Ngô Quốc Anh

August 13, 2010

An application of the (Moser-)Trudinger inequality to the mean field equations

Filed under: PDEs, Riemannian geometry — Tags: , , — Ngô Quốc Anh @ 5:05

Let (M,g) be a compact Riemannian surface with the volume |M|. The simplest form of the mean field equation studied in the contexts of the prescribing Gaussian curvature, statistical mechanics of many vortex points in the perfect fluid and self-dual gauss theories is given by

\displaystyle - {\Delta _g}u = \lambda \left( {\frac{{{e^u}}}{{\int_M {{e^u}d{v_g}} }} - \frac{1}{{|M|}}} \right), \quad \text{ on } M

with

\displaystyle\int_M {ud{v_g}} = 0

where \lambda is a real number.

The mean field equation has a variational structure, and u is a solution if and only if it is a critical point of

\displaystyle {J_\lambda }(v) = \frac{1}{2}\int_M {{{\left| {\nabla v} \right|}^2}d{v_g}} - \lambda \log \int_M {{e^v}d{v_g}}

defined for v \in H^1(M) with

\displaystyle\int_M {vd{v_g}} = 0.

It is worth noticing from this entry that so far our Moser-Trudinger’s inequality is just for \mathbb S^2

Theorem (Moser-Trudinger’s inequality for \mathbb S^2). There are constants \eta>0 and c=c(g)>0 such that for each p \geqslant 2

\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}

for all u \in W^{1,2}(\mathbb S^2).

(more…)

July 8, 2010

The Moser-Trudinger inequality

Filed under: PDEs — Tags: , , , — Ngô Quốc Anh @ 18:26

Followed by an entry where the Trudinger inequality had been discussed we now consider an important variant of it known as the Moser-Trudinger inequality.

Let us remind the Trudinger inequality

Theorem (Trudinger). Let \Omega \subset \mathbb R^n be a bounded domain and u \in W_0^{1, n}(\Omega) with

\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1.

Then there exist universal constants \beta>0, C_1>0 such that

\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega |.

The Trudinger inequality has lots of application. For application to the prescribed Gauss curvature equation, one requires a particular value for the best constant \beta_0. In connection with his work on the Gauss curvature equation, J. Moser [here] sharpended the above result of Trungdier as follows

Theorem (Moser). Let \Omega \subset \mathbb R^n be a bounded domain and u \in W_0^{1, n}(\Omega) with

\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1.

Then there exist sharp constants \beta_0=\beta(n)>0, C_1=C_1(n)>0 given by

\displaystyle \beta_0=n\omega_{n-1}^{\frac{1}{n}-1}

such that

\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega|, \quad \forall \beta \leqslant \beta_0.

The constant \beta_0 is sharp in the sense that for all \beta>\beta_0 there is a sequence of functions u_k \in W_0^{1,n}(\Omega) satisfying

\displaystyle\int_\Omega {{{\left| {\nabla u_k} \right|}^n}dx} \leqslant 1

but the integral

\displaystyle\int_\Omega {\exp (\beta {|u_k|^\frac{n}{n-1}})dx}

grow without bound.

For general compact closed manifold (M,g) the constant on the right hand side of the Moser-Trudinger inequality depends on the metric g. Working on a sphere (\mathbb S^2,g_c) with a canonical metric allows us to control the constants.

Theorem (Moser). There is a universal constant C_1>0 such that for all u \in W^{1,2}(\mathbb S^2) with

\displaystyle\int_{\mathbb S^2} {{{\left| {\nabla u} \right|}^n}dv_{g_c}} \leqslant 1

and

\displaystyle\int_{\mathbb S^2} u dv_{g_c}=0

we have

\displaystyle\int_{\mathbb S^2} \exp(4\pi u^2) \leqslant {C_1}.

Observe that

\displaystyle 4\pi = \int_{{\mathbb{S}^2}} {d{v_{{g_c}}}} <\int_{{\mathbb{S}^2}} {{e^{4\pi {u^2}}}d{v_{{g_c}}}} \leqslant {C_1}.

In the same way as we introduce in the entry concerning the Trudinger inequality one can show

Corollary. For

\displaystyle C_2 := \log C_1 +\log\frac{1}{4\pi}

one has

\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}

for all u \in W^{1,2}(\mathbb S^2).

Obviously, C_2 >0 since C_1 >4\pi. It turns out to determine the best constant C_2. This had been done by Onofri known as the Onofri inequality [here].

Theorem (Onofri).Let u \in W^{1,2}(\mathbb S^2) then we have

\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u

with the equality iff

\Delta u +e^{2u}=1.

The proof of the Onofri inequality relies on a result due to Aubin

Theorem (Aubin). For all \varepsilon>0 there exists a constant C_\varepsilon such that

\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\left( {\frac{1}{2} + \varepsilon } \right)\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_\varepsilon }

for any u belonging to the following class

\displaystyle S = \left\{ {u \in {W^{1,2}}({\mathbb{S}^2}):\int_{{\mathbb{S}^2}} {{e^{2u}}{x_j}d{v_{{g_c}}}} = 0,j = \overline {1,3} } \right\}.

Source: S-Y.A. Chang, Non-linear elliptic equations in conformal geometry, EMS, 2004.

July 1, 2010

The Trudinger inequality

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 10:53

In 1967, Neil S. Trudinger announced a result in J. Math. Mech. (now known as Indiana Univ. Math. J.) which can be seen as a limiting case of the Sobolev inequality  [here].

It is well-known from the Sobolev embedding theorem that

W_0^{\alpha, q}(\Omega) \hookrightarrow L^p(\Omega)

for

\displaystyle \frac{1}{p}=\frac{1}{q}-\frac{\alpha}{n}, \quad q\alpha<n.

The case q\alpha=n is commonly referred to the limitting case. If \alpha=1, n=2 and q<2 we obtain

W_0^{1, q}(\Omega) \hookrightarrow L^p(\Omega).

In general one cannot take the limits q \to 2 and p \to \infty, i.e.

W_0^{1, 2}(\Omega) \not\hookrightarrow L^\infty(\Omega).

A counter-example is given by

\displaystyle \log\left(1+\log\frac{1}{|x|}\right)

on the unit ball in \mathbb R^2. Instead, Trudinger proved exponential L^2-integrability in the following sense

Theorem (Trudinger). Let \Omega \subset \mathbb R^2 be a bounded domain and u \in W_0^{1, 2}(\Omega) with

\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^2}dx} \leqslant 1.

Then there exist universal constants \beta>0, C_1>0 such that

\displaystyle\int_\Omega {\exp (\beta {u^2})dx} \leqslant {C_1}|\Omega |.

We write

\displaystyle W_0^{1,2}(\Omega ) \hookrightarrow {e^{{L^2}}}(\Omega ).

Observe that the assumption

\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^2}dx} \leqslant 1

implies that inequality

\displaystyle\int_\Omega {\exp (\beta {u^2})dx} \leqslant {C_1}|\Omega |

is equivalent to

\displaystyle {\left\| u \right\|_p} \leqslant {C_2}\sqrt p {\left| \Omega \right|^{\frac{1}{p}}},\forall p \geqslant 2

for some universal constant C_2.

The way to see it is the following

  • For all k \in \mathbb N one has

    \displaystyle\frac{1}{{k!}}\int_\Omega {{{(\beta {u^2})}^k}dx} \leqslant {C_1}|\Omega |

    hence

    \displaystyle {\left( {\int_\Omega {{u^{2k}}dx} } \right)^{\frac{1}{{2k}}}} \leqslant {\left( {\frac{{k!}}{{{\beta ^k}}}{C_1}|\Omega |} \right)^{\frac{1}{{2k}}}} \leqslant {\widetilde C_2}\sqrt {2k} {\left| \Omega \right|^{\frac{1}{{2k}}}}.

    This proves the claim for p=2k, k \in \mathbb N. For odd p a simple use of the Holder inequality gives

    \displaystyle {\left( {\int_\Omega {{{\left| u \right|}^p}dx} } \right)^{\frac{1}{p}}} \leqslant {\left( {\int_\Omega {{u^{2p}}dx} } \right)^{\frac{1}{{2p}}}} \leqslant {\overline C _2}\sqrt p {\left| \Omega \right|^{\frac{1}{p}}}.

  • Now obviously

    \displaystyle\begin{gathered} \int_\Omega {\exp (\beta {u^2})dx} = \int_\Omega {\sum\limits_{k = 0}^\infty {\frac{1}{{k!}}{{(\beta {{\left| u \right|}^2})}^k}} dx} \hfill \\\qquad= \int_\Omega {\sum\limits_{k = 0}^\infty {\frac{{{\beta ^k}}}{{k!}}\left\| u \right\|_{2k}^{2k}} dx} \hfill \\ \qquad\leqslant \sum\limits_{k = 0}^\infty {\frac{{{\beta ^k}}}{{k!}}{{\left[ {{C_2}\sqrt {2k} {{\left| \Omega \right|}^{\frac{1}{{2k}}}}} \right]}^{2k}}} \hfill \\ \qquad= \sum\limits_{k = 0}^\infty {\frac{1}{{k!}}{{\left[ {2\beta C_2^2k} \right]}^k}\left| \Omega \right|} \hfill \\ \qquad\leqslant {C_1}|\Omega |. \hfill \\ \end{gathered}

    So if one chooses \beta so small that

    2\beta C_2^2<\frac{1}{e}

    which according to the Stirling formula implies that the infinie series

    \displaystyle\sum\limits_{k = 0}^\infty {\frac{1}{{k!}}{{\left[ {2\beta C_2^2k} \right]}^k}}

    is finite.

Proof. It suffices to prove

\displaystyle {\left\| u \right\|_p} \leqslant {C_2}\sqrt p {\left| \Omega \right|^{\frac{1}{p}}}, \quad \forall p \geqslant 2

for some constant C_2. By symmetric rearrangement

\displaystyle\int_\Omega {\exp (\beta {u^2})dx} \leqslant \int_{{B_1}(0)} {\exp (\beta {{({u^ \star })}^2})dx}

and

\displaystyle\int_{{B_1}(0)} {{{\left| {\nabla {u^ \star }} \right|}^2}dx} \leqslant \int_\Omega {{{\left| {\nabla u} \right|}^2}dx}

and scaling we may take \Omega=B_1(0). Furthermore, we may assume u \in C^\infty.

We can represent u as

\displaystyle u(y) = - \frac{1}{{2\pi }}\int_{{B_1}(0)} {\Delta u(y)\log |x - y|dy}

which after integration by parts leads to the estimate

\displaystyle\begin{gathered} |u(x)| \leqslant C\int_{{B_1}(0)} {\frac{{|\nabla u(y)|}}{{|x - y|}}dy} \hfill \\ \qquad\leqslant C{\left( {\int_{{B_1}(0)} {\frac{{|\nabla u(y){|^2}}}{{|x - y{|^a}}}dy} } \right)^{\frac{1}{p}}}{\left( {\int_{{B_1}(0)} {\frac{1}{{|x - y{|^a}}}dy} } \right)^{\frac{1}{2}}}{\left( {\int_{{B_1}(0)} {|\nabla u(y){|^2}dy} } \right)^{\frac{1}{2} - \frac{1}{p}}} \hfill \\ \end{gathered}

using the Holder inequality.

Now

\displaystyle {\int_{{B_1}(0)} {\frac{1}{{|x - y{|^a}}}dy} }

is finite since for any x,y \in B_1(0) one has B_1(0) \subset B_2(x) and then

\displaystyle\int_{{B_1}(0)} {\frac{1}{{|x - y{|^a}}}dy} \leqslant \int_{{B_2}(x)} {\frac{1}{{|x - y{|^a}}}dy} \leqslant C(p + 2).

Consequently

\displaystyle\int_{{B_1}(0)} {{{\left| u \right|}^p}dx} \leqslant C\left\| {\nabla u} \right\|_2^{p - 2}{(p + 2)^{\frac{p}{2}}}\int_{{B_1}(0)} {\left( {\int_{{B_1}(0)} {\frac{{|\nabla u(y){|^2}}}{{|x - y{|^a}}}dy} } \right)dx} \leqslant {(p + 2)^{\frac{p}{2} + 1}}\left\| {\nabla u} \right\|_2^p

where we uses the Fubini theorem to obtain the last inequality. By the assumption we have

\displaystyle {\left\| u \right\|_p} \leqslant {C_2}\sqrt p {\left| \Omega \right|^{\frac{1}{p}}},\forall p \geqslant 2

for some universal constant C_2>0.

The subscript 0 in the space of functions u can be dropped

Corollary. Let (M^2,g) be a compact and closed manifold. Then there exist constants \beta=\beta(g)>0 and C=C(g)>0 such that for all u \in W^{1,2}(M) with

\displaystyle\int_M {ud{v_g}} = 0, \quad \int_M {|\nabla u{|^2}d{v_g}} \leqslant 1

one has

\displaystyle\int_M {\exp (\beta {u^2})d{v_g}} \leqslant C{\rm vol}(M,g).

The proof replies on making use the unity of partition. Assumption \int_M {ud{v_g}} = 0 allows us to use the Poincare inequality.

There is one more corollary which is frequently used in the literature.

Corollary. For a compact and closed manifold (M^2, g) there are constants \eta>0 and c=c(g)>0 such that for each p \geqslant 2

\displaystyle\int_M {\exp (p(u - \overline u ))d{v_g}} \leqslant c\exp \left[ {\eta \frac{{{p^2}}}{4}\left\| {\nabla u} \right\|_2^2} \right]

for all u \in W^{1,2}(M) where

\displaystyle\overline u = \frac{1}{{\rm vol}(M,g)}\int_M {ud{v_g}} .

The proof relies on an elementary inequality, for \|\nabla u\|_2 \ne 0,

\displaystyle p(u - \overline u ) \leqslant \beta \frac{{{{(u - \overline u )}^2}}}{{\left\| {\nabla u} \right\|_2^2}} + \frac{1}{\beta }\frac{{{p^2}}}{4}\left\| {\nabla u} \right\|_2^2

where \beta>0 is the constant appeared in the previous corollary.

It is worth noticing that the above inequality can be rewritten as

\displaystyle\int_M {\exp (pu)d{v_g}} \leqslant c\exp \left[ {\eta \frac{{{p^2}}}{4}\left\| {\nabla u} \right\|_2^2} + \overline u\right].

Let us consider a simple application of the Trudinger inequality.

Example.  If {u_i} \rightharpoonup u in W^{1,2}(M) as i \to \infty and

\displaystyle\int_M {|\nabla u{|^2}d{v_g}} \leqslant c, \quad \int_M {{u_i}d{v_g}} = 0,\quad \int_M {|\nabla {u_i}{|^2}d{v_g}} \leqslant c

then for each f \in L^\infty(M)

\displaystyle\int_M {f{e^{p{u_i}}}d{v_g}} \to \int_M {f{e^{pu}}d{v_g}}

as i \to \infty.

Proof. Using the simple estimate |e^x-1| \leqslant |x|e^{|x|} we can write

\displaystyle\begin{gathered} \left| {\int_M {\left[ {f{e^{p{u_i}}} - f{e^{pu}}} \right]d{v_g}} } \right| \leqslant {\left\| f \right\|_\infty }\int_M {\left| {{e^{p{u_i}}} - {e^{pu}}} \right|d{v_g}} \hfill \\ \qquad= {\left\| f \right\|_\infty }\int_M {{e^{pu}}\left| {{e^{p({u_i} - u)}} - 1} \right|d{v_g}} \hfill \\ \qquad\leqslant {\left\| f \right\|_\infty }\int_M {{e^{pu}}p\left| {{u_i} - u} \right|{e^{p|{u_i} - u|}}d{v_g}} \hfill \\ \qquad\leqslant {\left\| f \right\|_\infty }{\left( {\int_M {{e^{4pu}}d{v_g}} } \right)^{\frac{1}{4}}}{\left( {\int_M {{{\left| {{u_i} - u} \right|}^2}d{v_g}} } \right)^{\frac{1}{2}}}{\left( {\int_M {{e^{4p|{u_i} - u|}}d{v_g}} } \right)^{\frac{1}{4}}} \hfill \\ \end{gathered}

using the Holder inequality.

There are some improvement of the Trudinger inequality in the literature. The most important one is the so-called Moser-Trudinger inequality which is a sharp version of a limiting case of the Sobolev inequality. This work had been done by Jürgen Moser around 1970. We will consider this inequality later. We end this entry by stating a version fo the n-dimension spaces, see also Chang and Yang Comm. Pure Appl. Math. 2003 [here].

Theorem (Trudinger). Let \Omega \subset \mathbb R^n be a bounded domain and u \in W_0^{1, n}(\Omega) with

\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1.

Then there exist universal constants \beta>0, C_1>0 such that

\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega |.

Source: S-Y.A. Chang, Non-linear elliptic equations in conformal geometry, EMS, 2004.

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