0000000000672329

AUTHOR

Pu-zhao Kow

showing 7 related works from this author

Optimality of Increasing Stability for an Inverse Boundary Value Problem

2021

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for the Schrödinger equation. The rigorous justification of increasing stability for the IBVP for the Schrödinger equation were established by Isakov [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631--640] and by Isakov et al. [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131--141]. In [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631--640] and [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131--141], the authors showed that the stability of this IBVP increases …

increasing stability phenomenaosittaisdifferentiaaliyhtälötinstabilityComputational MathematicsMathematics - Analysis of PDEsApplied Mathematics35J15 35R25 35R30FOS: MathematicsSchrödinger equationinverse boundary value probleminversio-ongelmatAnalysisAnalysis of PDEs (math.AP)SIAM Journal on Mathematical Analysis
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A minimization problem with free boundary and its application to inverse scattering problems

2023

We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some application to inverse scattering problem.

Mathematics - Analysis of PDEsFOS: MathematicsAnalysis of PDEs (math.AP)35J05 35J15 35J20 35R30 35R35
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Quadrature domains for the Helmholtz equation with applications to non-scattering phenomena

2022

In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.

metaharmonic functionsmatematiikkapartial balayageyhtälötmean value theoremMathematics::Numerical Analysis35J05 35J15 35J20 35R30 35R35quadrature domainnon-scattering phenomenaMathematics - Analysis of PDEsFOS: MathematicsHelmholtz equationacoustic equationAnalysisAnalysis of PDEs (math.AP)
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Landis-type conjecture for the half-Laplacian

2023

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schrödinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage’s Liouville-type theorem. peerReviewed

Landis conjecture half-Laplacian Caarelli- Silvestre extension Liouville-type theoremosittaisdifferentiaaliyhtälötMathematics - Analysis of PDEsApplied MathematicsGeneral Mathematicsunique continuation propertyPrimary: 35A02 35B40 35R11. Secondary: 35J05 35J15FOS: MathematicsAnalysis of PDEs (math.AP)
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Refined instability estimates for some inverse problems

2022

Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache [29]. In this work, based on Mandache's idea, we refine the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. Our aim is to derive explicitly the dependence of the instability estimates on key parameters. The first result of this work is to show how the instability depends on the depth of the hidden inclusion and the conductivity of the background medium. This work can be regarded as a counterpart of the depth-dependent and conductivity-dependent stability estim…

osittaisdifferentiaaliyhtälötimpedanssitomografiascattering theoryControl and Optimizationdepth-dependent instability of exponential-typeinverse problemsinversio-ongelmatincreasing stability phenomenainstabilityCalderón's problem35R30kuvantaminenRellich lemmaModeling and Simulation35J15Discrete Mathematics and CombinatoricssirontaHelmholtz equation35R25Analysiselectrical impedance tomographyInverse Problems and Imaging
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On the Landis conjecture for the fractional Schrödinger equation

2023

In this paper, we study a Landis-type conjecture for the general fractional Schrödinger equation ((−P)s+q)u=0. As a byproduct, we also prove the additivity and boundedness of the linear operator (−P)s for non-smooth coefficents. For differentiable potentials q, if a solution decays at a rate exp (−∣x∣1+), then the solution vanishes identically. For non-differentiable potentials q, if a solution decays at a rate exp (−∣x∣4s−14s+), then the solution must again be trivial. The proof relies on delicate Carleman estimates. This study is an extension of the work by Rüland and Wang (2019). peerReviewed

fractional Schrödinger equationLandis conjectureunique continuation at infinityStatistical and Nonlinear PhysicsGeometry and TopologyMathematical PhysicsJournal of Spectral Theory
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The Calderón problem for the fractional wave equation: Uniqueness and optimal stability

2021

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial di…

osittaisdifferentiaaliyhtälötApplied MathematicsnonlocalCalder´on problemfractional wave equationinversio-ongelmatComputational MathematicsperidynamicMathematics - Analysis of PDEslogarithmic stabilityFOS: Mathematicsstrong uniquenessfractional LaplacianRunge approximationAnalysisAnalysis of PDEs (math.AP)
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