no. 4
Uniform resolvent estimates for a non-dissipative Helmholtz equation
[Estimées résolvantes uniformes pour une équation de Helmholtz non dissipative]
Bulletin de la Société Mathématique de France, Tome 142 (2014) no. 4, pp. 591-633

We study the high frequency limit for a non-selfadjoint Helmholtz equation. This equation models the propagation of the electromagnetic field of a laser in an inhomogeneus material medium with non-constant absorption index. In this paper the absorption index can take negative values and we only use a damping condition on the classical limit of the problem. In this setting we first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies this damping assumption. Then we generalize the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures of the solution when the source term concentrates on a bounded submanifold of ℝn.

Publié le :
DOI : 10.24033/bsmf.2674
Classification : 35J10, 47A10, 47A55, 47B44, 47G30, 81Q20
Keywords: Non-selfadjoint operators, resolvent estimates, limiting absorption principle, Helmholtz equation, semiclassical measures. 
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     author = {Royer, Julien},
     title = {Uniform resolvent estimates for a non-dissipative {Helmholtz} equation},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {591--633},
     year = {2014},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {142},
     number = {4},
     doi = {10.24033/bsmf.2674},
     mrnumber = {3306871},
     zbl = {1315.35064},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2674/}
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Royer, Julien. Uniform resolvent estimates for a non-dissipative Helmholtz equation. Bulletin de la Société Mathématique de France, Tome 142 (2014) no. 4, pp. 591-633. doi: 10.24033/bsmf.2674

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