Decay estimates for higher-order elliptic operators

Feng, Hongliang; Soffer, Avy; Wu, Zhaojun; Yao, Xiaohua

doi:10.1090/tran/8010

Cited by 32 publications

(28 citation statements)

References 55 publications

(48 reference statements)

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“…In general, due to the presence of the potential function V pxq, the resolvent R V pκq may have poles on R `which are restricted in the set tx : 0 ă x ă C 0 u by Theorem 3.3. However, a recent result [11] shows that for a certain class of nonnegative potential functions the resolvent has no poles on R `. On the other hand, since the first eigenvalue µ 1 of H in B R increases as the radius R decreases, if the supports of the source f pxq and potential V pxq are small, we may shrink the ball B R to make µ 1 ě C 0 .…”

Section: Define a Differential Operatormentioning

confidence: 99%

Stability for an inverse source problem of the biharmonic operator

Li¹,

Yao²,

Zhao³

2021

Preprint

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In this paper, we study for the first time the stability of the inverse source problem for the biharmonic operator with a compactly supported potential in R 3 . Firstly, to connect the boundary data with the unknown source, we shall consider an eigenvalue problem for the bi-Schr: odinger operator ∆ 2 `V pxq on a ball which contains the support of the potential V . We prove a Weyl-type law for the upper bounds of spherical normal derivatives of both the eigenfunctions φ and their Laplacian ∆φ corresponding to the bi-Schr: odinger operator. This type of upper bounds was proved by Hassell and Tao for the Schr: odinger operator. Secondly, we investigate the meromorphic continuation of the resolvent of the bi-Schr: odinger operator and prove the existence of a resonance-free region and an estimate of L 2 comp ´L2loc type for the resolvent. As an application, we prove a bound of the analytic continuation of the data from the given data to the higher frequency data. Finally, we derive the stability estimate which consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases.

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Section: Define a Differential Operatormentioning

confidence: 99%

Stability for an inverse source problem of the biharmonic operator

Li¹,

Yao²,

Zhao³

2021

Preprint

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“…Recall that, if V is in the higher order Kato class K 2m (R n ) [see (2.4) below for its definition], then V is a Kato type perturbation of P(D) (see [18,66]). Further developments on the local estimates of the type (1.9) can be fund in [4,5,27,38].…”

Section: Introductionmentioning

confidence: 99%

Gaussian Estimates for Heat Kernels of Higher Order Schrödinger Operators with Potentials in Generalized Schechter Classes

Cao¹,

Liu²,

Yang³

et al. 2020

Preprint

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Let m ∈ N, P(D) := |α|=2m (−1) m a α D α be a 2m-order homogeneous elliptic operator with real constant coefficients on R n , and V a measurable function on R n . In this article, the authors introduce a new generalized Schechter class concerning V and show that the higher order Schrödinger operator L := P(D) + V possesses a heat kernel that satisfies the Gaussian upper bound and the Hölder regularity when V belongs to this new class. The Davies-Gaffney estimates for the associated semigroup and their local versions are also given. These results pave the way for many further studies on the analysis of L.

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“…the case of ǫ = −1 is open. Higher order Schrödinger operators (−∆) m + V , along with a criteria to rule out embedded eigenvalues are studied in [15].…”

Section: Introductionmentioning

confidence: 99%

Time integrable weighted dispersive estimates for the fourth order Schrödinger equation in three dimensions

Goldberg¹,

Green²

2020

Preprint

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We consider the fourth order Schrödinger operator H = ∆ 2 + V and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of H that the solution operator e −itH satisfies a large time integrable |t| − 5 4 decay rate between weighted spaces. This bound improves what is possible for the free case in two directions; both better time decay and smaller spatial weights. In the case of a mild resonance at zero energy, we derive the operator-valued expansion e −itH P ac (

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Decay estimates for higher-order elliptic operators

Cited by 32 publications

References 55 publications

Stability for an inverse source problem of the biharmonic operator

Stability for an inverse source problem of the biharmonic operator

Gaussian Estimates for Heat Kernels of Higher Order Schrödinger Operators with Potentials in Generalized Schechter Classes

Time integrable weighted dispersive estimates for the fourth order Schrödinger equation in three dimensions

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