Ngô Quốc Anh

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] or [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.

2 Comments »

  1. Hi, well done as usual.
    Chang’s book is a very nice and more or less complete work on this subject. I love it!
    Quickly reading, I have noted two misprints: 1) on the third embedding you have to put a cross, 2) implies that in equality turns in implies that inequality.

    Comment by fab — February 10, 2012 @ 17:59

    • Thanks Fab. You’re welcome. I have corrects those misprints as suggested. Thanks for your interest in my post(s).

      Comment by Ngô Quốc Anh — February 10, 2012 @ 18:05


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