An inverse boundary problem for fourth-order Schrödinger equations with partial data

Zhiwen Duan; Shuxia Han

doi:10.1515/ms-2017-0208

An inverse boundary problem for fourth-order Schrödinger equations with partial data

Mathematica Slovaca ◽

10.1515/ms-2017-0208 ◽

2019 ◽

Vol 69 (1) ◽

pp. 125-138

Author(s):

Zhiwen Duan ◽

Shuxia Han

Keyword(s):

Boundary Problem ◽

Fourth Order ◽

Schrodinger Equation ◽

Geometrical Optics ◽

Cauchy Data ◽

Carleman Estimates ◽

Schrödinger Equations ◽

Inverse Boundary ◽

Partial Data ◽

Inverse Boundary Problem

Abstract In this paper, we show that in dimension n ≥ 3, the knowledge of the Cauchy data for the fourth-order Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. The proof is based on the Carleman estimates and the construction of complex geometrical optics solutions.

Numerical Method for Solving an Inverse Boundary Problem with Unknown Initial Conditions for Parabolic PDE Using Discrete Regularization

Lecture Notes in Computer Science - Numerical Analysis and Its Applications ◽

10.1007/978-3-319-57099-0_87 ◽

2017 ◽

pp. 752-759 ◽

Cited By ~ 3

Author(s):

Natalia M. Yaparova

Keyword(s):

Numerical Method ◽

Boundary Problem ◽

Initial Conditions ◽

Inverse Boundary ◽

Parabolic Pde ◽

Inverse Boundary Problem

On an inverse boundary problem arising in brain imaging

Journal of Differential Equations ◽

10.1016/j.jde.2019.03.019 ◽

2019 ◽

Vol 267 (4) ◽

pp. 2471-2502 ◽

Cited By ~ 2

Author(s):

Youjun Deng ◽

Hongyu Liu ◽

Gunther Uhlmann

Keyword(s):

Brain Imaging ◽

Boundary Problem ◽

Inverse Boundary ◽

Inverse Boundary Problem

The inverse boundary problem for the Rayleigh system

Journal of Mathematical Physics ◽

10.1063/1.531182 ◽

1995 ◽

Vol 36 (12) ◽

pp. 6688-6708 ◽

Cited By ~ 5

Author(s):

Richard Beals ◽

G. M. Henkin ◽

N. N. Novikova

Keyword(s):

Boundary Problem ◽

Inverse Boundary ◽

Inverse Boundary Problem ◽

Rayleigh System

Abel-Lidskii bases in the non-self-adjoint inverse boundary problem

Journal of Mathematical Sciences ◽

10.1007/bf02673855 ◽

2000 ◽

Vol 102 (4) ◽

pp. 4237-4257 ◽

Cited By ~ 1

Author(s):

Ya. V. Kurylev ◽

M. Lassas

Keyword(s):

Boundary Problem ◽

Inverse Boundary ◽

Inverse Boundary Problem

On one nonlocal inverse boundary problem for the Benney – Luke equation with integral conditions

Journal of Physics Conference Series ◽

10.1088/1742-6596/1203/1/012100 ◽

2019 ◽

Vol 1203 ◽

pp. 012100

Author(s):

Yashar Mehraliyev ◽

Bahar Valiyeva

Keyword(s):

Boundary Problem ◽

Integral Conditions ◽

Inverse Boundary ◽

Inverse Boundary Problem

An inverse boundary problem for one-dimensional heat equation with a multilayer domain

Engineering Analysis with Boundary Elements ◽

10.1016/j.enganabound.2008.04.006 ◽

2009 ◽

Vol 33 (2) ◽

pp. 225-232 ◽

Cited By ~ 24

Author(s):

T. Wei ◽

Y.S. Li

Keyword(s):

Heat Equation ◽

Boundary Problem ◽

Inverse Boundary ◽

One Dimensional ◽

Inverse Boundary Problem

Boundary regularity for the Ricci equation, geometric convergence, and Gel?fand?s inverse boundary problem

Inventiones mathematicae ◽

10.1007/s00222-004-0371-6 ◽

2004 ◽

Vol 158 (2) ◽

pp. 261-321 ◽

Cited By ~ 52

Author(s):

Michael Anderson ◽

Atsushi Katsuda ◽

Yaroslav Kurylev ◽

Matti Lassas ◽

Michael Taylor

Keyword(s):

Boundary Problem ◽

Boundary Regularity ◽

Inverse Boundary ◽

Geometric Convergence ◽

Inverse Boundary Problem

An inverse boundary problem for the steady-diffusion equation with moving boundaries with applications to the pearlite$ndash$austenite transformation in steel

Inverse Problems ◽

10.1088/0266-5611/18/4/305 ◽

2002 ◽

Vol 18 (4) ◽

pp. 1013-1023 ◽

Cited By ~ 5

Author(s):

N D Aparicio ◽

C Atkinson

Keyword(s):

Diffusion Equation ◽

Boundary Problem ◽

Moving Boundaries ◽

Inverse Boundary ◽

Austenite Transformation ◽

Inverse Boundary Problem

On One Nonlocal Inverse Boundary Problem for the Second-Order Parabolic Equation

Bulletin of the South Ural State University series Mathematics Mechanics Physics ◽

10.14529/mmph170202 ◽

2017 ◽

Vol 9 (2) ◽

pp. 13-21

Author(s):

Ya.T. Mehraliev ◽

◽

A.N. Safarova ◽

Keyword(s):

Parabolic Equation ◽

Boundary Problem ◽

Second Order ◽

Inverse Boundary ◽

Order Parabolic Equation ◽

Inverse Boundary Problem

Structure theorems for 2D linear and nonlinear Schrödinger equations

Communications in Contemporary Mathematics ◽

10.1142/s0219199715500340 ◽

2016 ◽

Vol 18 (02) ◽

pp. 1550034

Author(s):

Hajer Bahouri

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Cauchy Data ◽

Nonlinear Schrodinger Equation ◽

Nonlinear Schrödinger Equations ◽

Schrödinger Equations ◽

Nonlinear Schrödinger ◽

Nonlinear Schrodinger Equations ◽

Linear And Nonlinear

This paper is devoted to the qualitative study of the nonlinear Schrödinger equation with exponential growth, where the Orlicz norm plays a crucial role. The approach we adopted in this paper which is based on profile decompositions consists of comparing the evolution of oscillations and concentration effects displayed by sequences of solutions to 2D linear and nonlinear Schrödinger equations associated to the same sequence of Cauchy data, up to small remainder terms both in Strichartz and Orlicz norms. The analysis we conducted in this work emphasizes the correlation between the nonlinear effect highlighted in the behavior of the solutions to the 2D nonlinear Schrödinger equation and the [Formula: see text]-oscillating component of the sequence of the Cauchy data.