Ngô Quốc Anh

May 13, 2010

Concentration-Compactness Principle: The loss of mass at infinity in the subcritical case

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 16:18

One of the main problems encountered in solving variational problems is to show the convergence of minimizing sequence (the direct method). In many problems, it is relatively easy to show the boundedness of a minimizing sequence in an appropriate Sobolev space.

Th aim of this section is to derive the concentration-compactness at infinity (just for subcritical case). Roughly speaking, for unbounded domains due to the lack of compact embedding in the Sobolev embedding theorem, the loss of mass of a weakly convergent sequence may occur. In fact, this can only occur at infinity.

Let us firstly consider the case \Omega \subset \mathbb R^n is a bounded domain. Let \{u_m\} be a sequence in W_0^{1,2}(\Omega) such that

u_m \rightharpoonup u

weakly in W_0^{1,2}(\Omega) (i.e. u \in W_0^{1,2}(\Omega) and f(u_m) \to f(u) for any f \in W_0^{-1,2}(\Omega)). Assume 2<p<2^\star. The Sobolev embedding tells us that

W_0^{1,2}(\Omega) \hookrightarrow L^p(\Omega)

is compact. Thus

u_m \to u

strongly in L^p(\Omega) in the sense that

\displaystyle\mathop {\lim }\limits_{m \to \infty } \int_\Omega {{{\left| {{u_m} - u} \right|}^p}dx} = 0.

It now follows from the Brezis-Lieb lemma or the use of the Minkowski inequality

\displaystyle {\left( {\int_\Omega {{{\left| {{u_m}} \right|}^p}dx} } \right)^{\frac{1}{p}}} \leqslant {\left( {\int_\Omega {{{\left| {{u_m} - u} \right|}^p}dx} } \right)^{\frac{1}{p}}} + {\left( {\int_\Omega {{{\left| u \right|}^p}dx} } \right)^{\frac{1}{p}}}

and

\displaystyle {\left( {\int_\Omega {{{\left| u \right|}^p}dx} } \right)^{\frac{1}{p}}} \leqslant {\left( {\int_\Omega {{{\left| {u - {u_m}} \right|}^p}dx} } \right)^{\frac{1}{p}}} + {\left( {\int_\Omega {{{\left| {{u_m}} \right|}^p}dx} } \right)^{\frac{1}{p}}}

that

\displaystyle \mathop {\lim }\limits_{m \to \infty } \int_\Omega | {u_m}{|^p}dx = \int_\Omega | u{|^p}dx

or

\displaystyle\mathop {\lim }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| < R} \right\}} | {u_m}{|^p}dx = \int\limits_{\Omega \cap \left\{ {|x| > R} \right\}} | u{|^p}dx.

However, when \Omega is unbounded, the situation is quite different. Actually, we are going to prove the following

Theorem. Let \Omega \in \mathbb R^n be an unbounded domain and 2<p<2^\star. Let \{u_m\} be a sequence in W_0^{1,2}(\Omega) such that u_m \rightharpoonup u in W_0^{1,2}(\Omega) and define

\displaystyle {\alpha_\infty } = \mathop {\lim }\limits_{R \to \infty } \mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| > R} \right\}} | {u_m}{|^p}dx

and

\displaystyle {\beta _\infty } = \mathop {\lim }\limits_{R \to \infty } \mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| > R} \right\}} | \nabla {u_m}{|^2}dx.

Then these quantities are well-defined and satisfy

\displaystyle\mathop {\lim \sup }\limits_{m \to \infty } \int_\Omega {{{\left| {{u_m}} \right|}^p}dx} = \int_\Omega {{{\left| u \right|}^p}dx} + {\alpha _\infty }

and

\displaystyle\mathop {\lim \sup }\limits_{m \to \infty } \int_\Omega {{{\left| {\nabla {u_m}} \right|}^2}dx} \geqslant \int_\Omega {{{\left| {\nabla u} \right|}^2}dx} + {\beta _\infty }.

Proof. For each R>0

\displaystyle\begin{gathered} \mathop {\lim \sup }\limits_{m \to \infty } \int_\Omega {{{\left| {{u_m}} \right|}^p}dx} = \mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| < R} \right\}} | {u_m}{|^p}dx + \mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| > R} \right\}} | {u_m}{|^p}dx \hfill \\ \qquad= \int\limits_{\Omega \cap \left\{ {|x| < R} \right\}} | u{|^p}dx + \mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| > R} \right\}} | {u_m}{|^p}dx. \hfill \\ \end{gathered}

Letting R \to \infty gives us the first relation.

Similarly,

\displaystyle\mathop {\lim \sup }\limits_{m \to \infty } \int_\Omega {{{\left| {\nabla {u_m}} \right|}^2}dx} = \mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| < R} \right\}} | \nabla {u_m}{|^2}dx + \mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| > R} \right\}} | \nabla {u_m}{|^2}dx.

Note that

\displaystyle \nabla u_m \rightharpoonup \nabla u

in L^2(\Omega\cap\{|x|<R\}). Using the weak lower semicontinuity of norms we get

\displaystyle\mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| < R} \right\}} | \nabla {u_m}{|^2}dx \geqslant \int\limits_{\Omega \cap \left\{ {|x| < R} \right\}} | \nabla u{|^2}dx

which implies

\displaystyle\mathop {\lim \sup }\limits_{m \to \infty } \int_\Omega {{{\left| {\nabla {u_m}} \right|}^2}dx} \geqslant \int\limits_{\Omega \cap \left\{ {|x| < R} \right\}} | \nabla u{|^2}dx + \mathop {\lim \sup }\limits_{m \to \infty } \int\limits_{\Omega \cap \left\{ {|x| > R} \right\}} | \nabla {u_m}{|^2}dx.

Thus by sending R\to\infty we obtain the second identity.

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