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.

Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media

1998 ◽  
Vol 128 (1) ◽  
pp. 173-192 ◽  
Author(s):  
Bo Zhang
Keyword(s):  
Unique Solution ◽  
Smooth Surface ◽  
Diffraction Problem ◽  
Propagation Speed ◽  
Radiation Condition ◽  
Resolvent Estimate ◽  
Limiting Absorption Principle ◽  
Low Frequencies ◽  
Unbounded Media

Consider the diffraction problem for perturbed acoustic propagators with perturbations decreasing slowly at infinity. The propagation speed is discontinuous at the interface of two unbounded media, and the interface may be an arbitrary and smooth surface locally. A Sommerfeld radiation condition is introduced for the acoustic propagator, and is then used to establish the limiting absorption principle and the resolvent estimate at low frequencies for such an operator. Furthermore, we prove the existence of a unique solution to the diffraction problem and the validity of the limiting amplitude principles for the acoustic propagator.

The three-dimensional inverse scattering problem for the Helmholtz equation

1973 ◽  
Vol 73 (3) ◽  
pp. 477-488 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Helmholtz Equation ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Radiation Condition ◽  
Polar Coordinates ◽  
Wave Number ◽  
Radial Coordinate ◽  
Cylindrical Polar Coordinates ◽  
Image Position

In this paper we are concerned with solutions of the three-dimensional Helmholtz equation which are of class C2 (i.e. regular) in the exterior of a bounded domain D. In cylindrical polar coordinates (r, z, φ) such solutions satisfy the equationin which we have dimensionalized the radial coordinate r so that the wave number is normalized to unity. If we further assume that u satisfies the Sommerfeld radiation conditionthen u may be regarded as being generated by volume sources, surface sources, or point singularities, all of which are contained in D.

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.

On transmission problems for wave propagation in two locally perturbed half-spaces

1994 ◽  
Vol 115 (3) ◽  
pp. 545-558 ◽  
Author(s):  
Bo Zhang
Keyword(s):  
Wave Propagation ◽  
Unique Solution ◽  
Radiation Condition ◽  
Absorption Method ◽  
Transmission Problems

AbstractIn this paper, we consider transmission problems for wave propagation in two inhomogeneous half-spaces with a locally perturbed hyperplane interface. A radiation condition is obtained for the problem in the framework of the spaces B and B*. It is then used, together with the spaces B, B* and the limiting absorption method to prove the existence of the unique solution to the problem.

Weighted estimates for solutions of a Sturm–Liouville equation in the space L1(ℝ)

2011 ◽  
Vol 141 (6) ◽  
pp. 1175-1206 ◽  
Author(s):  
N. A. Chernyavskaya ◽  
N. El-Natanov ◽  
L. A. Shuster
Keyword(s):  
Weight Function ◽  
Unique Solution ◽  
Liouville Equation ◽  
Positive Constant ◽  
Absolute Constant ◽  
A Priori ◽  
Weighted Estimates ◽  
Sturm Liouville ◽  
Image Position ◽  
The Space L1

We consider the equationwhereUnder these conditions, (1) is correctly solvable in L1(ℝ), i.e.(i) for any function f ∈ L1(ℝ), there exists a unique solution of (1), y ∈ L1(ℝ);(ii) there is an absolute constant c1 ∈ (0, ∞) such that the solution of (1),In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ Lloc1(ℝ) under which the solution of (1), y ∈ L1(ℝ), satisfies the estimatewhere c2 is some absolutely positive constant.

scholarly journals A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jinghuai Gao ◽  
Dehua Wang ◽  
Jigen Peng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Inverse Source Problem ◽  
A Posteriori ◽  
Modified Helmholtz Equation ◽  
Numerical Tests ◽  
Theoretical Frame ◽  
Set Up

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

scholarly journals The Neuroelectromagnetic Inverse Problem and the Zero Dipole Localization Error

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Rolando Grave de Peralta ◽  
Olaf Hauk ◽  
Sara L. Gonzalez
Keyword(s):  
Inverse Problem ◽  
Unique Solution ◽  
A Priori ◽  
Inverse Solution ◽  
Localization Error ◽  
A Priori Information ◽  
Dipole Localization ◽  
Inverse Solutions ◽  
Priori Information ◽  
Additional Constraints

A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP) is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA) attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.

Some Radical Properties of Jordan Matrix Rings

1979 ◽  
Vol 31 (1) ◽  
pp. 189-196
Author(s):  
Michael Rich
Keyword(s):  
Unique Solution ◽  
Identity Element ◽  
Alternative Ring ◽  
Matrix Rings ◽  
Jordan Ring ◽  
Jordan Matrix ◽  
Image Position ◽  
Canonical Involution

Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].

Discounted branching random walks

1985 ◽  
Vol 17 (01) ◽  
pp. 53-66
Author(s):  
K. B. Athreya
Keyword(s):  
Functional Equation ◽  
Random Walks ◽  
Unique Solution ◽  
Log P ◽  
Branching Random Walks ◽  
Probabilistic Construction ◽  
Image Position

Let F(·) be a c.d.f. on [0,∞), f(s) = ∑∞ 0 pjsi a p.g.f. with p 0 = 0, < 1 < m = Σj p j < ∞ and 1 < ρ <∞. For the functional equation for a c.d.f. H(·) on [0,∞] we establish that if 1 – F(x) = O(x –θ ) for some θ > α =(log m)/(log p) there exists a unique solution H(·) to (∗) in the class C of c.d.f.’s satisfying 1 – H(x) = o(x –α ). We give a probabilistic construction of this solution via branching random walks with discounting. We also show non-uniqueness if the condition 1 – H(x) = o(x –α ) is relaxed.

scholarly journals A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

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