Ngô Quốc Anh

October 17, 2009

The Brezis-Lieb lemma and several applications

Filed under: Các Bài Tập Nhỏ, Giải Tích 6 (MA5205), Nghiên Cứu Khoa Học — Tags: — Ngô Quốc Anh @ 14:34

What we did in this topic was just an L^p version of the Brezis-Lieb lemma. In this topic, we will discuss the generalization of this lemma.

Roughly speaking, what we are going to prove is the following:  If j : \mathbb C \to \mathbb C is a continuous function such that j(0) = 0, then, when f_n \to f a.e. and

\displaystyle\int |j(f_n(x))| d\mu(x) \leq C < \infty

we claim that

\displaystyle\lim\limits_{n \to \infty} \int \left[ j(f_n) - j(f_n - f)\right] = \int j(f)

under suitable conditions on j and/or \{f_n\}.

To be exact, in addition let j satisfy the following hypothesis:

For every sufficiently small \varepsilon>0, there exists two continuous, nonnegative functions \varphi_\varepsilon and \psi_\varepsilon such that

\displaystyle |j(a+b)-j(a)| \leq \varepsilon \varphi_\varepsilon(a) + \psi_\varepsilon(b)

for all a, b \in \mathbb C.

Theorem. Let j satisfy the above hypothesis and let f_n = f+g_n be a sequence of measurable functions from \Omega to \mathbb C such that

  1. g_n \to 0 a.e.
  2. j(f) \in L^1.
  3. \displaystyle\int \varphi_\varepsilon(g_n(x))d\mu(x) \leq C < \infty for some constant C, independent of \varepsilon and n.
  4. \displaystyle\int \psi_\varepsilon(f(x)) d\mu(x) < \infty for all \varepsilon >0.

Then, as n \to \infty,

\displaystyle\lim\limits_{n \to \infty} \int \left| j(f+g_n) - j(g_n) - j(f) \right| d\mu =0.

Proof. Fix \varepsilon >0 and let

\displaystyle W_{\varepsilon, n} (x) = \Big[ \big|j(f_n(x)) -j(g_n(x)) - j(f(x))\big| - \varepsilon \varphi_\varepsilon (g_n(x))\Big]_+,

where [a]_+ = \max\{a,0\}. As n \to \infty, W_{\varepsilon, n} (x) \to 0 a.e. On the other hand,

\displaystyle \big| j(f_n) - f(g_n) - j(f)\big| \leq |j(f_n) - j(g_n)| + |j(f)| \leq \varepsilon \varphi_\varepsilon(g_n) + \psi_\varepsilon(f) + |j(f)|.

Therefore, W_{\varepsilon, n} \leq \psi_\varepsilon(f) + |j(f)| \in L^1. By the Lebesgue Dominated Convergence theorem, \displaystyle\int W_{\varepsilon, n} d\mu \to 0 as n \to \infty. However,

\displaystyle |j(f_n) - j(g_n) - j(f)| \leq W_{\varepsilon, n} +\varepsilon \varphi_\varepsilon(g_n)

and thus

\displaystyle I_n \equiv \int \big| j(f_n) - j(g_n) - j(f) \big| d\mu\leq \int \big[ W_{\varepsilon, n} + \varepsilon \varphi_\varepsilon(g_n)\big] d\mu .

Consequently, \limsup_{n \to \infty} I_n \leq \varepsilon C. Now let \varepsilon \to 0.

Applications.

  • The simplest example is when we choose j(x)=|x|^p where 0< p<\infty. In this situation, one has

\displaystyle \int \Big(|f_n|^p - |f_n - f|^p - |f|^p \Big) d\mu \to 0.

  • We now assume u_n \rightharpoonup u in W^{1, 2}. As a consequence and up to a subsequence, u_n \to u in L^\alpha for every 1<\alpha<2^\star := \frac{2n}{n-2} and u_n \to u a.e. Therefore, for a fixed q \in (2, 2^\star), the fact that u_n \to u in L^q implies, by the Brezis-Lieb lemma, that

    \displaystyle u_n^{q-1} \to u^{q-1} in L^\frac{q}{q-1}.

    This is because \{u_n^{q-1}\}_n \subset L^\frac{q}{q-1} is bounded, u_n^{q-1} \to u^{q-1} a.e. and

\displaystyle\mathop {\lim }\limits_{n \to \infty } \int {\Big( {\underbrace {{{\left| {u_n^{q - 1}} \right|}^{\frac{q}{{q - 1}}}}}_{{{\left| {{u_n}} \right|}^q}} - {{\left| {u_n^{q - 1} - {u^{q - 1}}} \right|}^{\frac{q}{{q - 1}}}}} \Big)d\mu }=\int {\underbrace {{{\left| {{u^{q - 1}}} \right|}^{\frac{q}{{q - 1}}}}}_{{{\left| u \right|}^q}}d\mu }.

    The fact that u_n \to u strongly in L^p implies that \lim_{n\to \infty} \int |u_n|^p d\mu = \int |u|^p d\mu. Therefore,

\displaystyle \mathop {\lim }\limits_{x \to \infty }\int {{{\left| {u_n^{q - 1} - {u^{q - 1}}} \right|}^{\frac{q}{{q - 1}}}}d\mu } = 0.

    As a consequence, one has the following result

    \displaystyle \mathop {\lim }\limits_{x \to \infty } \int {\left( {u_n^{q - 1} - {u^{q - 1}}} \right)\left( {{u_n} - u} \right)d\mu } = 0.

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