July  2013, 12(4): 1587-1633. doi: 10.3934/cpaa.2013.12.1587

On Dirichlet, Poncelet and Abel problems

1. 

Institute of Applied Mathematics, Donetsk, 83114, Ukraine

2. 

Donetsk Institute for Physics and Technology, Donetsk, 83114, Ukraine

Received  June 2011 Revised  June 2011 Published  November 2012

We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary biquadratic algebraic curve is considered. It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem. Among them are the solvability problem of the algebraic Pell-Abel equation and the indeterminacy problem of a new moment problem that generalizes the well-known trigonometrical moment problem. Solvability criteria of the above-mentioned problems can be represented in form $\theta\in Q$ where number $\theta=m/n$ is built by means of data for a problem to solve. We also demonstrate close relations of the above-mentioned problems to such problems of modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Padé interpolation.
Citation: Vladimir P. Burskii, Alexei S. Zhedanov. On Dirichlet, Poncelet and Abel problems. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1587-1633. doi: 10.3934/cpaa.2013.12.1587
References:
[1]

N. I. Akhiezer, "Elements of the Theory of Elliptic Functions," 2nd edition, Nauka, Moscow, 1970. Translations Math. Monographs, 79, AMS, Providence, 1990.

[2]

N. I. Akhiezer, "Lectures on Approximation Theory," Nauka, M., 1965 (In Russian).

[3]

R. A. Alexandrjan, On the Dirichlet problem for the string equation and on completeness of a system of function in a disk, Doklady AN USSR., 73 (1950) (In Russian).

[4]

R. A. Alexandrjan, Spectral properties of operators generated by systems differential equations of Sobolev type, Trudy Mosc. Math. Obshchestva, 9 (1960), 455-505 (In Russian).

[5]

G. S. Akopyan and R. A. Aleksandryan, On the completeness of a system of eigen- and vector-polynomials of a linear differential operator pencil in ellipsoidal domains, Dokl. Akad. Nauk Arm. SSR, 86 (1988), 147-152 (In Russian).

[6]

V. I. Arnold, Small demominators. I, Izvestija AN SSSR, serija matematicheskaja, 25 (1961), 21-86.

[7]

R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), 1-55.

[8]

G. A. Baker and P. Graves-Morris, Padé approximants. Parts I and II, in "Encyclopedia of Mathematics and its Applications," 13, 14, Addison-Wesley Publishing Co., Reading, Mass., 1981.

[9]

H. Bateman and A. Erdélyi, "Higher Transcendental Functions," 3, McGraw-Hill, New York, 1955, Bateman manuscript project.

[10]

R. Baxter, "Exactly Solvable Models in Statistical Mechanics," London, Academic Press, 1982.

[11]

M. V. Beloglyadov, On the Dirichlet problem for the vibrating string equation in domain with a bi-quadratic boundary, Trudy IAMM NASU, 14 (2007), 14-29 (In Russian).

[12]

E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, "Algebro-geometrical Approach to Non-linear Integrable Equations," Springer Series in Nonlinear Dynamics, XII, Berlin: Springer-Verlag, 1994.

[13]

E. D. Belokolos and V. Z. Enolskii, Reduction of Abelian functions and algebraically integrable systems, Journal of Mathematical Sciences, Part I: 106 (2001), 3395-3486; Part II: 108 (2002), 295-374.

[14]

Yu. M. Berezanskii, "Expansion by Eigenfunctions of Selfadjoint Operators," Naukova Dumka, Kiev, 1965 (In Russian).

[15]

M. Berger, "Géométrie," CEDIC, Paris, 1978.

[16]

M. Berger, "Geometry Revealed, A Jacob's Ladder to Modern Higher Geometry," Springer, 2010.

[17]

D. Bourgin and R. Duffin, The Dirlchlet problem for the vibrating string equations, Bull. Am. Math. Soc., 45 (1939), 851-858.

[18]

A. B. Bogatyrev, Chebyshev representation for rational function, Sbornik Mathematics, 201 (2010), 1579-1598.

[19]

V. P. Burskii, On solution uniqueness of some boundary value problems for differential equations in domains with algebraic boundary, Ukr. Math. Journal, 45 (1993), 993-1003.

[20]

V. P. Burskii, On boundary value problems for differential equations with constant coefficients in a plane domain and a moment problem, Ukr. Math. Journal, 48 (1993), 1659-1668.

[21]

V. P. Burskii, "Investigation Methods of Boundary Value Problems for General Differential Equations," Kiev, Naukova dumka, 2002 (In Russian).

[22]

V. P. Burskii and A. S. Zhedanov, On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems, Ukr. Math. Journal, 58 (2006), 487-504.

[23]

V. P. Burskii and A. S. Zhedanov, Dirichlet and Neumann problems for string equation, Poncelet problem and Pell-Abel equation, Symmetry, Integrability and Geometry: Methods and Applications, 2006, V. 2, rec.No: 041.

[24]

V. P. Burskii and A. S. Zhedanov, Boundary value problems for string equation, Poncelet problem, and Pell-Abel equation: links and relations, Contemporary Mathematics. Fundamental Directions, 16 (2006), pp. 59.

[25]

A. A. Chernikov, R. Z. Sagdeev and G. M. Zaslavsky, Stochastic webs. Progress in chaotic dynamics, Phys. D, 33 (1988), 6576.

[26]

O. Egecioglu and C. K. Koc, A fast algorithm for rational interpolation via orthogonal polynomials, Math. Comp., 53 (1989), 249-264.

[27]

A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, "Higher Transcendental Functions. I," McGraw-Hill, New York, 1953 Bateman manuscript project.

[28]

M. V. Fokin, Solvability of the Dirichlet problem for the string equation, Doklady AN SSSR, 272 (1983), 801-805 (in Russian).

[29]

J. P. Francoise and O. Ragnisco, An iterative process on quartics and integrable symplectic maps, in "Symmetries and Integrability of Difference Equations," P. A. Clarkson and F. W. Nijhoff eds., Cambridge University Press, 1998.

[30]

Ya. I. Granovskii and A. S. Zhedanov, Integrability of the classical $XY$-chain, Pis'ma to Zh. Exp. Theor. Phys., 44 (1986), 237-239 (Russian).

[31]

P. Griffiths and J. Harris, Poncelet theorem in space, Comment. Math. Helvetici, 52 (1977), 145-160.

[32]

P. Griffiths and J. Harris, On a Cayley's explicit solution to Poncelet's porism, Enseign. Math., 24 (1978), 31-40.

[33]

P. Griffiths and J. Harris, "Principles of Algebraic Geometry," v. I, II, John Wiley and Sons, Inc., 1978.

[34]

J. Hadamard, Equations aux derivees partielles, L 'Enseignment Mathematique, 36 (1936), 25-42.

[35]

G. H. Halphen, "Traité des Fonctions Elliptiques et de Leures Applications," II, Gauthier朧illar, Paris, 1886.

[36]

A. Huber, Erste Randwertaufgabe fur geschlossene Bereiche bei der Gleichung $U_{xy}=f(x,y)$, Monatshefte für Mathematik und Physik, 39 (1932), 79-100.

[37]

E. L. Ince, Ordinary differential equations .

[38]

F. John, The Dirichlet problem for a hyperbolic equation, Am. J. Math., 63 (1941), 141-154.

[39]

A. Iatrou and J. A. G. Roberts, Integrable mappings of the plane preseving biquadratic invariants curves II, Nonlinearity, 15 (2002), 459-489.

[40]

A. Iatrou, Real Jacobian elliptic function parameterization for a genuinely asymmetric biquadratic curve, arXiv: nlin. SI/0306051 v1 25, Jun 2003.

[41]

S. M. Kerawala, Poncelet Porism in Two Circles, Bull. Calcutta Math. Soc., 39 (1947), 85-105.

[42]

J. L. King, Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609-628.

[43]

M. M. Lavrent'ev, Mathematical problems of tomography and hyperbolic mappings, Sib. Math. J., 42 (2001), 916-925.

[44]

V. F. Lazutkin, "KAM Theory and Semiclassical Approximation to Eigenfunctions," Springer Verlag, Berlin, Hei-delberg, New York (1993), Ergebnisse der Mathematik und ihrer Grenzgebiete: 3. Folge, Band 24.

[45]

A. Magnus, Rational interpolation to solutions of Riccati difference equations on elliptic lattices, Preprint http://www.math.ucl.ac.be/membres/magnus/.

[46]

V. A. Malyshev, Abel equation, Algebra and Analysis, 13 (2001), 1-55 (In Russian).

[47]

J. Meinguet, On the solubility of the Cauchy interpolation problem, Approximation Theory (Proc. Sympos., Lancaster, 1969), 137-163. Academic Press, London.

[48]

L. J. Mordell, "Diophantine Equations," Academic Press, 1969.

[49]

Z. Nitecki, "Differentiable Dinamics," MIT Press, Cambridge Mass - London, 1971.

[50]

S. G. Ovsepjan, On ergodisity of continuous automorphizms and solution uniqueness of the Dirichlet problem for the string equation. II, Izv. AN Arm. SSR., 2 (1967), 195-209.

[51]

B. Yo. Ptashnik, Incorrect boundary value problems for differential equations with partual derivatives, Kiev, Naukova dumka, 1984 (In Russian).

[52]

J. F. Ritt, Periodic functions with a multiplication theorem, Trans. Amer. Math. Soc., 23 (1922), 16-25.

[53]

I. J. Schoenberg, On Jacobi-Bertrand's proof of a theorem of Poncelet, in "Studies in Pure Mathematics, To the Memory of Paul Turan," 623-627, Birkhuser, Basel, 1983.

[54]

L. M. Sodin and P. M. Yuditskii, Functions least deviating from zero on closed sets of real axis, Algebra and Analysis, 4, 1-61 (In Russian).

[55]

V. Spiridonov and A. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions, Commun. Math. Phys., 210 (2000), 49-83.

[56]

V. P. Spiridonov and A. S. Zhedanov, To the theory of biorthogonal rational functions, RIMS Kokyuroku, 1302 (2003), 172-192.

[57]

V. Spiridonov and A. Zhedanov, Elliptic grids, rational functions, and Padé interpolation, Ramanujan J., 13 (2007), 285-310.

[58]

T. Stieltjes, Sur l'équation d'Euler, Bul.Sci.Math., Paris, sér. 2, 12 (1888), 222-227.

[59]

A. A. Telitsyna, The Dirichlet problem for wave equation in plane domain with biquadratic boundary, Trudy IAMM NASU, 13 (2007), 198-210 (In Russian).

[60]

M. Toda, "Theory of Nonlinear Lattices," Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin, 1989.

[61]

A. P. Veselov, Integrable systems with discrete time and difference operators, Functional Analysis and its Applications, 22 (1988), 1-13 (Russian).

[62]

A. P. Veselov, Integrable maps, Russian Math. Surveys, 46 (1991), 1-51.

[63]

L. Vinet and A. Zhedanov, Generalized Bochner theorem: characterization of the Askey-Wilson polynomials, J. Comput. Appl. Math., 211 (2008), 45-56.

[64]

T. I. Zelenjak, Selected topics of quality theory of equations with partial derivatives, Novosibirsk: NGU, 1970 (In Russian).

[65]

A. Zhedanov, Biorthogonal rational functions and the generalized eigenvalue problem, J. Approx. Theory, 101 (1999), 303-329.

[66]

A. Zhedanov, Padé interpolation table and biorthogonal rational functions, Proceedings of the Workshop on Elliptic Integrable Systems November 8-11, 2004, Kyoto, Rokko Lectures in Mathematics, No. 18, 323-363. http://www.math.kobe-u.ac.jp/publications/rlm18/20.pdf.

[67]

http:, //en.wikipedia.org/wiki/Archimedes .

show all references

References:
[1]

N. I. Akhiezer, "Elements of the Theory of Elliptic Functions," 2nd edition, Nauka, Moscow, 1970. Translations Math. Monographs, 79, AMS, Providence, 1990.

[2]

N. I. Akhiezer, "Lectures on Approximation Theory," Nauka, M., 1965 (In Russian).

[3]

R. A. Alexandrjan, On the Dirichlet problem for the string equation and on completeness of a system of function in a disk, Doklady AN USSR., 73 (1950) (In Russian).

[4]

R. A. Alexandrjan, Spectral properties of operators generated by systems differential equations of Sobolev type, Trudy Mosc. Math. Obshchestva, 9 (1960), 455-505 (In Russian).

[5]

G. S. Akopyan and R. A. Aleksandryan, On the completeness of a system of eigen- and vector-polynomials of a linear differential operator pencil in ellipsoidal domains, Dokl. Akad. Nauk Arm. SSR, 86 (1988), 147-152 (In Russian).

[6]

V. I. Arnold, Small demominators. I, Izvestija AN SSSR, serija matematicheskaja, 25 (1961), 21-86.

[7]

R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), 1-55.

[8]

G. A. Baker and P. Graves-Morris, Padé approximants. Parts I and II, in "Encyclopedia of Mathematics and its Applications," 13, 14, Addison-Wesley Publishing Co., Reading, Mass., 1981.

[9]

H. Bateman and A. Erdélyi, "Higher Transcendental Functions," 3, McGraw-Hill, New York, 1955, Bateman manuscript project.

[10]

R. Baxter, "Exactly Solvable Models in Statistical Mechanics," London, Academic Press, 1982.

[11]

M. V. Beloglyadov, On the Dirichlet problem for the vibrating string equation in domain with a bi-quadratic boundary, Trudy IAMM NASU, 14 (2007), 14-29 (In Russian).

[12]

E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, "Algebro-geometrical Approach to Non-linear Integrable Equations," Springer Series in Nonlinear Dynamics, XII, Berlin: Springer-Verlag, 1994.

[13]

E. D. Belokolos and V. Z. Enolskii, Reduction of Abelian functions and algebraically integrable systems, Journal of Mathematical Sciences, Part I: 106 (2001), 3395-3486; Part II: 108 (2002), 295-374.

[14]

Yu. M. Berezanskii, "Expansion by Eigenfunctions of Selfadjoint Operators," Naukova Dumka, Kiev, 1965 (In Russian).

[15]

M. Berger, "Géométrie," CEDIC, Paris, 1978.

[16]

M. Berger, "Geometry Revealed, A Jacob's Ladder to Modern Higher Geometry," Springer, 2010.

[17]

D. Bourgin and R. Duffin, The Dirlchlet problem for the vibrating string equations, Bull. Am. Math. Soc., 45 (1939), 851-858.

[18]

A. B. Bogatyrev, Chebyshev representation for rational function, Sbornik Mathematics, 201 (2010), 1579-1598.

[19]

V. P. Burskii, On solution uniqueness of some boundary value problems for differential equations in domains with algebraic boundary, Ukr. Math. Journal, 45 (1993), 993-1003.

[20]

V. P. Burskii, On boundary value problems for differential equations with constant coefficients in a plane domain and a moment problem, Ukr. Math. Journal, 48 (1993), 1659-1668.

[21]

V. P. Burskii, "Investigation Methods of Boundary Value Problems for General Differential Equations," Kiev, Naukova dumka, 2002 (In Russian).

[22]

V. P. Burskii and A. S. Zhedanov, On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems, Ukr. Math. Journal, 58 (2006), 487-504.

[23]

V. P. Burskii and A. S. Zhedanov, Dirichlet and Neumann problems for string equation, Poncelet problem and Pell-Abel equation, Symmetry, Integrability and Geometry: Methods and Applications, 2006, V. 2, rec.No: 041.

[24]

V. P. Burskii and A. S. Zhedanov, Boundary value problems for string equation, Poncelet problem, and Pell-Abel equation: links and relations, Contemporary Mathematics. Fundamental Directions, 16 (2006), pp. 59.

[25]

A. A. Chernikov, R. Z. Sagdeev and G. M. Zaslavsky, Stochastic webs. Progress in chaotic dynamics, Phys. D, 33 (1988), 6576.

[26]

O. Egecioglu and C. K. Koc, A fast algorithm for rational interpolation via orthogonal polynomials, Math. Comp., 53 (1989), 249-264.

[27]

A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, "Higher Transcendental Functions. I," McGraw-Hill, New York, 1953 Bateman manuscript project.

[28]

M. V. Fokin, Solvability of the Dirichlet problem for the string equation, Doklady AN SSSR, 272 (1983), 801-805 (in Russian).

[29]

J. P. Francoise and O. Ragnisco, An iterative process on quartics and integrable symplectic maps, in "Symmetries and Integrability of Difference Equations," P. A. Clarkson and F. W. Nijhoff eds., Cambridge University Press, 1998.

[30]

Ya. I. Granovskii and A. S. Zhedanov, Integrability of the classical $XY$-chain, Pis'ma to Zh. Exp. Theor. Phys., 44 (1986), 237-239 (Russian).

[31]

P. Griffiths and J. Harris, Poncelet theorem in space, Comment. Math. Helvetici, 52 (1977), 145-160.

[32]

P. Griffiths and J. Harris, On a Cayley's explicit solution to Poncelet's porism, Enseign. Math., 24 (1978), 31-40.

[33]

P. Griffiths and J. Harris, "Principles of Algebraic Geometry," v. I, II, John Wiley and Sons, Inc., 1978.

[34]

J. Hadamard, Equations aux derivees partielles, L 'Enseignment Mathematique, 36 (1936), 25-42.

[35]

G. H. Halphen, "Traité des Fonctions Elliptiques et de Leures Applications," II, Gauthier朧illar, Paris, 1886.

[36]

A. Huber, Erste Randwertaufgabe fur geschlossene Bereiche bei der Gleichung $U_{xy}=f(x,y)$, Monatshefte für Mathematik und Physik, 39 (1932), 79-100.

[37]

E. L. Ince, Ordinary differential equations .

[38]

F. John, The Dirichlet problem for a hyperbolic equation, Am. J. Math., 63 (1941), 141-154.

[39]

A. Iatrou and J. A. G. Roberts, Integrable mappings of the plane preseving biquadratic invariants curves II, Nonlinearity, 15 (2002), 459-489.

[40]

A. Iatrou, Real Jacobian elliptic function parameterization for a genuinely asymmetric biquadratic curve, arXiv: nlin. SI/0306051 v1 25, Jun 2003.

[41]

S. M. Kerawala, Poncelet Porism in Two Circles, Bull. Calcutta Math. Soc., 39 (1947), 85-105.

[42]

J. L. King, Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609-628.

[43]

M. M. Lavrent'ev, Mathematical problems of tomography and hyperbolic mappings, Sib. Math. J., 42 (2001), 916-925.

[44]

V. F. Lazutkin, "KAM Theory and Semiclassical Approximation to Eigenfunctions," Springer Verlag, Berlin, Hei-delberg, New York (1993), Ergebnisse der Mathematik und ihrer Grenzgebiete: 3. Folge, Band 24.

[45]

A. Magnus, Rational interpolation to solutions of Riccati difference equations on elliptic lattices, Preprint http://www.math.ucl.ac.be/membres/magnus/.

[46]

V. A. Malyshev, Abel equation, Algebra and Analysis, 13 (2001), 1-55 (In Russian).

[47]

J. Meinguet, On the solubility of the Cauchy interpolation problem, Approximation Theory (Proc. Sympos., Lancaster, 1969), 137-163. Academic Press, London.

[48]

L. J. Mordell, "Diophantine Equations," Academic Press, 1969.

[49]

Z. Nitecki, "Differentiable Dinamics," MIT Press, Cambridge Mass - London, 1971.

[50]

S. G. Ovsepjan, On ergodisity of continuous automorphizms and solution uniqueness of the Dirichlet problem for the string equation. II, Izv. AN Arm. SSR., 2 (1967), 195-209.

[51]

B. Yo. Ptashnik, Incorrect boundary value problems for differential equations with partual derivatives, Kiev, Naukova dumka, 1984 (In Russian).

[52]

J. F. Ritt, Periodic functions with a multiplication theorem, Trans. Amer. Math. Soc., 23 (1922), 16-25.

[53]

I. J. Schoenberg, On Jacobi-Bertrand's proof of a theorem of Poncelet, in "Studies in Pure Mathematics, To the Memory of Paul Turan," 623-627, Birkhuser, Basel, 1983.

[54]

L. M. Sodin and P. M. Yuditskii, Functions least deviating from zero on closed sets of real axis, Algebra and Analysis, 4, 1-61 (In Russian).

[55]

V. Spiridonov and A. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions, Commun. Math. Phys., 210 (2000), 49-83.

[56]

V. P. Spiridonov and A. S. Zhedanov, To the theory of biorthogonal rational functions, RIMS Kokyuroku, 1302 (2003), 172-192.

[57]

V. Spiridonov and A. Zhedanov, Elliptic grids, rational functions, and Padé interpolation, Ramanujan J., 13 (2007), 285-310.

[58]

T. Stieltjes, Sur l'équation d'Euler, Bul.Sci.Math., Paris, sér. 2, 12 (1888), 222-227.

[59]

A. A. Telitsyna, The Dirichlet problem for wave equation in plane domain with biquadratic boundary, Trudy IAMM NASU, 13 (2007), 198-210 (In Russian).

[60]

M. Toda, "Theory of Nonlinear Lattices," Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin, 1989.

[61]

A. P. Veselov, Integrable systems with discrete time and difference operators, Functional Analysis and its Applications, 22 (1988), 1-13 (Russian).

[62]

A. P. Veselov, Integrable maps, Russian Math. Surveys, 46 (1991), 1-51.

[63]

L. Vinet and A. Zhedanov, Generalized Bochner theorem: characterization of the Askey-Wilson polynomials, J. Comput. Appl. Math., 211 (2008), 45-56.

[64]

T. I. Zelenjak, Selected topics of quality theory of equations with partial derivatives, Novosibirsk: NGU, 1970 (In Russian).

[65]

A. Zhedanov, Biorthogonal rational functions and the generalized eigenvalue problem, J. Approx. Theory, 101 (1999), 303-329.

[66]

A. Zhedanov, Padé interpolation table and biorthogonal rational functions, Proceedings of the Workshop on Elliptic Integrable Systems November 8-11, 2004, Kyoto, Rokko Lectures in Mathematics, No. 18, 323-363. http://www.math.kobe-u.ac.jp/publications/rlm18/20.pdf.

[67]

http:, //en.wikipedia.org/wiki/Archimedes .

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