The Limiting Absorption Principle for Periodic Differential Operators and Applications to Nonlinear Helmholtz Equations

Abstract

We prove an Lp-version of the limiting absorption principle for a class of periodic elliptic differential operators of second order. The result is applied to the construction of nontrivial solutions of nonlinear Helmholtz equations with periodic coefficient functions.

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Acknowledgments

The author thanks the reviewer for several helpful remarks and corrections. Furthermore, he is grateful to TomášDohnal (Martin-Luther-UniversitätHalle-Wittenberg) for stimulating discussions about Floquet-Bloch theory during the past years and for providing Fig. 1. Additionally, he gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 ”Wave phenomena: analysis and numerics”.

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Correspondence to Rainer Mandel.

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Mandel, R. The Limiting Absorption Principle for Periodic Differential Operators and Applications to Nonlinear Helmholtz Equations. Commun. Math. Phys. 368, 799–842 (2019). https://doi.org/10.1007/s00220-019-03363-1

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