scholarly journals Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

2016 ◽  
Vol 106 (2) ◽  
pp. 342-374 ◽  
Author(s):  
Hoai-Minh Nguyen
Keyword(s):  
Helmholtz Equation ◽  
Well Posedness ◽  
Limiting Absorption Principle

scholarly journals A priori estimates for the Helmholtz equation with electromagnetic potentials in exterior domains

2013 ◽  
Vol 143 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Juan Antonio Barceló ◽  
Luca Fanelli ◽  
Alberto Ruiz ◽  
Maricruz Vilela
Keyword(s):  
Helmholtz Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Exterior Domains ◽  
Limiting Absorption Principle ◽  
Electromagnetic Potentials ◽  
Priori Estimates ◽  
Embedded Eigenvalues

We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension n ≥ 3. Using multiplier techniques in the style of Morawetz, we prove a family of a priori estimates from which the limiting absorption principle follows. Moreover, we give some standard applications to cases with an absence of embedded eigenvalues and zero resonances, under explicit conditions on the potentials.

scholarly journals Limiting absorption principle for the electromagnetic Helmholtz equation with singular potentials

2014 ◽  
Vol 144 (4) ◽  
pp. 857-890 ◽  
Author(s):  
Miren Zubeldia
Keyword(s):  
Helmholtz Equation ◽  
Unique Solution ◽  
A Priori ◽  
Radiation Condition ◽  
Singular Potentials ◽  
Absorption Method ◽  
Limiting Absorption Principle ◽  
Electric Potentials ◽  
Image Position

We study the Helmholtz equationin ℝd with magnetic and electric potentials that are singular at the origin and decay at ∞. We prove the existence of a unique solution satisfying a suitable Sommerfeld radiation condition, together with some a priori estimates. We use the limiting absorption method and a multiplier technique of Morawetz type.

scholarly journals A limiting absorption principle for the Helmholtz equation with variable coefficients

2018 ◽  
Vol 8 (4) ◽  
pp. 1349-1392 ◽  
Author(s):  
Federico Cacciafesta ◽  
Piero D'Ancona ◽  
Renato Lucà
Keyword(s):  
Helmholtz Equation ◽  
Variable Coefficients ◽  
Limiting Absorption Principle

scholarly journals Inverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization

2019 ◽  
Vol 53 (3) ◽  
pp. 1005-1030 ◽  
Author(s):  
Giovanni Alessandrini ◽  
Maarten V. De Hoop ◽  
Florian Faucher ◽  
Romina Gaburro ◽  
Eva Sincich
Keyword(s):  
Inverse Problem ◽  
Helmholtz Equation ◽  
Piecewise Linear ◽  
Cauchy Data ◽  
Tetrahedral Mesh ◽  
Normal Velocity ◽  
Well Posedness ◽  
Iterative Regularization ◽  
Element Discretization ◽  
Linear Representations

In this paper, we study the performance of Full Waveform Inversion (FWI) from time-harmonic Cauchy data via conditional well-posedness driven iterative regularization. The Cauchy data can be obtained with dual sensors measuring the pressure and the normal velocity. We define a novel misfit functional which, adapted to the Cauchy data, allows the independent location of experimental and computational sources. The conditional well-posedness is obtained for a hierarchy of subspaces in which the inverse problem with partial data is Lipschitz stable. Here, these subspaces yield piecewise linear representations of the wave speed on given domain partitions. Domain partitions can be adaptively obtained through segmentation of the gradient. The domain partitions can be taken as a coarsening of an unstructured tetrahedral mesh associated with a finite element discretization of the Helmholtz equation. We illustrate the effectiveness of the iterative regularization through computational experiments with data in dimension three. In comparison with earlier work, the Cauchy data do not suffer from eigenfrequencies in the configurations.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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