Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
  Quarterly of Applied Mathematics
Quarterly of Applied Mathematics
  
Online ISSN 1552-4485; Print ISSN 0033-569X
 

Mixed problem for the Laplace equation outside cuts in a plane with setting Dirichlet and skew derivative conditions on different sides of the cuts


Authors: P. A. Krutitskii and A. I. Sgibnev
Journal: Quart. Appl. Math. 64 (2006), 105-136
MSC (2000): Primary 35J05, 45E05
Published electronically: January 24, 2006
MathSciNet review: 2211380
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The mixed problem for the Laplace equation outside cuts in a plane is considered. The Dirichlet condition is posed on one side of each cut and the skew derivative condition is posed on the other side. This problem generalizes the mixed Dirichlet-Neumann problem. Integral representation for a solution of the boundary value problem is obtained in the form of potentials. The densities in the potentials satisfy the uniquely solvable Fredholm integral equation of the second kind and index zero. Uniqueness and existence theorems for a solution of the boundary value problem are proved. Singularities of the gradient of the solution of the boundary value problem at the tips of the cuts are studied. Asymptotic formulas for singularities are obtained. The effect of the disappearance of singularities is discussed.


References [Enhancements On Off] (What's this?)

  • 1. P. A. Krutitskiĭ and A. I. Sgibnev, The method of integral equations in the mixed Dirichlet-Neumann problem for the Laplace equation in the exterior of cuts on a plane, Differ. Uravn. 37 (2001), no. 10, 1299–1310, 1437 (Russian, with Russian summary); English transl., Differ. Equ. 37 (2001), no. 10, 1363–1375. MR 1945240 (2003g:35037), http://dx.doi.org/10.1023/A:1013399013691
  • 2. A. S. Gosteva, N. Ch. Krutitskaya, and P. A. Krutitskiĭ, A mixed problem in a magnetized semiconductor film with cuts along a straight line, Fundam. Prikl. Mat. 6 (2000), no. 4, 1061–1073 (Russian, with English and Russian summaries). MR 1813011 (2001k:78016)
  • 3. S. A. Gabov, Angular potential and some of its applications, Mat. Sb. (N.S.) 103(145) (1977), no. 4, 490–504, 630 (Russian). MR 0463457 (57 #3407)
  • 4. P. A. Krutitskiĭ, The Dirichlet problem for the Helmholtz equation in the exterior of cuts in the plane, Zh. Vychisl. Mat. i Mat. Fiz. 34 (1994), no. 8-9, 1237–1258 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 34 (1994), no. 8-9, 1073–1090. MR 1300397 (95f:35046)
  • 5. P. A. Krutitskiĭ, The Neumann problem for the Helmholtz equation in the exterior of cuts in the plane, Zh. Vychisl. Mat. i Mat. Fiz. 34 (1994), no. 11, 1652–1665 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 34 (1994), no. 11, 1421–1431 (1995). MR 1307611 (95i:45004)
  • 6. N.I. Muskhelishvili. Singular integral equations. Noordhoff, Groningen, 1972; 3rd Russian ed.: Nauka Publishers, Moscow, 1968.
  • 7. A.N. Kolmogorov, S.V. Fomin. Elements of the Theory of Functions and Functional Analysis. Dover, New York, 1999.
  • 8. S.G. Krein (editor). Functional analysis. Nauka Publishers, Moscow, 1964.
  • 9. P.A. Krutitskii. On the electric current from electrodes in a magnetized semiconductor film. IMA J. of Appl. Math., 1998, v. 60, p. 285-297.
  • 10. P. A. Krutitskiĭ, A mixed problem for the Laplace equation outside cuts on the plane, Differ. Uravn. 33 (1997), no. 9, 1181–1190, 1293 (Russian, with Russian summary); English transl., Differential Equations 33 (1997), no. 9, 1184–1193 (1998). MR 1638915 (99h:35030)
  • 11. Arnold F. Nikiforov and Vasilii B. Uvarov, Special functions of mathematical physics, Birkhäuser Verlag, Basel, 1988. A unified introduction with applications; Translated from the Russian and with a preface by Ralph P. Boas; With a foreword by A. A. Samarskiĭ. MR 922041 (89h:33001)
  • 12. P. K. Suetin, Klassicheskie ortogonalnye mnogochleny, “Nauka”, Moscow, 1979 (Russian). Second edition, augmented. MR 548727 (80h:33001)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35J05, 45E05

Retrieve articles in all journals with MSC (2000): 35J05, 45E05


Additional Information

P. A. Krutitskii
Affiliation: Dept. of Mathematics, Faculty of Physics, Moscow State University, Moscow 117234, Russia

A. I. Sgibnev
Affiliation: Dept. of Mathematics, Faculty of Physics, Moscow State University, Moscow 117234, Russia

DOI: http://dx.doi.org/10.1090/S0033-569X-06-00987-0
PII: S 0033-569X(06)00987-0
Received by editor(s): March 10, 2005
Published electronically: January 24, 2006
Additional Notes: This work was partly supported by the RFBR grant no. 05-01-00050.
Article copyright: © Copyright 2006 Brown University



Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2014 Brown University
Comments: qam-query@ams.org
AMS Website