Reflection principles for linear elliptic second order partial differential equations with constant coefflcients

Diaz, J. B.; Ludford, G. S. S.

doi:10.1007/bf02410764

Cited by 21 publications

(27 citation statements)

References 4 publications

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“…(c) In the case of the Robin boundary condition Bv := ∂ ν v + q v = 0 on Γ for some constant q ∈ C, we can choose (see [8] for the corresponding reflection principle)…”

Section: (B) Under the Neumann Boundary Conditionmentioning

confidence: 99%

Uniqueness and factorization method for inverse elastic scattering with a single incoming wave

Elschner

2019

Inverse Problems

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The first part of this paper is concerned with the uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern corresponding to a single incident plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern.

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“…(c) In the case of the Robin boundary condition Bv := ∂ ν v + q v = 0 on Γ for some constant q ∈ C, we can choose (see [8] for the corresponding reflection principle)…”

Section: (B) Under the Neumann Boundary Conditionmentioning

confidence: 99%

Uniqueness and factorization method for inverse elastic scattering with a single incoming wave

Elschner

2019

Inverse Problems

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“…The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ( [4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius.…”

Section: Introductionmentioning

confidence: 80%

“…" ^( M ) (3)(4)(5)(6)(7)(8) n =0m=-n \OJ forr^a,Q$,9^Tt,0<L<l>£2n,v/<i have (cf. [7, chapter 3], or [8, chapter 7]) that the singular points of h(r, 9, <f>) can be determined from a knowledge of the singular points of the analytic function of two complex variables [7].…”

Section: Kr9)= £ T A^kka)!-)mentioning

confidence: 99%

A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering problem

Colton¹

1977

Glasgow Math. J.

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. Introduction. A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space U 3 . We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).

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“…Ludford [5]. They considered a specific case, when Γ is a hyperplane and the coefficients α and β in (1.6) are constants.…”

Section: Introductionmentioning

confidence: 99%

The Schwarz reflection principle for harmonic functions in R2 subject to the Robin condition

Belinskiy

Savina

2008

Journal of Mathematical Analysis and Applications

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A non-local reflection formula for harmonic functions in R 2 satisfying the Robin boundary condition, α∂ n u +βu = 0, on a real-analytic curve is suggested. This formula generalizes the celebrated Schwarz reflection principle. It is also shown how the obtained formula reduces to well-known point-to-point reflection laws corresponding the Dirichlet and Neumann boundary conditions when one of the coefficients, α or β, vanishes.

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Reflection principles for linear elliptic second order partial differential equations with constant coefflcients

Cited by 21 publications

References 4 publications

Uniqueness and factorization method for inverse elastic scattering with a single incoming wave

Uniqueness and factorization method for inverse elastic scattering with a single incoming wave

A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering problem

The Schwarz reflection principle for harmonic functions in R2 subject to the Robin condition

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