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On the Unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics. / Kopylov, Anatolii Pavlovich.

In: Сибирские электронные математические известия, Vol. 14, 01.01.2017, p. 59-72.

Research output: Contribution to journal › Article › peer-review

Harvard

Kopylov, AP 2017, 'On the Unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics', Сибирские электронные математические известия, vol. 14, pp. 59-72. https://doi.org/10.17377/semi.2017.14.008

APA

Kopylov, A. P. (2017). On the Unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics. Сибирские электронные математические известия, 14, 59-72. https://doi.org/10.17377/semi.2017.14.008

Vancouver

Kopylov AP. On the Unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics. Сибирские электронные математические известия. 2017 Jan 1;14:59-72. doi: 10.17377/semi.2017.14.008

Author

Kopylov, Anatolii Pavlovich. / On the Unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics. In: Сибирские электронные математические известия. 2017 ; Vol. 14. pp. 59-72.

BibTeX

@article{b6839881de1c435cba96300ea58f26cb,
title = "On the Unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics",
abstract = "The article contains the results of the author's recent investigations of the rigidity problems of domains in Euclidean spaces undertaken for the development of a new approach to the classical problem about the unique determination of bounded closed convex surfaces. We prove a complete characterization of a plane domain U with smooth boundary (i.e., the Euclidean boundary frU of U is a one-dimensional manifold of class C1 without boundary) that is uniquely determined in the class of domains in ℝ2 with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics. In the case where U is bounded, a necessary and sufficient condition for the unique determination of the type under consideration in the class of all bounded plane domains with smooth boundary is the convexity of U. If U is unbounded then its unique determination in the class of all plane domains with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics is equivalent to its strict convexity.",
keywords = "Intrinsic metric, Local isometry of the boundaries, Relative metric of the boundary, Strict convexity, strict convexity, SUFFICIENT CONDITIONS, relative metric of the boundary, local isometry of the boundaries, intrinsic metric",
author = "Kopylov, {Anatolii Pavlovich}",
year = "2017",
month = jan,
day = "1",
doi = "10.17377/semi.2017.14.008",
language = "English",
volume = "14",
pages = "59--72",
journal = "Сибирские электронные математические известия",
issn = "1813-3304",
publisher = "Sobolev Institute of Mathematics",

}

RIS

TY - JOUR

T1 - On the Unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics

AU - Kopylov, Anatolii Pavlovich

PY - 2017/1/1

Y1 - 2017/1/1

N2 - The article contains the results of the author's recent investigations of the rigidity problems of domains in Euclidean spaces undertaken for the development of a new approach to the classical problem about the unique determination of bounded closed convex surfaces. We prove a complete characterization of a plane domain U with smooth boundary (i.e., the Euclidean boundary frU of U is a one-dimensional manifold of class C1 without boundary) that is uniquely determined in the class of domains in ℝ2 with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics. In the case where U is bounded, a necessary and sufficient condition for the unique determination of the type under consideration in the class of all bounded plane domains with smooth boundary is the convexity of U. If U is unbounded then its unique determination in the class of all plane domains with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics is equivalent to its strict convexity.

AB - The article contains the results of the author's recent investigations of the rigidity problems of domains in Euclidean spaces undertaken for the development of a new approach to the classical problem about the unique determination of bounded closed convex surfaces. We prove a complete characterization of a plane domain U with smooth boundary (i.e., the Euclidean boundary frU of U is a one-dimensional manifold of class C1 without boundary) that is uniquely determined in the class of domains in ℝ2 with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics. In the case where U is bounded, a necessary and sufficient condition for the unique determination of the type under consideration in the class of all bounded plane domains with smooth boundary is the convexity of U. If U is unbounded then its unique determination in the class of all plane domains with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics is equivalent to its strict convexity.

KW - Intrinsic metric

KW - Local isometry of the boundaries

KW - Relative metric of the boundary

KW - Strict convexity

KW - strict convexity

KW - SUFFICIENT CONDITIONS

KW - relative metric of the boundary

KW - local isometry of the boundaries

KW - intrinsic metric

UR - http://www.scopus.com/inward/record.url?scp=85074616185&partnerID=8YFLogxK

U2 - 10.17377/semi.2017.14.008

DO - 10.17377/semi.2017.14.008

M3 - Article

AN - SCOPUS:85074616185

VL - 14

SP - 59

EP - 72

JO - Сибирские электронные математические известия

JF - Сибирские электронные математические известия

SN - 1813-3304

ER -

ID: 22320188