Electron. J. Diff. Eqns.,
Vol. 1997(1997), No. 18, pp 1-20.

### Nonlinear weakly elliptic 2X2 systems of variational
inequalities with unilateral obstacle constraints

D.R. Adams & H.J. Nussenzveig Lopes

**Abstract:**

We study 2X2 systems of variational inequalities which are only
weakly elliptic; in particular, these systems are not necessarily monotone.
The prototype differential operator is the (vector-valued) p-Laplacian.
We prove, under certain conditions, the existence of solutions to the
unilateral obstacle problem. This work extends the results by the authors
in [Annali di Mat. Pura ed Appl., 169(1995), 183--201] to nonlinear
operators.

In addition, we address the question of determining function spaces on
which the p-Laplacian is a bounded nonlinear operator. This question arises
naturally when studying existence for these systems.

Submitted July 28, 1997. Published October 31, 1997.

Math Subject Classification: 35J85, 35J45, 31C45.

Key Words: p-Laplacian, obstacle problem,
non-monotone systems of variational inequalities.

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David R. Adams

Department of Mathematics, University of Kentucky,
Patterson Office Tower, Lexington, KY 40506, USA

e-mail: dave@ms.uky.edu
Helena J. Nussenzveig Lopes

Departamento de Matematica, IMECC-UNICAMP.
Caixa Postal 6065, Campinas, SP 13081-970, Brazil

e-mail: hlopes@ime.unicamp.br

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