Fourier-extension estimates for symmetric functions and applications to\n nonlinear Helmholtz equations

2020/05/26 by Tobias Weth, Weth, Tobias, Tolga YEŞİL +1
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2005.12589

openalex publication_date 2020/05/26 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We establish weighted L^p-Fourier-extension estimates for O(N-k) \×\nO(k)-invariant functions defined on the unit sphere mathbbS^N-1,\nallowing for exponents p below the Stein-Tomas critical exponent\n\(2(N+1))/(N-1). Moreover, in the more general setting of an arbitrary\nclosed subgroup G \⊂ O(N) and G-invariant functions, we study the\nimplications of weighted Fourier-extension estimates with regard to boundedness\nand nonvanishing properties of the corresponding weighted Helmholtz resolvent\noperator. Finally, we use these properties to derive new existence results for\nG-invariant solutions to the nonlinear Helmholtz equation\n -
Delta u - u = Q(x)|u|^p-2u,
quad u
in W^2,p(
mathbbR^N),\nwhere Q is a nonnegative bounded and G-invariant weight function.\n

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