Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-26T16:46:53.192Z Has data issue: false hasContentIssue false
  • English
  • Français

A global estimate for the gradient in a singular elliptic boundary value problem

Published online by Cambridge University Press:  14 November 2011

Manuel A. del Pino
Manuel A. del Pino
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

We investigate the singular problem

where Ω is a bounded smooth domain, k a bounded, nonnegative measurable function and v Ω 0. For the solution u to this problem, which is shown to exist if k(x) > 0 on some subset of Ω with positive measure, a uniform bound for |∇u| in Ω is derived when k(x) ≧ ψ (dist (x, ∂Ω)) with ψ (s)/sv ∈ Lp(0, a) for some a > 0, p > 1.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 3-4 , 1992 , pp. 341 - 352
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Amick, C. J. and Fraenkel, L. E.. The uniqueness of Hill's spherical vortex. Arch. Rational Mech. Anal. 92 (1986), 91–119.Google Scholar
2Crandall, M. G., Rabinowitz, P. H. and Tartar, L.. On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations 2 (1977), 193–222.Google Scholar
3Fink, A. M., Gatica, J. A., Hernandez, G. E. and Waltman, P.. Approximation of solutions of singular second order boundary value problems. SIAM J. Math. Anal. 22 (1991), 440–462.Google Scholar
4Gomes, S. M.. On a singular nonlinear elliptic problem. SIAM J. Math. Anal. 17 (1986), 1359–1369.CrossRefGoogle Scholar
5Gatica, J. A., Hernandez, G. E. and Waltman, P.. Radially symmetric solutions of a class of singular elliptic equations. Proc. Edinburgh Math. Soc. 33 (1990), 169–180.Google Scholar
6Gatica, J. A., Oliker, V. and Waltman, P.. Singular nonlinear boundary value problems for second order differential equations. J. Differential Equations 79 (1989), 62–78.Google Scholar
7Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, 2nd edn (Berlin: Springer, 1983).Google Scholar
8Lazer, A. C. and McKenna, P. J.. On a singular nonlinear elliptic boundary value problem (to appear).Google Scholar
9Stuart, C. A.. Existence and approximation of solutions of nonlinear elliptic equations. Math. Z. 147 (1976), 53–63.Google Scholar