We lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boundary value problems for the Helmholtz equation. This consists in introducing a quasi-periodic fundamental solution and the related layer potentials, showing how they are used to construct the solutions of quasi-periodic boundary value problems, and how they behave when we perform a singular perturbation of the domain. To show an application, we study a nonlinear quasi-periodic Robin problem in a domain with a set of holes that shrink to points.
Bramati R., Dalla Riva M., Luzzini P., Musolino P. (2024). THE FUNCTIONAL ANALYTIC APPROACH FOR QUASI-PERIODIC BOUNDARY VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION. ADVANCES IN DIFFERENTIAL EQUATIONS, 29(1-2), 27-68 [10.57262/ade029-0102-27].
THE FUNCTIONAL ANALYTIC APPROACH FOR QUASI-PERIODIC BOUNDARY VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION
Dalla Riva M.;
2024-01-01
Abstract
We lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boundary value problems for the Helmholtz equation. This consists in introducing a quasi-periodic fundamental solution and the related layer potentials, showing how they are used to construct the solutions of quasi-periodic boundary value problems, and how they behave when we perform a singular perturbation of the domain. To show an application, we study a nonlinear quasi-periodic Robin problem in a domain with a set of holes that shrink to points.
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