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Uniqueness for elliptic problems with Höldertype dependence on the solution
On Dirichlet, Poncelet and Abel problems
1.  Institute of Applied Mathematics, Donetsk, 83114, Ukraine 
2.  Donetsk Institute for Physics and Technology, Donetsk, 83114, Ukraine 
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), 455505 (In Russian). 
[5] 
G. S. Akopyan and R. A. Aleksandryan, On the completeness of a system of eigen and vectorpolynomials of a linear differential operator pencil in ellipsoidal domains, Dokl. Akad. Nauk Arm. SSR, 86 (1988), 147152 (In Russian). 
[6] 
V. I. Arnold, Small demominators. I, Izvestija AN SSSR, serija matematicheskaja, 25 (1961), 2186. 
[7] 
R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), 155. 
[8] 
G. A. Baker and P. GravesMorris, Padé approximants. Parts I and II, in "Encyclopedia of Mathematics and its Applications," 13, 14, AddisonWesley Publishing Co., Reading, Mass., 1981. 
[9] 
H. Bateman and A. Erdélyi, "Higher Transcendental Functions," 3, McGrawHill, 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 biquadratic boundary, Trudy IAMM NASU, 14 (2007), 1429 (In Russian). 
[12] 
E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, "Algebrogeometrical Approach to Nonlinear Integrable Equations," Springer Series in Nonlinear Dynamics, XII, Berlin: SpringerVerlag, 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), 33953486; Part II: 108 (2002), 295374. 
[14] 
Yu. M. Berezanskii, "Expansion by Eigenfunctions of Selfadjoint Operators," Naukova Dumka, Kiev, 1965 (In Russian). 
[15]  
[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), 851858. 
[18] 
A. B. Bogatyrev, Chebyshev representation for rational function, Sbornik Mathematics, 201 (2010), 15791598. 
[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), 9931003. 
[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), 16591668. 
[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, PellAbel equation, and some other related problems, Ukr. Math. Journal, 58 (2006), 487504. 
[23] 
V. P. Burskii and A. S. Zhedanov, Dirichlet and Neumann problems for string equation, Poncelet problem and PellAbel 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 PellAbel 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), 249264. 
[27] 
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, "Higher Transcendental Functions. I," McGrawHill, New York, 1953 Bateman manuscript project. 
[28] 
M. V. Fokin, Solvability of the Dirichlet problem for the string equation, Doklady AN SSSR, 272 (1983), 801805 (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), 237239 (Russian). 
[31] 
P. Griffiths and J. Harris, Poncelet theorem in space, Comment. Math. Helvetici, 52 (1977), 145160. 
[32] 
P. Griffiths and J. Harris, On a Cayley's explicit solution to Poncelet's porism, Enseign. Math., 24 (1978), 3140. 
[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), 2542. 
[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), 79100. 
[37]  
[38] 
F. John, The Dirichlet problem for a hyperbolic equation, Am. J. Math., 63 (1941), 141154. 
[39] 
A. Iatrou and J. A. G. Roberts, Integrable mappings of the plane preseving biquadratic invariants curves II, Nonlinearity, 15 (2002), 459489. 
[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), 85105. 
[42] 
J. L. King, Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609628. 
[43] 
M. M. Lavrent'ev, Mathematical problems of tomography and hyperbolic mappings, Sib. Math. J., 42 (2001), 916925. 
[44] 
V. F. Lazutkin, "KAM Theory and Semiclassical Approximation to Eigenfunctions," Springer Verlag, Berlin, Heidelberg, 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), 155 (In Russian). 
[47] 
J. Meinguet, On the solubility of the Cauchy interpolation problem, Approximation Theory (Proc. Sympos., Lancaster, 1969), 137163. 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), 195209. 
[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), 1625. 
[53] 
I. J. Schoenberg, On JacobiBertrand's proof of a theorem of Poncelet, in "Studies in Pure Mathematics, To the Memory of Paul Turan," 623627, 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. 
[55] 
V. Spiridonov and A. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions, Commun. Math. Phys., 210 (2000), 4983. 
[56] 
V. P. Spiridonov and A. S. Zhedanov, To the theory of biorthogonal rational functions, RIMS Kokyuroku, 1302 (2003), 172192. 
[57] 
V. Spiridonov and A. Zhedanov, Elliptic grids, rational functions, and Padé interpolation, Ramanujan J., 13 (2007), 285310. 
[58] 
T. Stieltjes, Sur l'équation d'Euler, Bul.Sci.Math., Paris, sér. 2, 12 (1888), 222227. 
[59] 
A. A. Telitsyna, The Dirichlet problem for wave equation in plane domain with biquadratic boundary, Trudy IAMM NASU, 13 (2007), 198210 (In Russian). 
[60] 
M. Toda, "Theory of Nonlinear Lattices," Springer Series in SolidState Sciences, vol. 20, SpringerVerlag, Berlin, 1989. 
[61] 
A. P. Veselov, Integrable systems with discrete time and difference operators, Functional Analysis and its Applications, 22 (1988), 113 (Russian). 
[62] 
A. P. Veselov, Integrable maps, Russian Math. Surveys, 46 (1991), 151. 
[63] 
L. Vinet and A. Zhedanov, Generalized Bochner theorem: characterization of the AskeyWilson polynomials, J. Comput. Appl. Math., 211 (2008), 4556. 
[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), 303329. 
[66] 
A. Zhedanov, Padé interpolation table and biorthogonal rational functions, Proceedings of the Workshop on Elliptic Integrable Systems November 811, 2004, Kyoto, Rokko Lectures in Mathematics, No. 18, 323363. http://www.math.kobeu.ac.jp/publications/rlm18/20.pdf. 
[67] 
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), 455505 (In Russian). 
[5] 
G. S. Akopyan and R. A. Aleksandryan, On the completeness of a system of eigen and vectorpolynomials of a linear differential operator pencil in ellipsoidal domains, Dokl. Akad. Nauk Arm. SSR, 86 (1988), 147152 (In Russian). 
[6] 
V. I. Arnold, Small demominators. I, Izvestija AN SSSR, serija matematicheskaja, 25 (1961), 2186. 
[7] 
R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), 155. 
[8] 
G. A. Baker and P. GravesMorris, Padé approximants. Parts I and II, in "Encyclopedia of Mathematics and its Applications," 13, 14, AddisonWesley Publishing Co., Reading, Mass., 1981. 
[9] 
H. Bateman and A. Erdélyi, "Higher Transcendental Functions," 3, McGrawHill, 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 biquadratic boundary, Trudy IAMM NASU, 14 (2007), 1429 (In Russian). 
[12] 
E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, "Algebrogeometrical Approach to Nonlinear Integrable Equations," Springer Series in Nonlinear Dynamics, XII, Berlin: SpringerVerlag, 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), 33953486; Part II: 108 (2002), 295374. 
[14] 
Yu. M. Berezanskii, "Expansion by Eigenfunctions of Selfadjoint Operators," Naukova Dumka, Kiev, 1965 (In Russian). 
[15]  
[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), 851858. 
[18] 
A. B. Bogatyrev, Chebyshev representation for rational function, Sbornik Mathematics, 201 (2010), 15791598. 
[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), 9931003. 
[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), 16591668. 
[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, PellAbel equation, and some other related problems, Ukr. Math. Journal, 58 (2006), 487504. 
[23] 
V. P. Burskii and A. S. Zhedanov, Dirichlet and Neumann problems for string equation, Poncelet problem and PellAbel 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 PellAbel 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), 249264. 
[27] 
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, "Higher Transcendental Functions. I," McGrawHill, New York, 1953 Bateman manuscript project. 
[28] 
M. V. Fokin, Solvability of the Dirichlet problem for the string equation, Doklady AN SSSR, 272 (1983), 801805 (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), 237239 (Russian). 
[31] 
P. Griffiths and J. Harris, Poncelet theorem in space, Comment. Math. Helvetici, 52 (1977), 145160. 
[32] 
P. Griffiths and J. Harris, On a Cayley's explicit solution to Poncelet's porism, Enseign. Math., 24 (1978), 3140. 
[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), 2542. 
[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), 79100. 
[37]  
[38] 
F. John, The Dirichlet problem for a hyperbolic equation, Am. J. Math., 63 (1941), 141154. 
[39] 
A. Iatrou and J. A. G. Roberts, Integrable mappings of the plane preseving biquadratic invariants curves II, Nonlinearity, 15 (2002), 459489. 
[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), 85105. 
[42] 
J. L. King, Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609628. 
[43] 
M. M. Lavrent'ev, Mathematical problems of tomography and hyperbolic mappings, Sib. Math. J., 42 (2001), 916925. 
[44] 
V. F. Lazutkin, "KAM Theory and Semiclassical Approximation to Eigenfunctions," Springer Verlag, Berlin, Heidelberg, 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), 155 (In Russian). 
[47] 
J. Meinguet, On the solubility of the Cauchy interpolation problem, Approximation Theory (Proc. Sympos., Lancaster, 1969), 137163. 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), 195209. 
[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), 1625. 
[53] 
I. J. Schoenberg, On JacobiBertrand's proof of a theorem of Poncelet, in "Studies in Pure Mathematics, To the Memory of Paul Turan," 623627, 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. 
[55] 
V. Spiridonov and A. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions, Commun. Math. Phys., 210 (2000), 4983. 
[56] 
V. P. Spiridonov and A. S. Zhedanov, To the theory of biorthogonal rational functions, RIMS Kokyuroku, 1302 (2003), 172192. 
[57] 
V. Spiridonov and A. Zhedanov, Elliptic grids, rational functions, and Padé interpolation, Ramanujan J., 13 (2007), 285310. 
[58] 
T. Stieltjes, Sur l'équation d'Euler, Bul.Sci.Math., Paris, sér. 2, 12 (1888), 222227. 
[59] 
A. A. Telitsyna, The Dirichlet problem for wave equation in plane domain with biquadratic boundary, Trudy IAMM NASU, 13 (2007), 198210 (In Russian). 
[60] 
M. Toda, "Theory of Nonlinear Lattices," Springer Series in SolidState Sciences, vol. 20, SpringerVerlag, Berlin, 1989. 
[61] 
A. P. Veselov, Integrable systems with discrete time and difference operators, Functional Analysis and its Applications, 22 (1988), 113 (Russian). 
[62] 
A. P. Veselov, Integrable maps, Russian Math. Surveys, 46 (1991), 151. 
[63] 
L. Vinet and A. Zhedanov, Generalized Bochner theorem: characterization of the AskeyWilson polynomials, J. Comput. Appl. Math., 211 (2008), 4556. 
[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), 303329. 
[66] 
A. Zhedanov, Padé interpolation table and biorthogonal rational functions, Proceedings of the Workshop on Elliptic Integrable Systems November 811, 2004, Kyoto, Rokko Lectures in Mathematics, No. 18, 323363. http://www.math.kobeu.ac.jp/publications/rlm18/20.pdf. 
[67] 
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