An inverse problem for the equation $\triangle u=-cu-d$

Vogelius, Michael

doi:10.5802/aif.1429

Cited by 26 publications

(35 citation statements)

References 9 publications

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“…Clearly disks have constant width. However there are plenty of smooth domains which have constant width but which are not disks: see [2], [5], [11] and [13]. Now we can state our main result.…”

Section: §1 Introductionmentioning

confidence: 54%

An Overdetermined Problem for an Elliptic Equation

Dalmasso¹

2010

Publ. Res. Inst. Math. Sci.

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We consider the following overdetermined boundary value problem: ∆u + λu + µ = 0 in Ω, u = 0 on ∂Ω and ∂u/∂n = c on ∂Ω, where c = 0, λ and µ are real constants and Ω ⊂ R 2 is a smooth bounded convex open set. We first show that it may happen that the problem has no solution. Then we study the existence of solutions for a wide class of domains.2010 Mathematics Subject Classification: 35J05, 35R30. Keywords: overdetermined elliptic boundary value problem, Schiffer property, Schiffer conjecture. §1. IntroductionLet Ω ⊂ R 2 be a smooth bounded simply-connected open set. We consider solutions of the following overdetermined elliptic boundary value problem:where λ, µ and c are real constants and ∂/∂n is the outward normal derivative.If c = 0 and µ = 0 (or equivalently µ = 1) we get as a special case Schiffer's problem (Yau [18, p. 688, problem 80]). If µ = 0 and c = 0 the problem was posed by Berenstein [1].In 1981 Williams [16] proved that if ∂Ω is Lipschitz and (1.1)-(1.3) has a solution for c = 0 and µ = 1, then ∂Ω is real analytic.

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Section: §1 Introductionmentioning

confidence: 54%

An Overdetermined Problem for an Elliptic Equation

Dalmasso¹

2010

Publ. Res. Inst. Math. Sci.

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“…A partial answer to this problem was first obtained by Vogelius in [11], and more recently we have also given a contribution [6].…”

Section: §1 Introductionmentioning

confidence: 99%

“…But even in the very particular case of an affine term the problem is difficult. It is well known that if Ω is a disk then such identification of (λ, µ) is completely impossible, even in the case where a sign is imposed on the right hand side of the equation: It is shown in [11] that there is a continuum of coefficient pairs (λ, µ λ ) ∈ R 2 , and therefore a continuum of affine functions, which give rise to the same normal derivative on the boundary. We refer the reader to paper [11] for a more detailed description of the problem in general and the difficulties encountered.…”

Section: §1 Introductionmentioning

confidence: 99%

“…We briefly describe the results obtained in [11]. In the case where a sign is imposed on the right hand side of the equation, Ω is a bounded, strictly convex C 3,α domain which is not a disk.…”

Section: §1 Introductionmentioning

confidence: 99%

“…Clearly disks have constant width. However there are plenty of smooth domains which have constant width but which are not disks: See [3], [8] and [11].…”

Section: §1 Introductionmentioning

confidence: 99%

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An Inverse Problem for an Elliptic Equation

Dalmasso¹

2004

Publ. Res. Inst. Math. Sci.

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We consider the following overdetermined boundary value problem: ∆u = −λu− µ in Ω, u = 0 on ∂Ω and ∂u ∂n = ψ on ∂Ω, where λ and µ are real constants and Ω is a smooth bounded planar domain. A very interesting problem is to examine whether one can identify the constants λ and µ from knowledge of the normal flux ∂u ∂n on ∂Ω corresponding to some nontrivial solution. It is well known that if Ω is a disk then such identification of (λ, µ) is completely impossible. Some partial results have already been obtained. The purpose of this paper is to extend and to improve these results. Moreover we also examine the interesting case where ψ is constant.

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Singularities almost always scatter: Regularity results for non‐scattering inhomogeneities

Cakoni

Vogelius

2023

Comm Pure Appl Math

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In this paper we examine necessary conditions for an inhomogeneity to be non‐scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary of the inhomogeneity. We examine broad classes of incident waves in both two and three dimensions. Our analysis is greatly influenced by the analysis carried out by Williams in order to establish that a domain, which does not possess the Pompeiu Property, has a real analytic boundary. That analysis, as well as ours, relies crucially on classical free boundary regularity results due to Kinderlehrer and Nirenberg, and Caffarelli.

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An inverse problem for the equation $\triangle u=-cu-d$

Cited by 26 publications

References 9 publications

An Overdetermined Problem for an Elliptic Equation

An Overdetermined Problem for an Elliptic Equation

An Inverse Problem for an Elliptic Equation

Singularities almost always scatter: Regularity results for non‐scattering inhomogeneities

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