Ngô Quốc Anh

September 28, 2011

Concentration-Compactness principle, II

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 9:35

In this entry, we continue to talk about the Concentration-Compactness Principle discovered by P.L. Lions [here]. In the previous entry, we already discussed two forms of non-compactness due to unbounded domains. Here we discuss what happens when passing to the limit on those functionals along weakly convergent subsequences.

Theorem (Lions). Let \{u_j\}_j be sequence in D^{1,p}(\mathbb R^n) weakly convergent to u and such that

  • \{|\nabla u_j\|^p\} converges weak* to a nonnegative measure \mu,
  • \{|u_j|^{p^\star}\} converges weak* to a nonnegative measure \nu.

Then there exists an at most countable index set J, sequence \{x_j\} \subset \mathbb R^n, \{\mu_j\}, \{\nu_j\} \subset (0,\infty), j \in J, such that

\displaystyle\nu = |u{|^{{p^ \star }}} + \sum\limits_{j \in J} {{\nu _j}{\delta _{{x_j}}}} ,

and

\displaystyle\mu \geqslant |\nabla u{|^p} + \sum\limits_{j \in J} {{\mu _j}{\delta _{{x_j}}}} ,

and

\displaystyle S\nu _j^{\frac{p}{{{p^ \star }}}} \leqslant {\mu _j},

where S is the best Sobolev constant and \delta_{x_j} are Dirac measures assigned to x_j. If u \equiv 0 and

\displaystyle \int_{{\mathbb{R}^n}} {d\mu } \leqslant S{\left( {\int_{{\mathbb{R}^n}} {d\nu } } \right)^{\frac{p}{{{p^ \star }}}}}

then J is a singleton and

\displaystyle\nu=\gamma\delta_{x_0}=\frac{1}{S}\gamma^\frac{p}{n}\mu

for some \gamma \geqslant 0.

Apparently, the theorem does not provide any information about possible loss of mass at infinity of a weakly convergent minimizing sequence. We shall consider that case in the forthcoming topic.

See also:

September 5, 2011

The Riemannian Penrose inequality

Filed under: Riemannian geometry — Ngô Quốc Anh @ 1:43

In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is the most important special case. Specifically, if (M, g) is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass m, and A is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts

\displaystyle m \geq \sqrt{\frac{A}{16\pi}}.

This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (M, g) having nonnegative scalar curvature is equivalent to the spacetime obeying the dominant energy condition.

This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 [here and here] in the case where A is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined inverse mean curvature flow, which they developed. In 1999, Hubert Bray [here] gave the first complete proof of the above inequality using a conformal flow of metrics. Both of the papers were published in 2001 in the Journal of Differential Geometry.

Source: Wiki

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