Small nonlinear Helmholtz equations have solutions with prescribed incoming and outgoing wave patterns at infinity, on Euclidean space and asymptotically conic manifolds, under a dimension-degree condition.
Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach
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abstract
We use a Lagrangian regularity perspective to discuss resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. In addition to the Lagrangian perspective we introduce and use a resolved pseudodifferential algebra to deal with zero energy degeneracies in a robust manner.
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Existence and asymptotics of nonlinear Helmholtz eigenfunctions
Small nonlinear Helmholtz equations have solutions with prescribed incoming and outgoing wave patterns at infinity, on Euclidean space and asymptotically conic manifolds, under a dimension-degree condition.