Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients

Zhu, Jiuyi

doi:10.1007/s00526-018-1328-8

Cited by 9 publications

(3 citation statements)

References 35 publications

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“…with the strongly singular terms b 1 , b 2 , V 1 and V 2 . We refer to [27,30,38] for the quantitative unique continuation properties of uniformly elliptic operators with higher order. Before describing our main results, we will explain the definition of vanishing of infinite order in the subelliptic setting.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

Liu

Yang

2020

Sci. China Math.

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In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation ∆ 2X u = V u, where ∆X = ∆x + |x| 2α ∆y (0 < α ≤ 1), with x ∈ R m , y ∈ R n , denotes the Baouendi-Grushin type subelliptic operators, and the potential V satisfies the strongly singular growth assumption |V | ≤ c 0 ρ 4 , whereis the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

Liu

Yang

2020

Sci. China Math.

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“…In the case of SUCP of higher‐order elliptic equations, we refer to previous works 23–26 and the related references therein for the strong unique continuation results of higher order elliptic equations by using Carleman estimates. In the case of three‐ball inequality and quantitative uniqueness of higher‐order elliptic equations, there are few references addressing these kind of problems (see previous works 27–30 ).…”

Section: Introductionmentioning

confidence: 99%

“…In Zhu, 29 the author considered the quantitative uniqueness of higher‐order elliptic equation

{(- Δ)}^{m} u (x) = V (x) u (x), in B_{10} .

Under some assumptions on the potential V and the solution u , based on a variant of frequency function, the author proved that for n ≥ 4 m , the vanishing order of u is less than

C false| false| V_{0} false| {false|}_{L^{\infty} false(normalΩ false)}

. In, 30 the author considered the quantitative uniqueness of general higher order elliptic equation with singular coefficients

{(- Δ)}^{m} u (x) + \sum_{false| α false| = 1}^{α_{0}} V_{α} (x) \cdot D^{α} u + V_{0} (x) u (x) = 0, in B_{10} .

It should be point out that in 30 if m is a positive even integer, the value α 0 ≤ [3 m /2] − 1, that is, in case of

m = 2

, α 0 ≤ 2. As far as we know, three‐ball inequality for bi‐Laplace equations () in the sense of L 2 ‐norm with third‐order terms and quantitative uniqueness havn't been discussed yet.…”

Section: Introductionmentioning

confidence: 99%

Quantitative unique continuation of solutions to the bi‐Laplace equations

Liao

2021

Math Methods in App Sciences

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In this paper, we prove a three‐ball inequality for y satisfying an equation of the form Δ2y=V0y+V1·∇y+V2Δy+V3·∇Δy in some open, connected set Ω of double-struckRn with V0,0.1emV2∈L∞false(normalΩ;double-struckCfalse) and V1,0.1emV3∈L∞false(normalΩ;0.1emdouble-struckCnfalse). The derivation of such estimate relies on a delicate Carleman estimate for the bi‐Laplace equation and some Caccioppoli inequalities to estimate the lower‐terms. Based on three‐ball inequality, we then derive the vanishing order of y is less than Cfalse|V0false|∞13+false|V1false|∞12+false|V2false|∞23+false|V3false|∞2, where | · |∞ means the L∞ norm, which is a quantitative version of the strong unique continuation property for y. Furthermore, under some priori assumptions on Vj and y, we prove that the nontrivial solution y satisfies the decay property e−CR2logR around the point at infinity. In particular, if V1=V3=false(0,…0.1em,0false), this decaying rate can be improved to e−CR4false/3logR.

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Doubling Inequality and Nodal Sets for Solutions of Bi-Laplace Equations

Zhu

2019

Arch Rational Mech Anal

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We investigate the doubling inequality and nodal sets for the solutions of bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for Laplace eigenfunctions by Logunov. In addition, we derive an implicit upper bound for the nodal sets of solutions. We show two types of doubling inequalities for the solutions of bi-Laplace equations. As a consequence, the rate of vanishing is given for the solutions.2010 Mathematics Subject Classification. 35J05, 35J30, 58J50, 35J05.

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Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients

Cited by 9 publications

References 35 publications

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

Quantitative unique continuation of solutions to the bi‐Laplace equations

Doubling Inequality and Nodal Sets for Solutions of Bi-Laplace Equations

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