Description of the lack of compactness of some critical Sobolev embedding

Bahouri, Hajer

Journées équations aux dérivées partielles (2011), article no. 1, 13 p.

In this text, we present two recent results on the characterization of the lack of compactness of some critical Sobolev embedding. The first one derived in [5] deals with an abstract framework including Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces. The second one established in [3] concerns the lack of compactness of $H^1\left({ℝ}^2\right)$ into the Orlicz space. Although the two results are expressed in the same manner (by means of defect measures) and rely on the defect of compactness due to concentration as in [17] and [18], they are actually of different nature. In fact, both in [5] and [3] it is proved that the lack of compactness can be described in terms of an asymptotic decomposition, but the elements involved in the decomposition are of completely different kinds in the two frameworks. We also highlight that contrary to semilinear cases like the wave equation studied in [2] and [9], the linearizability of the non linear wave equation with exponential growth is not directly related to the lack of compactness of $H^1\left({ℝ}^2\right)$ into the Orlicz space.

Mots clés : Critical Sobolev embedding, lack of compactness, BMO space, Orlicz space.

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