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 -