Stability estimate for the multidimensional elliptic obstacle problem with respect to the obstacle function

Naveed Ahmad
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan.
Malkhaz Shashiashvili
A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State Universirty, Tbilisi, Georgia.

\(^{1}\)Corresponding Author: nabhatti35@yahoo.com

Copyright © 2012 Naveed Ahmad, Malkhaz Shashiashvili. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: December, 2012.

Abstract

The stability estimate of the energy integral established by Danelia, Dochviri and Shashiashvili [1] for the solution of the multidimensional obstacle problem in case of the Laplace operator is generalized to the case of arbitrary linear second order self-adjoint elliptic operator. This estimate asserts that if two obstacle functions are close in the \(L^{∞}\)-norm, then the gradients of the solutions of the corresponding obstacle problem are close in the weighted \(L^{2}\) -norm.

Keywords:

Stability estimate, unilateral elliptic obstacle problem, energy integral.