Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762

 

 

On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations


Authors: Vasily Denisov and Andrey Muravnik
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 88-93
MSC (2000): Primary 35J25; Secondary 35B40, 35J60
Published electronically: September 29, 2003
MathSciNet review: 2029469
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the Dirichlet problem in half-space for the equation $\nobreak{\Delta u+g(u)\vert\nabla u\vert^2=0,}$ where $g$ is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as $y\to\infty,$ where $y$ denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function.


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

  • 1. V. N. Denisov and V. D. Repnikov, Stabilization of the solution of the Cauchy problem for parabolic equations, Differentsial′nye Uravneniya 20 (1984), no. 1, 20–41 (Russian). MR 731645
  • 2. V. N. Denisov and A. B. Muravnik, On stabilization of the solution of the Cauchy problem for quasilinear parabolic equations, Differ. Equ. 38, No. 3 (2002), 369-374.
  • 3. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR 0473443
  • 4. M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892.
  • 5. Ernesto Medina, Terence Hwa, Mehran Kardar, and Yi Cheng Zhang, Burgers’ equation with correlated noise: renormalization-group analysis and applications to directed polymers and interface growth, Phys. Rev. A (3) 39 (1989), no. 6, 3053–3075. MR 988871, 10.1103/PhysRevA.39.3053
  • 6. S. I. Pohožaev, Equations of the type Δ𝑢=𝑓(𝑥,𝑢,𝐷𝑢), Mat. Sb. (N.S.) 113(155) (1980), no. 2(10), 324–338, 351 (Russian). MR 594841
  • 7. Alexander Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differential Equations 110 (1994), no. 2, 322–355. MR 1278374, 10.1006/jdeq.1994.1070
  • 8. Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 0093649
  • 9. N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian). MR 563790
  • 10. A. K. Guščin and V. P. Mihaĭlov, The stabilization of the solution of the Cauchy problem for a parabolic equation, Differencial′nye Uravnenija 7 (1971), 297–311 (Russian). MR 0282044
  • 11. S. Kamin, On stabilisation of solutions of the Cauchy problem for parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 1, 43–53. MR 0440200
  • 12. V. V. Žikov, The stabilization of the solutions of parabolic equations, Mat. Sb. (N.S.) 104(146) (1977), no. 4, 597–616, 663 (Russian). MR 0473524
  • 13. A. V. Bitsadze, Nekotorye klassy uravnenii v chastnykh proizvodnykh, “Nauka”, Moscow, 1981 (Russian). MR 628612

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 35J25, 35B40, 35J60

Retrieve articles in all journals with MSC (2000): 35J25, 35B40, 35J60


Additional Information

Vasily Denisov
Affiliation: Moscow State University, Faculty of Computational Mathematics and Cybernetics, Leninskie gory, Moscow 119899, Russia
Email: V.Denisov@g23.relcom.ru

Andrey Muravnik
Affiliation: Department of Differential Equations, Moscow State Aviation Institute, Volokolamskoe shosse 4, Moscow, A-80, GSP-3, 125993, Russia
Email: abm@mailru.com

DOI: http://dx.doi.org/10.1090/S1079-6762-03-00115-X
Keywords: Asymptotic behaviour of solutions, BKPZ-type non-linearities
Received by editor(s): March 6, 2002
Published electronically: September 29, 2003
Additional Notes: The second author was supported by INTAS, grant 00-136 and RFBR, grant 02-01-00312.
Communicated by: Michael E. Taylor
Article copyright: © Copyright 2003 American Mathematical Society