Ngô Quốc Anh

January 24, 2015

Reversed Gronwall-Bellman’s inequality

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 23:01

In mathematics, Gronwall’s inequality (also called Grönwall’s lemma, Gronwall’s lemma or Gronwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. The differential form was proven by Grönwall in 1919. The integral form was proven by Richard Bellman in 1943. A nonlinear generalization of the Gronwall–Bellman inequality is known as Bihari’s inequality.

First, we consider the Gronwall inequality.

Type 1. Bounds by integrals based on lower bound a.

Let \beta and u be real-valued continuous functions defined on [a,b]. If u is differentiable in (a,b) and satisfies the differential inequality

\displaystyle u'(t) \leqslant \beta(t) u(t),

then u is bounded by the solution of the corresponding differential equation y'(t) = \beta (t)y(t), that is to say

\displaystyle \boxed{u(t) \leqslant u(a) \exp\biggl(\int_a^t \beta(s) ds\biggr)}

for all t \in [a,b].


January 7, 2015

The failure of compact Rellich-Kondrachov embedding: Unbounded domains and critical exponents

Filed under: Uncategorized — Ngô Quốc Anh @ 19:49

In a very old entry, I talked about an extension of Rellich-Kondrachov theorem for embeddings between Sobolev spaces. For the sake of convenience, here is the statement of this extension:

Theorem (Extension of Rellich-Kondrachov for bounded domains). Let \Omega \subset \mathbb R^n be an open, bounded Lipschitz domain, and let  1 \leqslant p \leqslant mn. Set

\displaystyle p^\star := \frac{np}{n - mp}.

Then we have

\displaystyle W^{j+m, p} (\Omega) \hookrightarrow W^{j, q} (\Omega) for  1 \leqslant q \leqslant p^\star


\displaystyle W^{j+m, p} (\Omega) \hookrightarrow \hookrightarrow W^{j,q} (\Omega) for  1 \leqslant q < p^\star.

Clearly, when q=p^\star=\frac{np}{n - mp}, the above embedding is not compact, in general. In this context, we call the failure of compact Rellich-Kondrachov embedding due to critical exponents.

There is an other example of the failure of compact Rellich-Kondrachov embedding which is basically due to the unbounded domains. In this entry, we address counter-examples for these two lacks of compactness.


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