Math. Z. https://doi.org/10.1007/s00209-018-2081-6 Mathematische Zeitschrift 1 1 2 Guher Camliyurt · Igor Kukavica · Fei Wang Received: 4 September 2017 / Accepted: 5 February 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract We address the question of quantitative uniqueness for the equation u = Vu with either periodic or Dirichlet boundary conditions in a disk. We construct solutions u 2/3 corresponding to potential functions V such that u vanishes of order constV .The example also shows sharpness of recently obtained bounds in the case of a parabolic equation u − u = Vu. 1 Introduction The main purpose of this paper is to construct smooth complex valued solutions to the problem u = Vu, (1.1) 2/3 which vanish of the order comparable with V in the unit disk of R with Dirichlet boundary condition. We start with construction of such solutions in the unit disk so that they satisfy a doubling type property. Then we modify the solutions to be periodic or that they satisfy the Dirichlet boundary condition. Finally, we construct solutions of a parabolic equation ∂ u − u = Vu (1.2) on R × (−1, 1) under some growth assumptions, showing
Mathematische Zeitschrift – Springer Journals
Published: May 29, 2018
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