Electron. J. Diff. Eqns., Vol. 2002(2002), No. 14, pp. 1-14.

A class of nonlinear elliptic variational inequalities: qualitative properties and existence of solutions

Luka Korkut, Mervan Pasic, & Darko Zubrinic

Abstract:
We study a class of nonlinear elliptic variational inequalities in divergence form. In the recent paper [6], we obtained results on the local control of essential infimum and supremum of solutions of quasilinear elliptic equations, and here we extend this point of view to the case of variational inequalities. It implies a new qualitative property of solutions in $W^{1,p}(\Omega )$ which we call ``jumping over the control obstacle.'' Using the Schwarz symmetrization technique, we give an existence and symmetrization theorems in $W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$ which agree completely with previous qualitative results. Also we consider generating singularities of weak solutions in $W^{1,p}(\Omega )$ of variational inequalities.

Submitted December 15, 2001. Published February 9, 2002.
Math Subject Classifications: 35J65, 35J85, 35B05.
Key Words: variational inequalities, double obstacle, qualitative properties, Schwarz symmetrization, generating singularities.

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  Luka Korkut
Department of mathematics
Faculty of Electrical Engineering and Computing
University of Zagreb
Unska 3, 10000 Zagreb, Croatia
e-mail: luka.korkut@fer.hr
Mervan Pasic
Department of mathematics
Faculty of Electrical Engineering and Computing
University of Zagreb
Unska 3, 10000 Zagreb, Croatia
e-mail: mervan.pasic@fer.hr
  Darko Zubrinic
Department of mathematics
Faculty of Electrical Engineering and Computing
University of Zagreb
Unska 3, 10000 Zagreb, Croatia
e-mail: darko.zubrinic@fer.hr

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