Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials
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https://doi.org/10.1007/s11425-020-1734-8
Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials Hairong Liu1,∗ & Xiaoping Yang2 1College
of Science, Nanjing Forestry University, Nanjing 210037, China; of Mathematics, Nanjing University, Nanjing 210093, China
2Department
Email: [email protected], [email protected] Received March 25, 2020; accepted July 4, 2020
Abstract
In this paper we prove the strong unique continuation property for a class of fourth order elliptic
equations involving strongly singular potentials. Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations. Keywords
strong unique continuation property, fourth order elliptic equation, singular potential, Hardy-
Rellich type inequality MSC(2010)
35J30, 35A07
Citation: Liu H R, Yang X P. Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-020-1734-8
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Introduction and main results
In this paper, we investigate the strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials. A differential operator L is said to have the strong unique continuation property in Ω if the only solution of Lu = 0 which vanishes to infinite order at a point x0 ∈ Ω is u ≡ 0. In the past decades the strong unique continuation property for the solutions to various kinds of partial differential equations has attracted a large number of researchers and induced many interesting and intensive results. By now the strong unique continuation problem for second order elliptic operators is well understood. The results for the second order operator −∆u = V0 u + V1 · ∇u
(1.1)
go back to the work of Carleman [5], who solved the uniqueness problem in R2 with bounded potentials. Cordes [9] and Aronszajn [4] extended the strong unique continuation property to second order equations in Rn . There is a large amount of work on the strong unique continuation for second order elliptic operators, achieved by different Carleman type estimates (see [15, 17, 18, 23]). * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Liu H R et al.
2
Sci China Math
On the other hand, Garofalo and Lin [10, 11] presented a geometric-variational approach to the strong unique continuation by using the frequency function. The frequency function they investigated for the Schr¨odinger equation −∆u + V u = 0 is given by ∫ r Br (|∇u|2 + V u2 )dx ∫ N (r) = . u2 dS ∂Br Their method is based on establishing the doubling estimate ∫ ∫ 2 u 6C u2 , B2r
Br
which in turn depends on the monotonicity property of the frequency function N (r). It is worth pointing out that the frequency function was first introduced
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