit{dy}.}\end{array}$

From (2.2.4) it follows

(2.2.5) $\begin{array}{l}{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }{r}^{\alpha -4}|u\left(x\right)-\tilde{u}|{}^2\mathit{dx}\le }\\ \le \frac{c}{2}\mathrm{\varrho }\cdot \mathit{meas}\mathrm{\Omega }{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }{r}^{\alpha -4}\left({\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x-y|{}^{1-N}|\nabla u\left(y\right)|{}^2\mathit{dy}}\right)}\mathit{dx}\le \\ \le c\cdot \mathit{meas}\mathrm{\Omega }{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }{r}^{\alpha -3}\left({\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x-y|{}^{1-N}|\nabla u\left(y\right)|{}^2\mathit{dy}}\right)}\mathit{dx}=\\ =c{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|\nabla u\left(y\right)|{}^2}\left({\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x|{}^{\alpha -3}|x-y|{}^{1-N}\mathit{dx}}\right)\mathit{dy}\le c{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }{r}^{\alpha -2}|\nabla u|{}^2\mathit{dx}\text{,}}\end{array}$

since

$\begin{array}{l}{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x|{}^{\alpha -3}|x-y|{}^{1-N}\mathit{dx}}\le c{\mathrm{\varrho }}^{\alpha -3}{\displaystyle \underset{G_{\mathrm{\varrho }/2}^{\mathrm{\varrho }}}{\int }|x-y|{}^{1-N}\mathit{dx}={\mathrm{\varrho }}^{\alpha -3}\cdot \frac{c}{2}}\mathrm{\varrho }\cdot \mathit{meas}\mathrm{\Omega }\le \\ \le c{\mathrm{\varrho }}^{\alpha -2}.\end{array}$

Replacing in (2.2.5) ϱ by 2^-k ϱ we can (2.2.5) rewrite so

${\displaystyle \underset{G^{\left(k\right)}}{\int }{r}^{\alpha -4}|u-\tilde{u}|{}^2\mathit{dx}\le c}{\displaystyle \underset{G^{\left(k\right)}}{\int }{r}^{\alpha -2}|\nabla u|{}^2\mathit{dx},\forall \mathrm{\varrho }\in \left(0,d\right)\text{,}}$

whence by summing over all k  =   0,1,·   ·   · we get the required (2.2.2).

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Weak Poincaré Inequalities and Convergence of Semigroups

Feng-Yu Wang , in Functional Inequalities, Markov Semigroups and Spectral Theory, 2005

Proof.

ByTheorem 4.3.3 , the Poincaré inequality (1.1.4) holds provided ψ(∞) > 0. Hence we only consider the case where ψ(∞) = 0. In this case one has ϕ(R) → ∞ as R→ ∞. Then there exist ε > 0 and R(ε) ≥ r ₀+ 1 such that

(4.3.8) $\psi {\left(R\right)}^2\mu \left(B_{\varphi \left(R\right)}\right){\left(\varphi \left(R\right)-r_0\right)}^2⩾8\left(1+ℰ\right)\mu \left(B_{\varphi \left(R\right)}^c\right),R⩾R\left(ℰ\right).$

Next, since $L_{0\rho }⩽-\psi \left(R\right)$ in B_R \ B _{r 0}in the distribution sense, by Corollary 2.5.4we obtain

(4.3.9) ${\lambda }_0\left(B_R\backslash B_{r0}\right):=\inf \left\{\mu (|\nabla f{|}^2:f\in C_0^{\infty }\left(B_R\backslash B_{ro}\right),\mu \left(f^2\right)=1\right\}⩾\frac{\psi {\left(R\right)}^2}{4}.$

Next, the proof of [194, Theorem 1.1] yields that

$\begin{array}{ll}\overline{\lambda }\left(R\right)\hfill & :=\inf \left\{\mu \left(|\nabla f{|}^2\right):f\in C_0^{\infty }\left(B_R\right),\mu \left(f^2\right)-\mu {\left(f\right)}^2=1\right\}\hfill \\ \hfill & ⩾\frac{{\lambda }_0\left(B_R\backslash B_{r0}\right)\lambda \left(r\right)\mu \left(B_r\right){\left(r-r_0\right)}^2-2\lambda \left(r\right)\mu \left(B_{\tau }^c\right)}{2\lambda \left(r\right){\left(r-r_0\right)}^2+{\lambda }_0\left(B_R\backslash B_{r0}\right){\left(r-r_0\right)}^2\mu \left(B_r\right)+2\mu \left(B_r\right)},\hfill \end{array}$

Combining this with (4.3.5), (4.3.8) and (4.3.9), we obtain

$\begin{array}{ll}\overline{\lambda }\left(R\right)\hfill & \ge \frac{c_1\lambda (\varphi (R))\mu (B_{\varphi (R)})\Psi {(R)}^2{(\varphi (R)-r_0)}^2}{\lambda (\varphi (R)){(\varphi (R)-r_0)}^2+\Psi {(R)}^2{(\varphi (R)-r_0)}^2\mu (B_{\varphi (R)})+\mu (B_{\varphi (R)})}\hfill \\ \hfill & \ge c_2\zeta (R),\hfill \end{array}$

for some c ₁, c ₂ > 0 and all R ≥ R(ε) such that ϕ(R)< R. If ϕ(R) ≥ R, then $\overline{\lambda }\left(R\right)\left(⩾\lambda \left(R\right)\right)⩾c_2\zeta \left(R\right)$ still holds for some c ₂>0 according to (4.3.5). Then for any $f\in C_b^{\infty }\left(M\right)$ with μ(f) = 0 and any $R⩾R_{ℰ},leth={\left(R-\rho \right)}^{+}\wedge 1$ , we have

$\begin{array}{ll}\mu \left(f^2\right)\hfill & ⩽\mu \left(f^2{h}^2\right)+||f|{|}_{\infty }^2\mu \left(B_{R-1}^c\right)\hfill \\ \hfill & ⩽\frac{2\mu \left(|\nabla f{|}^2\right)+2||f|{|}_{\infty }^2\mu \left(B_{R-1}^c\right)}{c_2\zeta \left(R\right)}+\frac{||f|{|}_{\infty }^2\mu \left(B_{R-1}^c\right)}{\mu \left(B_{R-1}\right)}.\hfill \end{array}$

This proves the desired result by taking c= 2/ c ₂ for small r>0 such that R_r ≥ R(ε).

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Nonlinear Spectral Problems for Degenerate Elliptic Operators

Peter Takáč , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2004

Proof of Theorem 6.7

Our proof of this theorem combines the improved Poincaré inequality ( 6.3) with a generalized Rayleigh quotient formula. To this end, we may assume that f ∈ ${{\mathcal{D}}^{\prime }}_{{\phi }_1}$ satisfies f ≢ 0 in Ω and 〈f, φ ₁ = 0. Define the number M_f (0 ⩽ M_f ⩽ ∞) by

(6.13) $M_f\stackrel{\text{def}}{=}\underset{\begin{array}{c}v\in W_0^{1,p}(\Omega )\\ v\notin \{\kappa {\phi }_1:\kappa \in ℝ\}\end{array}}{\sup }\frac{|〈f,v〉{|}^p}{{\displaystyle {\int }_{\Omega }|\nabla v}{|}^p\text{d}x-{\text{λ}}_1{\displaystyle {\int }_{\Omega }|v{|}^p\text{d}x}}.$

Clearly, M_f > 0. Moreover, (6.3) entails

$\begin{array}{ll}{\left|〈f,v〉\right|}^p& ⩽{\Vert f\Vert }_{W^{-1,p^{\prime }}(\Omega )}^p{\Vert v^{\top }\Vert }_{W_0^{1,p}(\Omega )}^p\\ & ⩽C_f\left({\displaystyle {\int }_{\Omega }|\nabla v{|}^p}\text{d}x-{\text{λ}}_1{\displaystyle {\int }_{\Omega }|v{|}^p\text{d}x}\right)\end{array}$

for all v ∈ $W_0^{1,p}$ (Ω), where C_f = c⁻¹ ‖f‖ ^p _{W ^−1,p′ (Ω}is a constant. This shows that M_f ⩽ C_f <∞. In a similar way we arrive at

(6.14) $\begin{array}{l}{\left|v^{\left|\right|}\right|}^{p-2}{\left|〈f,v〉\right|}^2\\ ⩽{\left|v^{\left|\right|}\right|}^{p-2}\left|\right|f|{|}_{{{\mathcal{D}}^{\prime }}_{{\phi }_1}}^2{\Vert v^{\top }\Vert }_{\mathcal{D}{\phi }_1}^2\\ \begin{array}{cc}⩽{C^{\prime }}_f\left({\displaystyle {\int }_{\Omega }|\nabla v{|}^p}\text{d}x-{\text{λ}}_1{\displaystyle {\int }_{\Omega }|v{|}^p\text{d}x}\right)& \text{for\hspace{0.17em}all\hspace{0.17em}v}\in W_0^{1,p}(\Omega )\end{array},\end{array}$

where C′_f = c ⁻¹ ${\Vert f\Vert }_{{{\mathcal{D}}^{\prime }}_{{\phi }_1}}^2$ is a constant, and ${\Vert \cdot \Vert }_{{{\mathcal{D}}^{\prime }}_{{\phi }_1}}$ stands for the dual norm on ${{\mathcal{D}}^{\prime }}_{{\phi }_1}$ . From (6.13) and inequality (6.14) we can draw the following conclusion: If v ∈ $W_0^{1,p}$ (Ω) is such that v ^⊤ ≢ 0 in Ω and

$\frac{{\left|〈f,v〉\right|}^p}{{\displaystyle {\int }_{\Omega }|\nabla v{|}^p}\text{d}x-{\text{λ}}_1{\displaystyle {\int }_{\Omega }|v{|}^p\text{d}x}}⩾\frac{1}{2}{M}_f,$

then 〈f, v〉 ≠ 0 and

${\left|v^{\left|\right|}\right|}^{p-2}⩽2\frac{{C^{\prime }}_f}{M_f}{\left|〈f,v〉\right|}^{p-2}⩽{\left({C^{″}}_f\right)}^{p-2}{\Vert v^{\top }\Vert }_{W_0^{1,p}(\Omega )}^{p-2},$

where C″_f = [2(C′_f /M_f )]^1/(p−2)‖f‖ _{W ^−1,p′ (Ω)} is a constant, i.e.,

(6.15) $\left|v^{\left|\right|}\right|⩽{C^{″}}_f{\Vert v^{\top }\Vert }_{W_0^{1,p}(\Omega )}.$

Next, take any maximizing sequence {v_n }^∞ _n=1 in $W_0^{1,p}$ (Ω) for the generalized Rayleigh quotient (6.13), that is, v_n ^⊤ ≢0 in Ω and

(6.16) $\begin{array}{cc}\frac{{\left|〈f,v_n〉\right|}^p}{{\displaystyle {\int }_{\Omega }|\nabla v_n{|}^p\text{d}x-{\text{λ}}_1{\displaystyle {\int }_{\Omega }|v_n{|}^p\text{d}x}}}\to M_f& \text{as}n\to \infty .\end{array}$

Since both, the numerator and the denominator are p-homogeneous, we may assume ‖v_n ‖ _{W ^1,p 0(Ω)} = 1 for all n ⩾ 1. The Sobolev space W ^1,p ₀(Ω) being reflexive, we may pass to a convergent subsequence v_n ⇀ w weakly in $W_0^{1,p}$ (Ω); hence, also v_n → w strongly in L^p (Ω), by Rellich's theorem, and 〈f, v_n 〉 → 〈f, w〉 as n → ∞. We insert these limits into (6.16) to obtain

(6.17) ${\displaystyle {\int }_{\Omega }|\nabla w{|}^p\text{d}x-{\text{λ}}_1}{\displaystyle {\int }_{\Omega }|w{|}^p\text{d}x⩽1-{\text{λ}}_1}{\displaystyle {\int }_{\Omega }|w{|}^p\text{d}x=M_f^{-1}{\left|〈f,w〉\right|}^p}.$

In particular, we have w≡ 0 in Ω, therefore also w ^⊤ ≡ 0 by (6.15), and consequently |〈f, w〉| ≠ 0 by (6.17). We combine (6.13) with (6.17) to get ∫_ω |∇w| ^p dx = 1. Hence, the supremum M_f in (6.13) is attained at w in place of v.

Finally, we can apply the calculus of variations to the inequality

$\begin{array}{cc}{\displaystyle {\int }_{\Omega }|\nabla v{|}^p\text{d}x-{\text{λ}}_1}{\displaystyle {\int }_{\Omega }|v{|}^p\text{d}x}-M_f^{-1}{\left|〈f,v〉\right|}^p⩾0& \text{for}v\in W_0^{1,p}(\Omega )\end{array}$

to derive

$\{\begin{array}{l}\begin{array}{cc}-{\Delta }_{p}w-{\text{λ}}_1|w{|}^{p-2}w=M_f^{-1}{\left|〈f,w〉\right|}^{p-2}〈f,w〉f(x)& \text{in}\Omega \end{array},\hfill \\ \begin{array}{cc}w=0& \text{on}\partial \Omega \end{array}.\hfill \end{array}$

It follows that u $\stackrel{\text{def}}{=}$ M_f ^1/(p−1)〈f,w〉⁻¹ · w is a weak solution of problem (6.12).

Theorem 6.7 is proved.

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Functional Inequalities and Essential Spectrum

Feng-Yu Wang , in Functional Inequalities, Markov Semigroups and Spectral Theory, 2005

Corollary 3.4.15

In the situation of Theorem 1.4.11.Ifξ(r) → 0 as r→ ∞,then(3.3.3)holds for someβ.

Finally, let us consider the isoperimetric constants for the super Poincaré inequalities in the context of diffusion processes on a connected complete Riemannian manifold M. Let $\mu \left(dx\right):=e^{V\left(x\right)}dx$ for some V∈C(M), and let $ℰ\left(f,f\right):=\mu \left({\left|\nabla f\right|}^2\right).$ Define

$k\left(r\right)=\underset{\mu \left(A\right)⩽r}{\inf }\frac{{\mu }_{\partial }\left(\partial A\backslash \partial M\right)}{\mu \left(A\right)},r>0,$

(3.4.21) $k\left(r\right)=\underset{\mu \left(A\right)⩽r}{\inf }\frac{{\mu }_{\partial }\left(\partial A\backslash \partial M\right)}{\mu \left(A\right)},r>0,$

where μ_∂is the (d– 1)-dimensional measure induced by μ, and A ranges over all open smooth domains.

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Special Volume: Mathematical Modeling and Numerical Methods in Finance

Olivier Pironneau , Yves Achdou , in Handbook of Numerical Analysis, 2009

Theorem 1.2

•

The space D(R ₊) is dense in V.

•

(Poincaré's inequality) If v ɛ V, then

(1.15) ${\Vert \upsilon \Vert }_{L^2(R_{+})}\le 2{\Vert s\frac{\text{d}\upsilon }{\text{d}S}\Vert }_{L^2(R_{+})}$

so the seminorm ${|v|}_V={\Vert S\frac{\text{d}\upsilon }{\text{d}S}\Vert }_{L^2(R_{+})}$ is also a norm on V, equivalent to ||.||V.

•

For any w ɛ L (R ₊), the function S → v(S) = $\frac{1}{S}$ ${\int }_0^S$ w(s)ds belongs to V, and ||v||V ≤ C||w|| _{L ²(R +)} for some positive constant C is independent of w.

We denote by V′ the topological dual space of V, and for $w\in V^{\prime },||w|{|}_{V^{\prime }}={sup}_{\upsilon \in V\backslash \left\{0\right\}}\frac{(w,\upsilon )}{|\upsilon |V}.$

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Variational Methods

Paul Sacks , in Techniques of Functional Analysis for Differential and Integral Equations, 2017

15.1 The Dirichlet Quotient

We have earlier introduced the concept of the Rayleigh quotient

(15.1.1) $\begin{array}{l}\hfill J(u)=\frac{〈Tu,u〉}{〈u,u〉}\end{array}$

for a linear operator T on a Hilbert space H. In the previous discussion we were mainly concerned with the case that T was a bounded or even a compact operator, but now we will allow for T to be unbounded. In such a case, J(u) is defined at least for u ∈ D(T)∖{0}, and possibly has a natural extension to some larger domain. The principal case of interest to us here is the case of the Dirichlet Laplacian discussed in Section 13.4,

(15.1.2) $\begin{array}{l}\hfill Tu=-\mathrm{\Delta }u\text{on}D(T)=\{u\in H_0^1(\mathrm{\Omega }):\mathrm{\Delta }u\in L^2(\mathrm{\Omega })\}\end{array}$

In this case

(15.1.3) $\begin{array}{l}\hfill J(u)=\frac{〈-\mathrm{\Delta }u,u〉}{〈u,u〉}=\frac{{\int }_{\mathrm{\Omega }}|\nabla u{|}^{2}dx}{{\int }_{\mathrm{\Omega }}|u{|}^{2}dx}=\frac{\parallel u{\parallel }_{H_0^1(\mathrm{\Omega })}^2}{\parallel u{\parallel }_{L^2(\mathrm{\Omega })}^2}\end{array}$

which we may evidently regard as being defined on all of $H_0^1(\mathrm{\Omega })$ except the origin. We will refer to any of these equivalent expressions as the Dirichlet quotient (or Dirichlet form) for −Δ. Throughout this section we take Eq. (15.1.3) as the definition of J, and denote by {λ _n , ψ _n } the eigenvalues and eigenfunctions of T, where we may choose the ψ _n 's to be an orthonormal basis of L ²(Ω), according to the discussion of Section 13.4. It is immediate that

(15.1.4) $\begin{array}{l}\hfill J({\psi }_n)={\lambda }_n\end{array}$

for all n.

If we define a critical point of J to be any $u\in H_0^1(\mathrm{\Omega })\setminus \{0\}$ for which

(15.1.5) $\begin{array}{l}\hfill {\left.\frac{d}{d\alpha }J(u+\alpha v)\right|}_{\alpha =0}=0\forall v\in H_0^1(\mathrm{\Omega })\end{array}$

then precisely as in Eq. (12.3.40) and the following discussion we find that at any critical point there holds

(15.1.6) $\begin{array}{l}\hfill {\int }_{\mathrm{\Omega }}\nabla u\cdot \nabla vdx=J(u){\int }_{\mathrm{\Omega }}uvdx\forall v\in H_0^1(\mathrm{\Omega })\end{array}$

In other words, Tu = λu must hold with λ = J(u). Conversely, by straightforward calculation, any eigenfunction of T is a critical point of J. Thus the set of eigenfunctions of T coincides with the set of critical points of the Dirichlet quotient, and by Eq. (15.1.4) the eigenvalues are exactly the critical values of J.

Among these critical points, one might expect to find a point at which J achieves its minimum value, which must then correspond to the critical value λ ₁, the least eigenvalue of T. We emphasize, however, that the existence of a minimizer of J must be proved—it is not immediate from anything we have stated so far. We give one such proof here, and indicate another one in Exercise 15.3.

Theorem 15.1

There exists $\psi \in H_0^1(\mathrm{\Omega })\setminus \{0\}$ such that J(ψ) ≤ J(ϕ) for all $\varphi \in H_0^1(\mathrm{\Omega })\setminus \{0\}$ .

Proof

Let

(15.1.7) $\begin{array}{l}\hfill \lambda =\underset{\varphi \in H_0^1(\mathrm{\Omega })}{inf}J(\varphi )\end{array}$

so λ > 0 by the Poincaré inequality. Therefore there exists ${\psi }_n\in H_0^1(\mathrm{\Omega })$ such that J(ψ _n ) → λ. Without loss of generality we may assume $\parallel {\psi }_n{\parallel }_{L^2(\mathrm{\Omega })}=1$ for all n, in which case $\parallel {\psi }_n{\parallel }_{H_0^1(\mathrm{\Omega })}^2\to \lambda$ . In particular {ψ _n } is a bounded sequence in $H_0^1(\mathrm{\Omega })$ , so by Theorem 12.1 there exists $\psi \in H_0^1(\mathrm{\Omega })$ such that ${\psi }_{n_k}\stackrel{w}{\to }\psi$ in $H_0^1(\mathrm{\Omega })$ , for some subsequence. By Theorem 13.4 it follows that ${\psi }_{n_k}\to \psi$ strongly in L ²(Ω), so in particular $\parallel \psi {\parallel }_{L^2(\mathrm{\Omega })}=1$ . Finally, using the lower semicontinuity property of weak convergence (Proposition 12.2)

(15.1.8) $\begin{array}{l}\hfill \lambda \le J(\psi )=\parallel \psi {\parallel }_{H_0^1(\mathrm{\Omega })}^2\le \underset{n_k\to \infty }{liminf}\parallel {\psi }_{n_k}{\parallel }_{H_0^1(\mathrm{\Omega })}^2=\underset{n_k\to \infty }{liminf}J({\psi }_{n_k})=\lambda \end{array}$

so that J(ψ) = λ, that is, J achieves its minimum at ψ.

Note that by its very definition, the minimum λ ₁ of the Rayleigh quotient J, gives rise to the best constant in the Poincaré inequality, namely Eq. (13.4.82) is valid with $C=\frac{1}{\sqrt{{\lambda }_1}}$ and no smaller C works.

The above argument provides a proof of the existence of one eigenvalue of T, namely the smallest eigenvalue λ ₁, with corresponding eigenfunction ψ ₁, which is completely independent from the proof given in Chapter 12. It is natural to ask then whether the existence of the other eigenvalues can be obtained in a similar way. Of course they can no longer be obtained by minimizing the Dirichlet quotient (nor is there any maximum to be found), but we know in fact that J has other critical points, since other eigenfunctions exist. Consider, for example the case of λ ₂, for which there must exist an eigenfunction orthogonal in L ²(Ω) to the eigenfunction already found for λ ₁. Thus it is a natural conjecture that λ ₂ may be obtained by minimizing J over the orthogonal complement of ψ ₁. Specifically, if we set

(15.1.9) $\begin{array}{l}\hfill H_1=\left\{\varphi \in H_0^1(\mathrm{\Omega }):{\int }_{\mathrm{\Omega }}\varphi {\psi }_{1}dx=0\right\}\end{array}$

then the existence of a minimizer of J over H ₁ can be proved just as in Theorem 15.1. If the minimum occurs at ψ ₂, with λ ₂ = J(ψ ₂) then the critical point condition amounts to

(15.1.10) $\begin{array}{l}\hfill {\int }_{\mathrm{\Omega }}\nabla {\psi }_2\cdot \nabla vdx={\lambda }_2{\int }_{\mathrm{\Omega }}{\psi }_{2}vdx\forall v\in H_1\end{array}$

Furthermore, if v = ψ ₁ then

(15.1.11) $\begin{array}{l}\hfill {\int }_{\mathrm{\Omega }}\nabla {\psi }_2\cdot \nabla {\psi }_{1}dx=-{\int }_{\mathrm{\Omega }}{\psi }_2\mathrm{\Delta }{\psi }_1=-{\lambda }_1{\int }_{\mathrm{\Omega }}{\psi }_2{\psi }_1=0\end{array}$

since ψ ₂ ∈ H ₁. It follows that Eq. (15.1.10) holds for every $v\in H_0^1(\mathrm{\Omega })$ , so ψ ₂ is an eigenvalue of T for eigenvalue λ ₂. Clearly λ ₂ ≥ λ ₁, since λ ₂ is obtained by minimization over a smaller set.

We may continue this way, successively minimizing the Rayleigh quotient over the orthogonal complement in L ²(Ω) of the previously obtained eigenfunctions, to obtain a variational characterization of all eigenvalues.

Theorem 15.2

We have

(15.1.12) $\begin{array}{l}\hfill {\lambda }_n=J({\psi }_n)=\underset{u\in H_{n-1}}{min}J(u)\end{array}$

where

(15.1.13) $\begin{array}{l}\hfill H_n=\left\{u\in H_0^1(\mathrm{\Omega }):{\int }_{\mathrm{\Omega }}u{\psi }_{k}dx=0,k=1,2,\dots ,n\right\}{H}_0=H_0^1(\mathrm{\Omega })\end{array}$

This proof is essentially a mirror image of the proof of Theorem 12.10, in which a compact operator has been replaced by an unbounded operator, and maximization has been replaced by minimization. One could also look at critical points of the reciprocal of J in order to maintain it as a maximization problem, but it is more common to proceed as we are doing. Similar results can be obtained for a larger class of unbounded self-adjoint operators, see for example [39]. The eigenfunctions may be interpreted as saddle points of J, that is, critical points which are not local extrema.

The characterization of eigenvalues and eigenfunctions stated in Theorem 15.2 is unsatisfactory, in the sense that the minimization problem to be solved in order to obtain an eigenvalue λ _n requires knowledge of the eigenfunctions corresponding to smaller eigenvalues. We next discuss two alternative characterizations of eigenvalues, which may be regarded as advantageous from this point of view.

If E is a finite dimensional subspace of $H_0^1(\mathrm{\Omega })$ , we define

(15.1.14) $\begin{array}{l}\hfill \mu (E)=\underset{u\in E}{max}J(u)\end{array}$

and set

(15.1.15) $\begin{array}{l}\hfill S_n=\{E\subset H_0^1(\mathrm{\Omega }):\text{E is a subspace,}dim(E)=n\}n=0,1,\dots \end{array}$

Note that μ(E) exists and is finite for E ∈ S _n , since if we choose any orthonormal basis $\{{\zeta }_1,\dots ,{\zeta }_n\}$ of S _n then

(15.1.16) $\begin{array}{l}\hfill \underset{u\in E}{max}J(u)=\underset{\sum _{k=1}^n|c_k{|}^2=1}{max}{\int }_{\mathrm{\Omega }}{\left|\sum _{k=1}^n{c}_k\nabla {\zeta }_k\right|}^{2}dx\end{array}$

Thus finding μ(E) amounts to maximizing a continuous function over a compact set.

Theorem 15.3

(Poincaré Min-Max formula) We have

(15.1.17) $\begin{array}{l}\hfill {\lambda }_n=\underset{E\in S_n}{min}\mu (E)=\underset{E\in S_n}{min}\underset{u\in E}{max}J(u)\end{array}$

for $n=1,2,\dots$ .

Proof

J is constant on any one-dimensional subspace, that is, μ(E) = J(ϕ) if E = span{ϕ}, so the conclusion is equivalent to the statement of Theorem 15.1 for n = 1. For n ≥ 2, if E ∈ S _n we can find w ∈ E, w≠0 such that w ⊥ ψ _k for $k=1,\dots ,n-1$ , since this amounts to n − 1 linear equations for n unknowns (here {ψ _n } still denotes the orthonormalized Dirichlet eigenfunctions). Thus w ∈ H _n−1 and so by Theorem 15.2.

(15.1.18) $\begin{array}{l}\hfill {\lambda }_n\le J(w)\le \underset{u\in E}{max}J(u)=\mu (E)\end{array}$

It follows that

(15.1.19) $\begin{array}{l}\hfill {\lambda }_n\le \underset{E\in S_n}{inf}\mu (E)\end{array}$

On the other hand, if we choose $E=\text{span}\{{\psi }_1,\dots ,{\psi }_n\}$ note that

(15.1.20) $\begin{array}{l}\hfill J(u)=\frac{\sum _{k=1}^n{\lambda }_k{c}_k^2}{\sum _{k=1}^n{c}_k^2}\end{array}$

for any $u={\sum }_{k=1}^n{c}_k{\psi }_k\in E$ . Thus

(15.1.21) $\begin{array}{l}\hfill \mu (E)=J({\psi }_n)={\lambda }_n\end{array}$

and so the infimum in Eq. (15.1.19) is achieved for this E. The conclusion (15.1.17) then follows.

A companion result with a similar proof (e.g., Theorem 5.2 of [39]) is the following.

Theorem 15.4

(Courant-Weyl Max-Min formula) We have

(15.1.22) $\begin{array}{l}\hfill {\lambda }_n=\underset{E\in S_{n-1}}{max}\underset{u\perp E}{min}J(u)\end{array}$

for $n=1,2,\dots$ .

An interesting application of the variational characterization of the first eigenvalue is the following monotonicity property. We use temporarily the notation λ ₁(Ω) to denote the smallest Dirichlet eigenvalue of −Δ for the domain Ω.

Theorem 15.5

If $\mathrm{\Omega }\subset {\mathrm{\Omega }}^{\prime }$ then λ ₁(Ω′) ≤ λ ₁(Ω).

Proof

By the density of $C_0^{\infty }(\mathrm{\Omega })$ in $H_0^1(\mathrm{\Omega })$ and Theorem 15.1, for any ϵ > 0 there exists $u\in C_0^{\infty }(\mathrm{\Omega })$ such that

(15.1.23) $\begin{array}{l}\hfill J(u)\le {\lambda }_1(\mathrm{\Omega })+ϵ\end{array}$

But extending u to be zero outside of Ω we may regard it as also belonging to $C_0^{\infty }({\mathrm{\Omega }}^{\prime })$ , and the value of J(u) is the same whichever domain we have in mind. Therefore

(15.1.24) $\begin{array}{l}\hfill {\lambda }_1({\mathrm{\Omega }}^{\prime })\le J(u)\le {\lambda }_1(\mathrm{\Omega })+ϵ\end{array}$

and so the conclusion follows by letting ϵ → 0.

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Handbook of Differential Equations

Henghui Zou , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2008

Lemma 2.3.2.

For every φ ɛ C ( $\overline{\Omega }$ ) and u ₀ɛ C ( $\overline{\Omega }$ ), the BVP (2.3.2) has a unique solution $u\in X\cap C\overline{\left(\Omega \right)}$ , where X = W ₀ ^{1, m} (Ω).

Proof. We first show the existence in X. For any ϕ ɛ C ( $\overline{\Omega }$ ), define

$L\left(\upsilon \right)={\int }_{\Omega }\varphi \upsilon ,\upsilon \in X.$

Clearly L ɛ X′. Therefore, it suffices to show that A: X ↦ X ′ is surjective. In view of Lemma 2.3.1, it remains to show that A is coercive since A is bounded, continuous and monotone. By the Poincare inequality, there exists C = C (Ω, n, m) > 0 such that

$\left|\right|\text{u}|{|}_X^m\le C\int {\left|\nabla \text{u}\right|}^m.$

By the monotonicity of h,

$\left[h\left(u\right)-h\left(0\right)\right]u\ge 0.$

Finally, by the Hölder inequality,

$\left|h\left(0\right)\right|\int \left|u\right|\le C{\left(\int {\left|u\right|}^m\right)}^{1/m}\le C\left|\right|\text{u}\left|\right|X.$

It follows that, by Lemma 2.2.1, for || u || _X sufficiently large

$\begin{array}{l}\left(\mathcal{A}\left(u\right),u\right)=\int \text{A}\left(x,u,\nabla \text{u}\right)·\nabla \text{u}+\int \left[h\left(u\right)-h\left(0\right)\right]u+h\left(0\right)\int \text{u}\\ \ge \int (\text{A}\left(x,u,\nabla \text{u}\right)-\text{A}\left(x,u,0\right))·\nabla \text{u}-C|\left|\text{u}\right||X\\ \ge C\int {\left(\delta +\left|\nabla \text{u}\right|\right)}^{m-2}{\left|\nabla \text{u}\right|}^2-C\left|\right|\text{u}\left|\right|X\ge C\left|\right|\text{u}|{|}_X^m-C|\left|\text{u}\right|{|}_X,\end{array}$

where we have used (A2). In turn,

$\underset{\left|\right|\text{u}\left|\right|X\to \infty }{\lim }\frac{\left(\mathcal{A}\left(u\right),u\right)}{\left|\right|\text{u}\left|\right|X}\ge \underset{\left|\right|\text{u}\left|\right|X\to \infty }{\lim \inf }\left[C\left|\right|\text{u}|{|}_X^{m-1}-C\right]=\infty ,$

since m >1. Hence the BVP (2.3.2) has a solution u ɛ X.

Next, with the aid of Lemma 2.2.1, using | u |^γ u as a test function in (2.3.2) to get

$\begin{array}{l}C\gamma \int {\left(\delta +|\nabla \text{u}|\right)}^{m-2}\left|\nabla \text{u}\right|{\left|u\right|}^{\gamma }\le \int \text{A}\left(x,u,\nabla \text{u}\right)·\nabla \text{u}\\ =\int {\left|u\right|}^{\gamma }u\varphi -\int h\left(u\right){\left|u\right|}^{\gamma }u\\ \le \int {\left|u\right|}^{\gamma }u\left[\varphi -h\left(0\right)\right]\le C\int {\left|u\right|}^{\gamma +1},\end{array}$

since [h (u) – h (0)]| u |^γ u ≥ 0 by the monotonicity of h. One then readily derives an a priori W ₀ ^{1, m} (Ω)-bound for u and a standard boot-strap argument (cf. [47], see also Section 3.2 for details and it may require slightly more work when m < 2 and δ = 1) implies that there exists C = C (n, m, K, h, ||ϕ||) > 0 such that

$\left|\right|\text{u}|{|}_{L^{\infty }}\left(\Omega \right)C.$

Now the classical De Giorgi estimate implies that u is actually Holder-continuous on $\overline{\Omega }$ , i.e., $u\in X\cap C\overline{\left(\Omega \right)}$ .

The uniqueness follows directly from the weak comparison Lemma 2.2.2, in view of the monotonicity of h.

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Concentration, Results and Applications

Gideon Schechtman , in Handbook of the Geometry of Banach Spaces, 2003

2.4 Spectral methods

Let $\left(\Omega ,\mathcal{F},\mu \right)$ be a probability space, $\mathcal{A}$ some set of measurable functions and $\mathcal{E}:\mathcal{A}\to {ℝ}^{+}$ some function (which we shall refer to as energy function). For ƒ ∈ L ₂(Ω) denote by σ ²(ƒ) the variance of ƒ,

(49) ${\sigma }^2\left(ƒ\right)={\displaystyle \int {\left(ƒ-\mathbb{E}ƒ\right)}^2}\text{d}\mu ={\displaystyle \int {ƒ}^2\text{d}\mu }-{\left({\displaystyle \int ƒ\text{d}\mu }\right)}^2$

and, for ƒ ∈ L ² log L $\left(\text{i}.\text{e}.,{\displaystyle \int {ƒ}^2{\log }^{+}ƒ\text{d}\mu \text{<}\infty }\right)$ , denote by ε(ƒ) the entropy of ƒ²,

(50) $\epsilon \left(ƒ\right)={\displaystyle \int {ƒ}^2\log {ƒ}^2\text{d}\mu }-{\displaystyle \int {ƒ}^2\text{d}\mu \log \left({\displaystyle \int {ƒ}^2\text{d}\mu }\right)}$

(which is necessarily finite). We say that $\left(\mathcal{A},\mathcal{E}\right)$ satisfy a Poincaré inequality with constant C if

(51) $\begin{array}{cc}{\sigma }^2\left(ƒ\right)⩽C\mathcal{E}\left(ƒ\right)& \text{for}\text{all}ƒ\in \mathcal{A}.\end{array}$

We say that $\left(\mathcal{A},\mathcal{E}\right)$ satisfy a logarithmic Sobolev inequality with constant C if

(52) $\begin{array}{cc}\epsilon \left(ƒ\right)⩽C\mathcal{E}\left(ƒ\right)& \text{for all}ƒ\in \mathcal{A}.\end{array}$

The main example of an energy function $\mathcal{E}$ is related to the gradient or generalization of it. If d isa metric on Ω (and $\mathcal{F}$ the Borel σ-field), define the norm of the gradient at x ∈ Ω by

(53) $\left|\nabla ƒ\left(x\right)\right|=\lim \underset{y\to x}{\sup }\frac{\left|ƒ\left(x\right)-ƒ\left(y\right)\right|}{d\left(x,y\right)}.$

Note that ∇ ƒ(x) by itself is not defined. The reason for this notation is of course that if (Ω, d) is a Riemannian manifold (in particular if it is ℝ ⁿ with the Euclidean distance) and if ƒ is differentiable at x then |∇ ƒ(x)| is the Euclidean norm of the gradient of ƒ at x. Define now

(54) $\mathcal{E}\left(ƒ\right)={\displaystyle \int {\left|\nabla ƒ\left(x\right)\right|}^2\text{d}\mu \left(x\right)}.$

The classical Poincaré (or Rayleigh–Ritz) inequality says that, in the case of a compact Riemannian manifold, (51) is satisfied with $C={\lambda }_1^{-1}$ , λ₁ being the first positive eigenvalue of the Laplacian on L ₂(Ω, μ).

We shall only deal here with the energy function (54). [33] contains many other examples and a comprehensive treatment of the subject of this section.

If $\mathcal{A}$ is the set of bounded Lipschitz functions on (Ω, d), the norm of the gradient satisfies the chain rule: if ϕ ∈ C ¹ (ℝ) and ƒ ∈ Ω then ϕ ∘ ƒ ∈ Ω and

(55) $\left|\nabla \left(\varphi \circ ƒ\right)\left(x\right)\right|⩽\left|\nabla ƒ\left(x\right)\right|\left|{\varphi }^{\prime }\left(ƒ\left(x\right)\right)\right|$

and consequently

(56) $\mathcal{E}\left(\varphi \circ ƒ\right)⩽{\Vert ƒ\Vert }_{\text{Lip}}^2{\displaystyle \int {\left|{\varphi }^{\prime }\left(ƒ\left(x\right)\right)\right|}^2\text{d}\mu \left(x\right)},$

where ∥ƒ∥_Lip denotes the Lipschitz constant of ƒ. The next theorem, basically due to Gromov and Milman, shows that Poincaré inequality implies concentration.

Theorem 14

Let $\left(\Omega ,\mathcal{F},\mu ,d\right)$ be a probability metric space. Let $\mathcal{A}$ be the set of bounded Lipschitz functions on (Ω, d) and let $\mathcal{E}$ be defined by (54). Assume that $\left(\mathcal{A},\mathcal{E}\right)$ satisfies the Poincaré inequality (51). Then for all $\left|\lambda \right|<2/\sqrt{C}$ and every bounded ƒ with Lipschitz constant 1

(57) $\mathbb{E}{\text{e}}^{\lambda \left(ƒ-\mathbb{E}ƒ\right)}⩽\frac{240}{4-C{\lambda }^2}.$

In particular

(58) $\begin{array}{cc}P\left(\left|ƒ-\mathbb{E}ƒ\right|>t\right)⩽240{\text{e}}^{-\sqrt{\frac{2}{C}t}}& forallt>0.\end{array}$

Proof

By (51) and (56)

$\mathbb{E}{\text{e}}^g-{\left(\mathbb{E}{\text{e}}^{g/2}\right)}^2⩽C\mathcal{E}\left({\text{e}}^{g/2}\right)⩽\frac{C}{4}{\Vert g\Vert }_{\text{Lip}}\mathbb{E}{\text{e}}^g$

for any $g\in \mathcal{A}$ . In particular, for any λ,

$\mathbb{E}{\text{e}}^{\lambda ƒ}-{\left(\mathbb{E}{\text{e}}^{\frac{\lambda }{2}ƒ}\right)}^2⩽\frac{C{\lambda }^2}{4}\mathbb{E}{\text{e}}^{\lambda ƒ}$

or

$\mathbb{E}{\text{e}}^{\lambda ƒ}⩽\frac{1}{1-\frac{C{\lambda }^2}{4}}{\left(\mathbb{E}{\text{e}}^{\frac{\lambda }{2}ƒ}\right)}^2.$

Iterating we get for every n,

$\mathbb{E}{\text{e}}^{\lambda ƒ}⩽{\displaystyle \prod _{k=0}^{n-1}{\left(\frac{1}{1-\frac{C{\lambda }^2}{4^{k+1}}}\right)}^{2^k}{\left(\mathbb{E}{\text{e}}^{\frac{\lambda }{2^n}ƒ}\right)}^{2^n}}$

which tends to

${\displaystyle \prod _{k=o}^{\infty }{\left(\frac{1}{1-\frac{C{\lambda }^2}{4^{k+1}}}\right)}^{2^k}{\text{e}}^{\lambda \mathbb{E}ƒ}}.$

Remark 15

(1)

A simple limiting argument shows now that the assumption that ƒ is bounded is superfluous.

(2)

The simple example of the exponential distribution on ℝ shows that (except for the absolute constants involved) one can't improve the concentration function e^−ct . As we shall see below, what looks like a slight change, logarithmic Sobolev inequality instead of Poincaré inequality, changes the behavior of the concentration function from e^−ct to e^{−ct 2} .

The next theorem is apparently due to Herbst.

Theorem 16

Let $\left(\Omega ,\mathcal{F},\mu ,d\right)$ be a probability metric space. Let $\mathcal{A}$ be the set of bounded Lipschitz functions on (Ω, d) and let $\mathcal{E}$ be defined by (54). Assume that $\left(\mathcal{A},\mathcal{E}\right)$ satisfies the logarithmic Sobolev inequality (52) then for all λ ∈ ℝ and every bounded ƒ with Lipschitz constant 1

(59) $\mathbb{E}{\text{e}}^{\lambda \left(ƒ-\mathbb{E}ƒ\right)}⩽{\text{e}}^{C{\lambda }^2/4}.$

In particular

(60) $\begin{array}{cc}P\left(\left|ƒ-\mathbb{E}ƒ\right|>t\right)⩽2{\text{e}}^{-t^2/C}& forallt>0.\end{array}$

Proof

Put $h\left(\lambda \right)=\mathbb{E}{\text{e}}^{\lambda ƒ}$ , then

(61) $\epsilon \left({\text{e}}^{\lambda ƒ\text{/2}}\right)=\mathbb{E}\lambda ƒ{\text{e}}^{\lambda ƒ}-\mathbb{E}{\text{e}}^{\lambda ƒ}\log \left(\mathbb{E}{\text{e}}^{\lambda ƒ}\right)=\lambda h^{\prime }\left(\lambda \right)-h\left(\lambda \right)\log \left(h\left(\lambda \right)\right).$

Also, from (56), we get,

(62) $\mathcal{E}\left({\text{e}}^{\lambda ƒ/2}\right)⩽\frac{{\lambda }^2}{4}\mathbb{E}{\text{e}}^{\lambda ƒ}=\frac{{\lambda }^2}{4}h\left(\lambda \right).$

Combining (61), (62) and (52) we get

$\lambda h^{\prime }\left(\lambda \right)-h\left(\lambda \right)\log \left(h\left(\lambda \right)\right)⩽\frac{{\lambda }^{2}C}{4}h\left(\lambda \right)$

or, putting k(λ) = λ⁻¹ log h(λ) (and, by continuity, $k\left(0\right)=\mathbb{E}ƒ$ ,

$\begin{array}{cc}{k}^{\prime }\left(\lambda \right)=\frac{1}{\lambda }\frac{h^{\prime }\left(\lambda \right)}{h\left(\lambda \right)}-\frac{1}{{\lambda }^2}\log h\left(\lambda \right)⩽\frac{C}{4},& \text{for}\text{all}\lambda \in ℝ.\end{array}$

It follows that k(λ) − k(0) ⩽ Cλ/4 and thus

$\mathbb{E}{\text{e}}^{\lambda \left(ƒ-\mathbb{E}ƒ\right)}={\text{e}}^{\lambda \left(k\left(\lambda \right)-k\left(0\right)\right)}⩽{\text{e}}^{C{\lambda }^2/4}.$

Remark 17

A simple limiting argument shows that here too the assumption that ƒ is bounded is superfluous.

Both Poincaré inequality and logarithmic Sobolev inequality carry over nicely to product spaces in the following sense: for i = 1, 2, …, n, let $\left({\Omega }_i,{\mathcal{F}}_i,{\mu }_i\right)$ be a probability space, ${\mathcal{A}}_i$ some set of measurable functions on Ω_i and ${\mathcal{E}}_i:{\mathcal{A}}_i\to {ℝ}^{+}$ some energy function. Put $\left(\Omega ,P\right)={\displaystyle \prod {}_{i=1}^n\left({\Omega }_i,{\mu }_i\right)}$ Given a function ƒ on Ω we denote by ƒ _i the same function considered as a function of the i-th variable only, keeping all other variables fixed. Define $\mathcal{E}\left(ƒ\right)={\mathbb{E}}_P{\displaystyle \sum {}_{i=1}^n{\mathcal{E}}_i\left({ƒ}_i\right)}$ Let $\mathcal{A}$ denote the set of all functions ƒ such that (for all x ₁, …, x_n ) and for all i, ƒ _i is in ${\mathcal{A}}_i$ .

One can prove that

(63) $\begin{array}{ccc}{\sigma }^2\left(ƒ\right)⩽{\mathbb{E}}_p{\displaystyle \sum _{i=1}^n{\sigma }^2\left({ƒ}_i\right)}& \text{and}& \epsilon \left(ƒ\right)⩽{\mathbb{E}}_p{\displaystyle \sum _{i=1}^n\epsilon \left({ƒ}_i\right)},\end{array}$

from which the following proposition easily follows.

Proposition 18

Assume $\left({\mathcal{A}}_i,{\mathcal{E}}_i\right)$ , i = 1, …, n, all satisfy Poincaré inequality (resp. logarithmic Sobolev inequality) with a common constant, C. Then $\left(\mathcal{A},\mathcal{E}\right)$ satisfies Poincaré inequality (resp. logarithmic Sobolev inequality) with the same constant, C.

Example 19

The symmetric exponential measure on ℝ, i.e., the measure with density $\frac{1}{2}{\text{e}}^{-\left|t\right|}$ , satisfies Poincaré inequality with constant 4. Consequently, the same is true for the measure on ℝ ⁿ which is the n-fold product of this measure.

The canonical Gaussian measure on ℝ and thus on ℝ ⁿ satisfies logarithmic Sobolev inequality with constant 2.

The proof of both statements can be found in [33]. The second one is due to Gross and, in view of Theorem 16, implies the concentration inequality for γ_n , the Gaussian measure on ℝ ⁿ : if ƒ : ℝ ⁿ → ℝ is Lipschitz with constant one with respect to the Euclidean metric then

${\gamma }_n\left(\left|ƒ-{\displaystyle \int ƒ\text{d}{\gamma }_n}\right|>t\right)⩽C{\text{e}}^{-c{t}^2}.$

From this it is not hard to get the concentration inequality for S ⁿ⁻¹. One uses Lemma 22 below.

We would also like to state a theorem first proved by Talagrand [61] which "interpolates" between the last two theorems. See [8] and [33] for a relatively simple proof along the lines of the proofs of the last two theorems. We state it only for a specific probability measure P on ℝ ⁿ , the product of the measures with density $\frac{1}{2}{\text{e}}^{-\left|t\right|}$ on ℝ. See [33] for generalizations.

Theorem 20

Let ƒ : ℝ ⁿ → ℝ be a function satisfying

(64) $\begin{array}{ccc}\left|ƒ\left(x\right)-ƒ\left(y\right)\right|⩽\alpha {\Vert x-y\Vert }_2& and& \left|ƒ\left(x\right)-ƒ\left(y\right)\right|⩽\beta {\Vert x-y\Vert }_1.\end{array}$

Then, with the probability introduced above,

(65) $P\left(\left|ƒ\left(x\right)-\mathbb{E}ƒ\right|>r\right)⩽C\exp \left(-c\min \left(r/\beta ,r^2/{\alpha }^2\right)\right)$

for some absolute positive constants C, c and all r > 0.

Remark 21

Although it deals with a different probability measure, Theorem 20 also implies the concentration inequality for the Gaussian measure on ℝ ⁿ (and thus, via Lemma 22 below, also for the Haar measure on S ⁿ⁻¹). This follows from a simple transference of the Gaussian measure to the product of the symmetrized exponential measure discussed above. Thus, Theorem 20 can be considered as a strengthening of these inequalities. We refer to [61] and [33] for that and further discussion.

Although the methods in this and the previous section are specialized to product measures, there is a way to transfer such results to some other situations. In particular to the case of unit balls of ${\ell }_p^n$ spaces equipped with the normalized Lebesgue measure. The basic tool is the following simple result: consider the measure $\mu \left(A\right)=\frac{\left|\left\{tA;0⩽t⩽1\right\}\right|}{\left|\left\{x;{\Vert x\Vert }_p⩽1\right\}\right|}$ on the surface of the ${\ell }_p^n$ ball, $0<p<\infty$ . Consider also n independent random variables X ₁, X ₂, …, X_n each with density function $c_p{\text{e}}^{-{\left|t\right|}^p}$ , t ∈ ℝ. (Note that necessarily c_p = p/2Γ(1/p).)

Lemma 22

Put $S={\left({\displaystyle \sum {}_{i=1}^n{\left|X_i\right|}^p}\right)}^{1/p}$ . Then $\left(\frac{X_1}{S},\frac{X_2}{S},\dots ,\frac{X_n}{S}\right)$ induces the measure μ on $\partial B_p^n$ . Moreover, $\left(\frac{X_1}{S},\frac{X_2}{S},\dots ,\frac{X_n}{S}\right)$ is independent of S.

See [54] for a proof. This lemma is used there to compute the tail behavior of the ℓ_q norm on the ${\ell }_p^n$ ball. Recently ([55]) this result was strengthen, in the case p = 1, q = 2, to give a concentration inequality for general Lipschitz functions, with respect to the Euclidean metric, on the ${\ell }_1^n$ ball $B_1^n$ . The proof combines most of the results of this section and we shall not give it here.

Theorem 23

There exist positive constants C,c such that if $ƒ:\partial B_1^n\to ℝ$ satisfies |ƒ(x) − ƒ(y)| ⩽ ∥x − y∥₂ for all x, y, ∈ ∂ $B_1^n$ then, for all t > 0,

(66) $\mu \left(\left\{x;\left|ƒ\left(x\right)-\mathbb{E}ƒ\right|>t\right\}\right)⩽C\exp \left(-ctn\right).$

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Navier–Stokes Equations

In Studies in Mathematics and Its Applications, 1977

The set Ω

Let Ω be an open set of R ⁿ with boundary Γ. In general we shall need some kind of smoothness property for Ω. Sometimes we shall assume that Ω is smooth in the following sense:

(1.3) $\mathrm{The}\mathrm{boundary}\lceil \mathrm{is}\text{a}\left(n–1\right)–\mathrm{dimensional}\mathrm{manifold}\mathrm{of}\mathrm{class}{\ell }^r\left(r⩾1\mathrm{which}\mathrm{must}\mathrm{be}\mathrm{specified}\right)\mathrm{and}\Omega \mathrm{is}\mathrm{locally}\mathrm{located}\mathrm{on}\mathrm{one}\mathrm{side}\mathrm{of}\lceil .$

We will say that a set Ω satisfying (1.3) is of class C ^r . However this hypothesis is too strong for practical situations (such as a flow in a square) and all the main results will be proved under a weaker condition:

(1.4) $\mathrm{The}\mathrm{boundary}\mathrm{of}\Omega \mathrm{is}\mathrm{locally}\mathrm{Lipschitz}.$

This means that in a neighbourhood of any point x ∈ Γ, Γ admits a representation as a hypersurface y _n = θ(y ₁, …, y _n-1) where θ is a Lipschitz function, and (y ₁, …, y _n ) are rectangular coordinates in R ⁿ in a basis that may be different from the canonical basis e ₁, …, e _n .

Of course if Ω is of class C ¹, then Ω is locally Lipschitz.

Fig. 1.

It is useful for the sequel of this section to note that a set Ω satisfying (1.4) is "locally star-shaped". This means that each point x _j ∈ Γ, has an open neighbourhood O _j , such that $O_j^0=\Omega \cap O_j$ is star-shaped with respect to one of its points. According to (1.4) we may, moreover, suppose that the boundary of O _j is Lipschitz.

If Γ is bounded, it can be covered by a finite family of such sets O _j , j ∈ J; if Γ is not bounded, the family (O _j ) _{j ∈ J} can be chosen to be locally finite.

It will be assumed that Ω will always satisfy (1.4), unless we explicitly state that Ω is any open set in R ⁿ or that some other smoothness property is required.

LP and Sobolev Spaces

Let Ω be any open set in R ⁿ . We denote by L ^p (Ω), 1 < p < + ∞ (or L ^∞ (Ω)) the space of real functions defined on Ω with the p-th power absolutely integrable (or essentially bounded real functions) for the Lebesgue measure dx = dx ₁ … dx _n . This is a Banach space with the norm

(1.5) ${║u║}_{L^P\left(\Omega \right)}={\left({\int }_{\Omega }{\left|u\left(x\right)\right|}^p\text{dx}\right)}^{1/p}$

or

${║u║}_{L^{\infty }{}_{\Omega }}=\underset{\Omega }{\mathrm{ess}.\sup }\left|u\left(x\right)\right|).$

For p = 2, L ² (Ω) is a Hilbert space with the scalar product

(1.6) $\left(u,v\right)={\displaystyle {\int }_{\Omega }u\left(x\right)v\left(x\right)\text{dx}.}$

The Sobolev space W ^{m, p} (Ω) is the space of functions in L ^p (Ω) with derivatives of order less than or equal to m in L ^p (Ω) (m an integer, 1 ⩽ p ⩽ + ∞). This is a Banach space with the norm

(1.7) ${║u║}_{W^{m.p}{}_{\Omega }}={\left({\displaystyle \sum _{\left[j\right]<m}{║D^{i}u║}_{L^p\left(\Omega \right)}^p}\right)}^{1/p}$

When p = 2, W ^m,2(Ω) = H ^m (Ω) is a Hilbert space with the scalar product

(1.8) ${\left(\left(u,v\right)\right)}_{H^m\left(\Omega \right)}={\displaystyle \sum _{\left[j\right]⩽m}\left(D^{j}u,D^{j}v\right)}$

Let $D\left(\Omega \right)\left(\mathrm{or}D\left(\overline{\Omega }\right)\right)$ be the space of C ^∞ functions with compact support contained in $\Omega \left(\mathrm{or}\overline{\Omega }\right)$ . The closure of D(Ω) in W ^{m, p} (Ω) is denoted by $W_0^{m,p}\left(\Omega \right)\left(H_0^m\left(\Omega \right)\mathrm{when}p=2\right).$

We recall, when needed, the classical properties of these spaces such as the density or trace theorems (assuming regularity properties for Ω).

We shall often be concerned with n-dimensional vector functions with components in one of these spaces. We shall use the notation

$L^p\left(\Omega \right)={\left\{L^p\left(\Omega \right)\right\}}^n,W^{m,p}\left(\Omega \right)={\left\{W^{m,p}\left(\Omega \right)\right\}}^n$

$H^m\left(\Omega \right)={\left\{H^m\left(\Omega \right)\right\}}^n,D\left(\Omega \right)={\left\{D\left(\Omega \right)\right\}}^n,$

and we suppose that these product spaces are equipped with the usual product norm or an equivalent norm (except D(Ω) and $D\left(\overline{\Omega }\right)$ , which are not normed spaces).

The following spaces will appear very frequently

$L^2\left(\Omega \right),L^2\left(\Omega \right),H_0^1\left(\Omega \right),H_0^1\left(\Omega \right).$

The scalar product and the norm are denoted by (·, ·) and |·| on L ²(Ω) or L ²(Ω) (or [·, ·] and [·] on $H_0^1\left(\Omega \right)$ or $H_0^1\left(\Omega \right)$ )).

We recall that if Ω is bounded in some direction ⁽¹⁾ then the Poincaré inequality holds:

(1.9) ${║u║}_{L^2\left(\Omega \right)}⩽c\left(\Omega \right){║Du║}_{L^2\left(\Omega \right)},\forall u\in H_0^1\left(\Omega \right),$

where D is the derivative in that direction and c(Ω) is a constant depending only on Ω which is bounded by 2l, l = the diameter of Ω or the thickness of Ω in any direction. In this case the norm [·] on $H_0^1\left(\Omega \right)$ (or $H_0^1\left(\Omega \right)$ ) is equivalent to the norm:

(1.10) $║u║={\left({\displaystyle \sum _{i=1}^n{\left|D_{i}u\right|}^2}\right)}^{1/2}$

The space $H_0^1\left(\Omega \right)\left(or{H}_0^1\left(\Omega \right)\right)$ is also a Hilbert space with the associated scalar product

(1.11) $\left(\left(u,v\right)\right)={\displaystyle \sum _{i=1}^n\left(D_{i}u,D_{i}v\right).}$

This scalar product and this norm are denoted by ((·, ·)) and ||·|| on $H_0^1\left(\Omega \right)$ and $H_0^1\left(\Omega \right)$ (Ω bounded in some direction).

Let V be the space (without topology)

(1.12) $V=\left\{u\in D\left(\Omega \right):\mathrm{div}u=0\right\}.$

The closures of V in L ² (Ω) and in $H_0^1\left(\Omega \right)$ are two basic spaces in the study of the Navier-Stokes equations; we denote them by H and V. The results of this section will allow us to give a characterization of H and V.

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Poincaré Inequality and Spectral Gap

Feng-Yu Wang , in Functional Inequalities, Markov Semigroups and Spectral Theory, 2005

Proof.

(1)

LetP^b _t be the C ₀ Markov semigroup generated by (L_b , D ₁(L_b )). For any $f\in C_0^{\infty }\left(M\right)$ andt> 0, one has $P_t^{b}f-\mu \left(P_t^{b}f\right)\in D_1\left(L_b\right)$ since 1 ∈ D ₁(L_b ) (becauseP^b _t is Markovian). ByLemma 1.4.14, there is uniformly bounded $\left\{f_n\right\}\subset C_0^{\infty }\left(M\right)$ such that $f_n\to P_t^{b}f-\mu \left(P_t^{b}f\right)$ inL ²(μ) asn→ ∞, and (1.4.20) holds for $P_t^{b}f-\mu \left(P_t^{b}f\right)$ in place off . Then it follows from the Poincaré inequality that

(1.4.22) $\begin{array}{ll}\frac{\text{d}}{\text{d}t}\mu \left({\left[P_t^{b}f-\mu \left(P_t^{b}f\right)\right]}^2\right)& =2\mu \left(\left(P_t^{b}f-\mu \left(P_t^{b}f\right)\right)\left(L_b{P}_t^{b}f-\mu \left(L_b{P}_t^{b}f\right)\right)\right)\\ & =2\mu \left(\left(P_t^{b}f-\mu \left(P_t^{b}f\right)\right)L_b\left(P_t^{b}f-\mu \left(P_t^{b}f\right)\right)\right)\\ & ⩽\alpha \mu \left({\left[P_t^{b}f-\mu \left(P_t^{b}f\right)\right]}^2\right)-2\underset{n\to \infty }{\overline{\lim }}E\left(f_n,f_n\right)\\ & ⩽\alpha \mu \left({\left[P_t^{b}f-\mu \left(P_t^{b}f\right)\right]}^2\right)-2\underset{n\to \infty }{\overline{\lim }}[\mu (f_n^2)-\mu {\left(f_n\right)}^2]\\ & =-\left(2\lambda -\alpha \right)\mu \left({\left[P_t^{b}f-\mu \left(P_t^{b}f\right)\right]}^2\right).\end{array}$

Therefore, ${\Vert P_t^{b}f-\mu \left(P_t^{b}f\right)\Vert }_2^2⩽e^{-\left(2\lambda -\alpha \right)t}{\Vert f-\mu \left(f\right)\Vert }_2^2.$ This estimate is true for allf∈L ²(μ) since $C_0^{\infty }\left(M\right)$ is dense inL ²(μ).

(2)

According toTheorem 3.1.7below,P^b _t (resp. the resolvent) is compact inL^p (μ) provided, it isL^p -uniformly integrable and has integrable density with respect to μ. So, it suffices to prove the L^p -uniform integrability ofP^b _t (hence of the resolvent). Moreover, byProposition 1.4.16below, we only consider the case wherep∈ (1, 2]. To this end, we make use of [93], Theorem 4.3 which says that (σ_ess(L) = ∅ implies the following super-Poincaré inequality (see [§3.2] below for details)

(1.4.23) $\mu \left(f^2\right)⩽rE\left(f,f\right)+\beta \left(r\right)\mu {\left(\left|f\right|\right)}^2,r>0,f\in D\left(E\right).$

Let $f\in C_0^{\infty }\left(M\right)$ be nonnegative. We have $P_t^{b}f\in D_1\left(L_b\right).$ ByLemma 1.4.14(2), there is a nonnegative and uniformly bounded sequence $\left\{f_n\right\}\subset C_0^{\infty }\left(M\right)$ such that $f_n\to {\left(P_t^{b}f\right)}^{p/2}$ inL ¹(μ) asn→ ∞ and (1.4.21) holds forP^b _t f in place off. Then it follows from (1.4.22) that (recall thatp/2 ⩽ 1)

Moreover, (1.4.21) implies that $\mu \left({\left(P_{t}f\right)}^p\right)⩽e^{\alpha t}\mu \left(f^p\right).$ for allp> 1. Thus, letting p→ 1 we obtain $\mu \left(P_{t}f\right)⩽e^{\alpha t}\mu \left(f\right).$ Combining this with (1.4.23) we arrive at

(1.4.24) $\mu \left({\left[P_t^{b}f\right]}^p\right)⩽e^{-\left[4\left(p-1\right)/\left(pr\right)-\alpha \right]t}\mu \left(|f{|}^p\right)+C\left(t,r,p,\alpha \right)\mu {\left(\left|f\right|\right)}^p,0<r<\frac{4\left(p-1\right)}{p\alpha },$

where For $C\left(t,r,p,\alpha \right):=\frac{4\left(p-1\right)\beta \left(r\right)e^{p\alpha t}}{4\left(p-1\right)-p\alpha r}.$ For f ∈ L ^p(μ) we have a sequence of nonnegative functions $\left\{f_n\right\}\subset C_0^{\infty }\left(M\right)$ such thatf_n → |f| in $L^p\left(\mu \right)$ (hence inL ¹(μ)) asn→ ∞, then $\mu \left(f_n^p\right)\to \mu \left({\left|f\right|}^p\right)$ and $\mu \left(f_n\right)\to \mu \left(\left|f\right|\right)$ asn→ ∞. Moreover, sinceP^b _t is bounded inL^P (μ), one has $\mu \left(|P_t^b{f}_n{|}^p\right)\to \mu \left({\left(P_t^b\left|f\right|\right)}^p\right)\left(⩾\mu \left(|P_t^{b}f{|}^p\right)\right)$ asn→ ∞. Therefore, (1.4.24) holds for allf∈L^p (μ). Now, forf⩾ 0 with μ(f^p ) = 1, let

$g_s:={\left(P_t^{b}f-s\right)}^{+},h_s:=P_t^b{\left(f-s\right)}^{+},s⩾0.$

One hash_s ⩾g_s sinceP^b _t is sub-Markovian. By(1.4.24),

$\begin{array}{ll}\mu \left(h_s^p\right)& ⩽e^{-\left[4\left(p-1\right)/\left(pr\right)-\alpha \right]t}+C\left(t,r,p,\alpha \right)\mu {\left({\left(f-s\right)}^{+}\right)}^p\\ & ⩽e^{-\left[4\left(p-1\right)/\left(pr\right)-\alpha \right]t}+C\left(t,r,p,\alpha \right)s^{-p\left(p-1\right)}=:I_t\left(r,s\right),\\ & s>0,0<r,\frac{4\left(p-1\right)}{p\alpha },\end{array}$

This implies that

$\begin{array}{ll}\underset{s\to \infty }{\lim }\underset{\left|\right|f|{|}_p⩽1}{\sup }\mu \left({\left(\left|p_t^{p}f\right|-s\right)}^{+p}\right)& =\underset{s\to \infty }{\lim }\underset{f⩾0,\mu \left(f^p\right)⩽1}{\sup }\mu \left({\left(p_t^{b}f-s\right)}^{+p}\right)\\ & ⩽\underset{r\to \infty }{\lim }\underset{s\to \infty }{\lim }{I}_t\left(r,s\right)=0.\end{array}$

Therefore,P^b _t is uniformly integrable inL^p (μ).

Finally, byTheorem 0.3.9(2), the compactness ofP^b _t (or of the resolvent) is equivalent to ${\sigma }_{ess}^p\left(L_b\right)=\varnothing .$

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