TY - JOUR

T1 - On the Stability of the Spectrum in the Pompeiu Problem

AU - Agranovsky, M. L.

PY - 1993

Y1 - 1993

N2 - Let Ω be a Jordan domain in the complex plane with smooth boundary. We call the Pompeiu spectrum σ(Ω) the set of all λ such that there exists a nontrivial solution of overdetermined Dirichlet-Neumann boundary-value problem. [formula presented] (ν is the normal vector to the boundary ∂Ω). Let Ωt, t ∈ [0, T) be a family of Jordan domains in the complex plane with real-analytic boundaries. Suppose that Ωt analytically depends on the parameter t and Ω0 = {z ∈ C : |z| ≤ 1}. It is proved that if there exists a real-analytic function λ(t), such that λ(t) ∈ σ(Ωt), t ∈ [0, T), then all domains Ωt are discs.

AB - Let Ω be a Jordan domain in the complex plane with smooth boundary. We call the Pompeiu spectrum σ(Ω) the set of all λ such that there exists a nontrivial solution of overdetermined Dirichlet-Neumann boundary-value problem. [formula presented] (ν is the normal vector to the boundary ∂Ω). Let Ωt, t ∈ [0, T) be a family of Jordan domains in the complex plane with real-analytic boundaries. Suppose that Ωt analytically depends on the parameter t and Ω0 = {z ∈ C : |z| ≤ 1}. It is proved that if there exists a real-analytic function λ(t), such that λ(t) ∈ σ(Ωt), t ∈ [0, T), then all domains Ωt are discs.

UR - http://www.scopus.com/inward/record.url?scp=38248999361&partnerID=8YFLogxK

U2 - 10.1006/jmaa.1993.1305

DO - 10.1006/jmaa.1993.1305

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AN - SCOPUS:38248999361

SN - 0022-247X

VL - 178

SP - 269

EP - 279

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -