TY - JOUR

T1 - Existence and asymptotics of nonlinear helmholtz eigenfunctions

AU - Gell-Redman, Jesse

AU - Hassell, Andrew

AU - Shapiro, Jacob

AU - Zhang, Junyong

N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

PY - 2020

Y1 - 2020

N2 - We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form (Δ-λ2)u = N[u], where Δ = Σj ∂ 2j is the Laplacian on Rn, λ is a positive real number, and N[u] is a nonlinear operator depending polynomially on u and its derivatives of order up to order two. Nonlinear Helmholtz eigenfunctions with N[u] = ±|u|p-1u were first considered by Gutierrez [Math. Ann., 328 (2004), pp. 1-25]. We show that for suitable nonlinearities and for every f ϵ Hk+4(Sn-1) of sufficiently small norm, there is a nonlinear Helmholtz function taking the form u(r,ω) = r-(n 1)/2(e -iλ r f(ω)+e+iλ rb(ω)+O(r -ϵ )), as r → ∞, ϵ > 0, for some b ∞ Hk(Sn-1). Moreover, we prove the result in the general setting of asymptotically conic manifolds. The proof uses an elaboration of anisotropic Sobolev spaces defined by Vasy [A minicourse on microlocal analysis for wave propagation, in Asymptotic Analysis in General Relativity, London Math. Soc. Lecture Note Ser. 443, Cambridge University Press, Cambridge, 2018, pp. 219-374], between which the Helmholtz operator Δ λ2 acts invertibly. These spaces have a variable spatial weight I±, varying in phase space and distinguishing between the two "radial sets"corresponding to incoming oscillations, e -iλr, and outgoing oscillations, e+iλr. Our spaces have, in addition, module regularity with respect to two different "test modules"and have algebra (or pointwise multiplication) properties that allow us to treat nonlinearities N[u] of the form specified above.

AB - We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form (Δ-λ2)u = N[u], where Δ = Σj ∂ 2j is the Laplacian on Rn, λ is a positive real number, and N[u] is a nonlinear operator depending polynomially on u and its derivatives of order up to order two. Nonlinear Helmholtz eigenfunctions with N[u] = ±|u|p-1u were first considered by Gutierrez [Math. Ann., 328 (2004), pp. 1-25]. We show that for suitable nonlinearities and for every f ϵ Hk+4(Sn-1) of sufficiently small norm, there is a nonlinear Helmholtz function taking the form u(r,ω) = r-(n 1)/2(e -iλ r f(ω)+e+iλ rb(ω)+O(r -ϵ )), as r → ∞, ϵ > 0, for some b ∞ Hk(Sn-1). Moreover, we prove the result in the general setting of asymptotically conic manifolds. The proof uses an elaboration of anisotropic Sobolev spaces defined by Vasy [A minicourse on microlocal analysis for wave propagation, in Asymptotic Analysis in General Relativity, London Math. Soc. Lecture Note Ser. 443, Cambridge University Press, Cambridge, 2018, pp. 219-374], between which the Helmholtz operator Δ λ2 acts invertibly. These spaces have a variable spatial weight I±, varying in phase space and distinguishing between the two "radial sets"corresponding to incoming oscillations, e -iλr, and outgoing oscillations, e+iλr. Our spaces have, in addition, module regularity with respect to two different "test modules"and have algebra (or pointwise multiplication) properties that allow us to treat nonlinearities N[u] of the form specified above.

KW - Anisotropic Sobolev spaces

KW - Asymptotic expansions

KW - Incoming boundary data

KW - Module regularity

KW - Nonlinear Helmholtz equation

KW - Nonlinear eigenfunctions

UR - http://www.scopus.com/inward/record.url?scp=85098747426&partnerID=8YFLogxK

U2 - 10.1137/19M1307238

DO - 10.1137/19M1307238

M3 - Article

SN - 0036-1410

VL - 52

SP - 6180

EP - 6221

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 6

ER -