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Comparison of H-measures and semiclassical measures

Semiclassical measures were introduced by Patrick Gérard (1991), but are also known under the name of Wigner measures as Pierre Louis Lions and Thierry Paul gave a different construction using the Wigner transform. They are mathematical objects used in the study of high-frequency limits in continuum and quantum mechanics. In contrast to the H-measures, they are tailored to deal with problems which have a characteristic length (e.g. thickness of a plate). Recently, Luc Tartar extended semiclassical measures to a functional on continuous functions on a compactification of $R^d\setminus\{; ; ; 0\}; ; ; $. Our aim is to study more deeply this extension and its relation to H-measures. It is known that for an $\epsilon_n$-oscillatory sequence we can obtain that the corresponding H-measure and the semiclassical measure are equal, but for the above extension such assumption is superfluous, while the proof is simpler. In particular, we shall study PDEs with characteristic length tending to zero to emphasize situations where H-measures and semiclassical measures do not give the same information, and where with semiclassical measures we indeed get something more.

H-measures ; semiclassical measures

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