Journal of Elliptic and Parabolic Equations

, Volume 4, Issue 1, pp 107–139 | Cite as

Solutions of nonlinear equations of divergence type in domains having corner points

  • E. E. Perepelkin
  • B. I. Sadovnikov
  • N. G. Inozemtseva
Article
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Abstract

An algorithm has been suggested for finding exact solutions of a nonlinear equation in partial derivatives of a divergence type which appears in the formulation of magnetostatics, hydro-and aerodynamics, quantum mechanics (stationary Schrödinger equation). The properties of smoothness of solutions in domains with corner points (piecewise smooth boundary) have been considered. The solutions with unbounded derivatives in the corner point domain have been presented on the basis of a new class of special functions.

Keywords

Rigorous result Exact solution Nonlinear equations in partial derivatives Corner point Magnetostatics problem Equation of a divergent type Special functions 

Mathematical Subject Classification

35J15 35J25 35J60 35C10 35Q61 

References

  1. 1.
    Perepelkin, E., ATLAS collaboration: The ATLAS experiment at the CERN large Hadron collider. Aad. JINST S08003(3), 437 (2008)Google Scholar
  2. 2.
    Perepelkin, E., ATLAS collaboration: Commissioning of the magnetic field in the ATLAS muon spectrometer. Nucl. Phys. Proc. Suppl. 177178, 265–266 (2008). (ISBN: 0920-5632) Google Scholar
  3. 3.
    Perepelkin, E.E., Sadovnikov, B.I., Inozemtseva, N.G.: The properties of the first equation of the Vlasov chain of equations. J. Stat. Mech. 5, P05019 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second, p. 530. Springer, Berlin (1998). (ISSN 1431-0821) Google Scholar
  5. 5.
    Perepelkin, E.E., Sadovnikov, B.I., Inozemtseva, N.G.: Riemann surface and quantization. Ann. Phys. 376, 194–217 (2017)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    DiBenedetto, E.: Degenerate Parabolic Equations, p. 403. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  7. 7.
    Fufaev, V.V.: Dirichlet problem for regions having corners. Doklady Akademii Nauk SSSR. 131(1), 37–39 (1960)MathSciNetMATHGoogle Scholar
  8. 8.
    Volkov, E.A.: On the solution by the grid method of the inner Dirichlet problem for the Laplace equation. Transl. Am. Math. Soc. 24, 279–307 (1963)MATHGoogle Scholar
  9. 9.
    Volkov, E.A.: Differentiability properties of solutions of boundary value problems for the Laplace and Poisson equations on a rectangle. Proc. Steklov Inst. Math. 77, 101–126 (1965)MATHGoogle Scholar
  10. 10.
    Volkov, E.A.: Differentiability properties of solutions of boundary value problems for the Laplace equation on a polygon. Proc. Steklov Inst. Math. 77, 127–159 (1965)MATHGoogle Scholar
  11. 11.
    Volkov, E.A.: The net-method for finite and infinite regions with piecewise smooth boundary. Sov. Math. Dokl. 7, 744–747 (1966)MATHGoogle Scholar
  12. 12.
    Èskin, G.I.: General boundary-value problems for equations of principal type in a planar domain with angle points. Uspekhi Matematicheskikh Nauk. 18(3), 241–242 (1963)Google Scholar
  13. 13.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and quasilinear elliptic equations, p. 538. Nauka, Moscow (1964)MATHGoogle Scholar
  14. 14.
    Višik, M.I., Ėskin, G.I., Višik, M.I., Èskin, G.I.: Variable order Sobolev–Slobodeckii spaces with weighted norms and their applications to mixed boundary value problems. Sibirskij matematiceskij zurnal 95, 973–997 (1968)MATHGoogle Scholar
  15. 15.
    Adams Robert, A.: Sobolev Spaces. Academic Press, Boston, MA (1975). ISBN 978-0-12-044150-1MATHGoogle Scholar
  16. 16.
    Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Tr. Mosk. Mat. Obs. 16, 209–292 (1967)MathSciNetGoogle Scholar
  17. 17.
    Oganesyan, L.A., Rukhovets, L.A.: Variational-difference schemes for second order linear elliptic equations in a two-dimensional region with a piecewise-smooth boundary. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki. 8(1), 97–114 (1968)MathSciNetMATHGoogle Scholar
  18. 18.
    Babuska, J.: Finite element method for elliptic equations with corners. Computing 6, N3 (1970)CrossRefGoogle Scholar
  19. 19.
    Babuska, J., Rozenzweig, M.B.: A finite scheme for domains with corners. Numer. Mathem. 20, N1 (1972)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fix, G.: Higher order Rayleigh–Ritz approximations. J. Math. Mech. 18, N7 (1969)MathSciNetMATHGoogle Scholar
  21. 21.
    Oganesyan, L.A., Rukhovets, L.A., Rivkind, V.J.: Variational-difference methods for solving elliptic equations, Part I. In: Differential equations and their applications, vol. 5, Vilnius (1973) (Russian)Google Scholar
  22. 22.
    Oganesyan, L.A., Rukhovets, L.A., Rivkind, V.J.: Variational-difference methods for solving elliptic equations, Part II. In: Differential equations and their applications, Vol. 8, Vilnius (1974) (Russian)Google Scholar
  23. 23.
    Samarskii, A.A., Fryazinov, I.V.: Difference schemes for the solution of the Dirichlet problem in an arbitrary domain for an elliptic equation with variable coefficients. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki. 11(2), 385–410 (1971)MathSciNetGoogle Scholar
  24. 24.
    Fryazinov, I.V.: Difference schemes for the Laplace equation in step-domains. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki. 18(5), 1170–1185 (1978)MathSciNetMATHGoogle Scholar
  25. 25.
    Akulov N. Theorie der Feinstruktur der Magnetisie rungskurven der Einkristalle. – «Zeitschr. Phys.», 1931, Bd 69Google Scholar
  26. 26.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, p. 422. Pitman Publishing Inc, NewYork (1985). (ISBN 0-273-08647-2) MATHGoogle Scholar
  27. 27.
    Maz’ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations Grundlehren der Mathematischen Wissenschaften 342 (2nd revised and augmented ed.), p. 866. Springer Verlag, Berlin–Heidelberg–New York (2011). (ISBN 978-3-642-15563-5) Google Scholar
  28. 28.
    Ercolani, N., Siggia, E.D.: Painleve property and geometry. Physica D 34, 303–346 (1989)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ercolani, N., Siggia, E.D.: Painleve property and integrability. Phys Lett A. 119(3), 112–116 (1986)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kruskal, M.D., Clarkson, P.A.: The Painleve-Kovalevski and Poly-Painleve test for integrability. Studies Appl Math 86, 87–165 (1992)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kruskal, M.D., Ramani, A., Grammaticos, B.: Singularity analysis and its relation to complete, partial and non-integrability. In: Conte, R., Boccara, N. (eds.) Partially integrable evolution equations in physics. Kluwer Academic Publishers, Dordrecht (1990)Google Scholar
  32. 32.
    Ibragimov, N.K.: Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie). Russian Math Surv 47(4), 89 (1992)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Olver, P.J.: Applications of Lie Groups to differential equations, p. 513. Springer, Berlin (2000)MATHGoogle Scholar
  34. 34.
    Painleve, P.: Lecons sur la theorie analytique des equation s differentielles 9 professes a Stokholm, Paris, 189Google Scholar
  35. 35.
    Painleve, P.: Memoire sur les equations differentielles dont Vintegrale generale est uniforme. Bull. Soc. Math. France 28, 201–261 (1900)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Painleve, P.: Sur les equations differentielles du second ordre et d’ordre superieure dont Vintegrdble generale est uniforme. Acta Math. 25, 1–85 (1902)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)CrossRefMATHGoogle Scholar
  38. 38.
    Adler, V.E., Shabat, A.B., Yamilov, R.I.: Symmetry approach to the integrability problem. Theor Math Phys 125(3), 1603–1661 (2000)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Newell, A.C.: Solitons in Mathematics and Physics, Society for Industrial and Applied Mathematics (1985). (ISBN 978-0-898711-96-7)Google Scholar
  40. 40.
    Boiti, M., Pompinelli, F.: Nonlinear Schrodinger equation, Backlund transformations and Painleve transcendents. Nuovo Cimento B 71, 253–264 (1982)CrossRefGoogle Scholar
  41. 41.
    Lakshmanan, M., Kaliappan, P.: Lie transformations, nonlinear evolution equations, and Painleve forms. J. Math. Phys. 24, 795–805 (1983)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Musette, M.: Painleve analysis for nonlinear partial differential equations. In: Conte, R. (ed.) The Painleve Property, One Century Later. CRM series in Mathematical Physics, pp. 517–572. Springer, New York (1999)CrossRefGoogle Scholar
  43. 43.
    Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Marvan Mivchal Scalar second order evolution equations possessing an irreducible SL2-valued zero curvature representation. Preprint DIPS-4, March, (2002)Google Scholar
  45. 45.
    Sakovich, S.Y.: On zero-curvature representation of evolution equations. J. Phys. A. Math. Gen. 28, 2861–2869 (1995)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Mikhailov, A.V., Shabat, A.B., Yamilov, R.I.: The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems. Russian Math Surv 42(4), 1–63 (1987)CrossRefMATHGoogle Scholar
  47. 47.
    Ablowitz, M.J., Кaиp, D.J., Newell, A.C., Segur, H.: Inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315 (1974)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Ablowitz, M.J., Кaиp, D.J., Newell, A.C., Segur, H.: Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30, 1262–1264 (1973)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Ablowitz, M.J., Clarkson, P.A.: Solitons Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)CrossRefMATHGoogle Scholar
  50. 50.
    Zhidkov, E.P., Perepelkin, E.E.: An analytical approach for quasi-linear equation in secondary order. CMAM 3, 285–297 (2001)MATHGoogle Scholar
  51. 51.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Vol. II: Partial Differential Equations (Vol. II by R. Courant). Interscience, New York, London (1962)Google Scholar

Copyright information

© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • E. E. Perepelkin
    • 1
  • B. I. Sadovnikov
    • 1
  • N. G. Inozemtseva
    • 2
  1. 1.Faculty of PhysicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Dubna State UniversityMoscowRussia

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