Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-29T08:10:00.183Z Has data issue: false hasContentIssue false

An oscillation estimate to a variational inequality

Published online by Cambridge University Press:  17 April 2009

Hyeong-Ohk Bae
Affiliation:
Department of MathematicsKAISTTaejonRepublic of Korea, e-mail: hobae@mathx.kaist.ac.kr, ch@math.kaist.ac.kr
Hi Jun Choe
Affiliation:
Department of MathematicsKAISTTaejonRepublic of Korea, e-mail: hobae@mathx.kaist.ac.kr, ch@math.kaist.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that solutions for elliptic equations and variational inequalities are continuous pointwisely if the obstacle is continuous pointwisely. The continuity of weakly monotone functions in a high Sobolev space is crucial. Also a comparison principle is useful in estimating oscillations of solutions.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Choe, H.J., ‘A regularity theory for a more general class of quasilinear elliptic partial differential equations and obstacle problems’, Arch. Rational Mech. Anal. 114 (1991), 383394.CrossRefGoogle Scholar
[2]Giaquinta, M., ‘Remarks on the regularity of weak solutions to some variational inequalities’, Math. Z. 177 (1981), 1531.CrossRefGoogle Scholar
[3]Lieberman, G., ‘Local and boundary regularity for some variational inequalities involving p-Laplacian-type operators’, (preprint).Google Scholar
[4]Lindquist, P., ‘Regularity for the gradient of the solution to a nonlinear obstacle problem with degenerate ellipticity’, Nonlinear Anal. 12 (1988), 12451255.CrossRefGoogle Scholar
[5]Manfredi, J., ‘Weakly monotone functions’, J. Geom. Anal. 4 (1994), 393402.CrossRefGoogle Scholar
[6]Michael, J. and Ziemer, W., ‘Interior regularity for solutions to obstacle problems’, Nonlinear Anal. 10 (1986), 14271448.CrossRefGoogle Scholar
[7]Mu, J. and Ziemer, W., ‘Smooth regularity of solutions of double obstacle problems involving degenerate elliptic equations’, Comm. Partial Differential Equations 16 (1991), 821843.CrossRefGoogle Scholar