An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potentia
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Mathematische Annalen
An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential Rainer Mandel1 · Dominic Scheider1 Received: 16 March 2020 / Revised: 9 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider the Helmholtz equation −u + V u − λ u = f on Rn where the potential V : Rn → R is constant on each of the half-spaces Rn−1 × (−∞, 0) and Rn−1 × (0, ∞). We prove an L p − L q -Limiting Absorption Principle for frequencies λ > max V with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and nonlinear Helmholtz equations. As a main analytical tool we develop new L p − L q estimates for a singular Fourier multiplier supported in an annulus. Mathematics Subject Classification Primary 35J05; Secondary 35Q60
1 Introduction In this paper we are interested in the Limiting Absorption Principle (LAP) for the Helmholtz equation on Rn involving a step potential of the form V1 if x ∈ Rn−1 , y > 0, V (x, y) = (1) if x ∈ Rn−1 , y < 0 V2 where V1 = V2 are two fixed real numbers. We will without loss of generality assume V1 > V2 in the following. Examples for elliptic problems involving interfaces modelled by potentials of this kind can be found in [14, Theorem 1], [15, Theorem 2] or
Communicated by Loukas Grafakos.
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Dominic Scheider [email protected] Rainer Mandel [email protected]
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Karlsruhe Institute of Technology, Institute for Analysis, Englerstraße 2, 76131 Karlsruhe, Germany
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R. Mandel, D. Scheider
[28]. To explain the motivation behind our study, we recall the interesting phenomenon called “double scattering”. In the context of the Schrödinger equation it means that for sufficiently regular and fast decaying right hand sides f the unique solution of the initial value problem i∂t ψ − ψ + V ψ = f in Rn , ψ(0) = ψ0 , with V as in (1) splits up into two pieces as t → ±∞ that correspond to the two different values of V at infinity. This phenomenon is mathematically understood in the one-dimensional case n = 1 [24, Theorem 1.2], see also [12,13]. One byproduct of our results is that such a splitting into two pieces may as well be observed for the solutions of the corresponding Helmholtz equations in Rn which are obtained through the Limiting Absorption Principle, see for instance the formula (17) where the two parts f (x, y)1(0,∞) (±y) of the right hand side contribute differently to the LAP-solution of the Helmholtz equation. Notice that solutions u of such Helmholtz equations provide monochromatic solutions ψ(x, t) = eiλt u(x) of the Schrödinger equation where λ belongs to the L 2 -spectrum of the selfadjoint operator − + V with domain H 2 (Rn ). We prove our LAP in the topology of Lebesgue spaces in order to treat both linear and nonlinear Helmholtz equations. As far as we can see, the more classical results in weighted L 2 spaces resp. B(Rn ), B ∗ (Rn ) (for the definition, cf. [4, page 4]) by Agmon [1–3] and Agmon–Hörmander [4] do not appl
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