From Heisenberg uniqueness pairs to properties of the Helmholtz and Laplace equations

Aingeru Fernandez-Bertolin (1) , Karlheinz Gröchenig (2) , Philippe Jaming (1)

1 IMB - Institut de Mathématiques de Bordeaux
2 Universität Wien = University of Vienna

The aim of this paper is to establish uniqueness properties of solutions of the Helmholtz and Laplace equations. In particular, we show that if two solutions of such equations on a domain of R d agree on two intersecting d − 1-dimensional submanifolds in generic position, then they agree everywhere.

Mots clés: Harmonic functions; Nodal set; Helmholtz–Laplace equation; Unique continuation; Heisenberg uniqueness pair; Schwarz reflection principle

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]; Variables complexes [math.CV]

Aingeru Fernandez-Bertolin, Karlheinz Gröchenig, Philippe Jaming. From Heisenberg uniqueness pairs to properties of the Helmholtz and Laplace equations. Journal of Mathematical Analysis and Applications, 2019, 469 (1), pp.202-219. ⟨10.1016/j.jmaa.2018.09.008⟩. ⟨hal-01634903v2⟩