Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary | Zendy

Oana Ivanovici | Zendy

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Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary

Author(s) -

Oana Ivanovici

Publication year - 2012

Publication title -

journal of the european mathematical society

Language(s) - English

Resource type - Journals

eISSN - 1435-9863

pISSN - 1435-9855

DOI - 10.4171/jems/335

Subject(s) - mathematics , counterexample , wave equation , boundary (topology) , mathematical analysis , inequality , boundary value problem , pure mathematics , algebra over a field , discrete mathematics

In this paper we consider a smooth and bounded domain O?R d of dimension d=2 with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains in R d .

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