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

@article{Ivanovici2012,
abstract = {In this paper we consider a smooth and bounded domain $\Omega \subset \mathbb \{R\}^d$ of dimension $d\ge 2$ 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 $\mathbb \{R\}^d$.},
author = {Ivanovici, Oana},
journal = {Journal of the European Mathematical Society},
keywords = {microlocal analysis; wave equation; Dirichlet boundary condition; Strichartz estimates; propagation and reflection of singularities; conormal waves with cusps; caustics; gliding point; Dirichlet boundary conditions; propagation and reflection of singularities; conormal waves with cusps; caustics; gliding point},
language = {eng},
number = {5},
pages = {1357-1388},
publisher = {European Mathematical Society Publishing House},
title = {Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary},
url = {http://eudml.org/doc/277363},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Ivanovici, Oana
TI - Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 5
SP - 1357
EP - 1388
AB - In this paper we consider a smooth and bounded domain $\Omega \subset \mathbb {R}^d$ of dimension $d\ge 2$ 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 $\mathbb {R}^d$.
LA - eng
KW - microlocal analysis; wave equation; Dirichlet boundary condition; Strichartz estimates; propagation and reflection of singularities; conormal waves with cusps; caustics; gliding point; Dirichlet boundary conditions; propagation and reflection of singularities; conormal waves with cusps; caustics; gliding point
UR - http://eudml.org/doc/277363
ER -