Ngô Quốc Anh

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

and

\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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