Die erste Randwertaufgabe für geschlossene Bereiche bei der Gleichung $$\frac{{\partial ^2 z}}{{\partial x\partial y}} = f(x,y)$$

Huber, Adam S.

doi:10.1007/bf01699054

Cited by 19 publications

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The investigations carried out by Huber [2] and then by Bourgin and Duffin [3] resulted in the establishment of a condition for the violation of the uniqueness of a solution of the homogenous Dirichlet problem for the equation u tt -u xx = 0 in the rectangle { 0 ≤ t ≤ T; 0 ≤ x ≤ X }. It turned out that the uniqueness of a solution of this problem in classical spaces is violated if and only if the ratio T / X is rational.…”

Section: Historical Informationmentioning

confidence: 99%

On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

Burskii

Zhedanov

2006

Ukr Math J

View full text Add to dashboard Cite

UDC 517.95In a plane domain bounded by a biquadratic curve, we consider the problem of the uniqueness of a solution of the Dirichlet problem for the string equation. We show that this problem is equivalent to the classical Poncelet problem in projective geometry for two appropriate ellipses and also to the problem of the solvability of the Pell -Abel algebraic equation; some other related problems are also considered.In the present paper, we describe the relationships between properties of boundary-value problems for the string equation in a domain bounded by a biquadratic curve and some classical problems in algebra, geometry, and analysis that have recently been established by the authors. Boundary-Value Problems1.1. Historical Information. Investigations of ill-posed boundary-value problems date back to Hadamard [1], who first noted that the homogeneous Dirichlet problem for the string equation in a certain rectangle admits a nontrivial solution. The investigations carried out by Huber [2] and then by Bourgin and Duffin [3] resulted in the establishment of a condition for the violation of the uniqueness of a solution of the homogenous Dirichlet problem for the equation u tt -u xx = 0 in the rectangle { 0 ≤ t ≤ T; 0 ≤ x ≤ X }. It turned out that the uniqueness of a solution of this problem in classical spaces is violated if and only if the ratio T / X is rational. In the papers indicated, the method of Fourier expansion of a solution was used. In the works of Ptashnik and his disciples, this method was used for the investigation of properties of various boundary-value problems for partial differential equations in a rectangle and a parallelepiped [4]. In [5], John studied the problem of the violation of the uniqueness of a solution of the homogeneous Dirichlet problem for the string equation in a general bounded plane domain convex with respect to both families of characteristics in connection with a certain mapping of the boundary of the domain into itself (a so-called characteristic billiard, see below) used earlier in the mentioned works of Hadamard and Huber. The investigation of this relationship was continued by Aleksandryan and his disciples in the case of the Sobolev problem of oscillations of the surface of a liquid filling a cavity in a flying body [6 -8]. In [9], Arnol'd showed the relationship between this Dirichlet problem in an ellipse and the problem of small denominators in the context of the influence of the rate of approximation of a certain number associated with the problem by rational numbers on the smoothness of a solution. Some questions related to these problems were studied by Berezanskii [10], Zelenyak [11], Fokin [12], etc.Let us describe a characteristic billiard. Consider the Dirichlet problem for the string equation

show abstract

Section: Historical Informationmentioning

confidence: 99%

On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

Burskii

Zhedanov

2006

Ukr Math J

View full text Add to dashboard Cite

show abstract

“…Snadno se nahlédne, jak ukázal A. HUBEK [6], že Dirichletova úloha není obecně řešitelná, má-H oblast pouze dva nebo tři vrcholy (obr. 2b, c), neboť hodnoty funkce na jedné části hranice ur čují její hodnoty na druhé části.…”

Section: Dx* + Dy 2 + \ + Xf Dz % "unclassified

On a spectral problem and on Dirichlet problem for wave equation

Denčev¹

1960

Časopis Pěst. Mat.

View full text Add to dashboard Cite

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Časopis pro pěstování matematiky, roč. 85 (1960), Praha

show abstract

Small denominators. I. Mapping of the circumference onto itself

2009

Collected Works

View full text Add to dashboard Cite

Die erste Randwertaufgabe für geschlossene Bereiche bei der Gleichung $$\frac{{\partial ^2 z}}{{\partial x\partial y}} = f(x,y)$$

Cited by 19 publications

References 0 publications

On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

On a spectral problem and on Dirichlet problem for wave equation

Small denominators. I. Mapping of the circumference onto itself

Contact Info

Product

Resources

About