The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations

TL;DR

This paper establishes an $L^p$-based limiting absorption principle for periodic elliptic operators and uses it to construct solutions for nonlinear Helmholtz equations with periodic coefficients.

Contribution

It introduces an $L^p$-version of the limiting absorption principle for second-order periodic elliptic operators, enabling new solution constructions.

Findings

Proved an $L^p$-limiting absorption principle for periodic elliptic operators.

Constructed nontrivial solutions to nonlinear Helmholtz equations with periodic coefficients.

Extended the theoretical framework for analyzing periodic differential operators.

Abstract

We prove an $L^p$-version of the limiting absoprtion 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.