Nonlinear Weakly Elliptic 2 × 2 Systems Of Variational Inequalities With Unilateral Obstacle Constraints

We study 2 × 2 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.1997120Adams, D.R., Sets and functions of finite L p-capacity (1978) Ind. Univ. Math. J., 27, pp. 611-627Adams, D.R., (1992) Weakly Elliptic Systems with Obstacle Constraints: Part I - a 2 × 2 Model Problem, 42, pp. 1-14. , Partial Differential Equations with Minimal Smoothness and Applications, IMA Volumes in Math, Springer-VerlagAdams, D.R., Hedberg, L., (1996) Function Spaces and Potential Theory, , Springer-VerlagAdams, D.R., Nussenzveig Lopes, H.J., Weakly elliptic systems of variational inequalities: A 2 x 2 model problem with obstacles in both components (1995) Annali di Mat. Pura ed Appl., 169, pp. 183-201Berman, A., Plemmons, R.J., (1979) Nonnegative Matrices in the Mathematical Sciences, , Computer Science and Applied Math. Series, Academic PressDe Figueiredo, D., Mitidieri, E., Maximum principles for linear elliptic systems (1990) Rendiconti Mat. Trieste, 22, pp. 36-66Fuchs, M., Existence via partial regularity for degenerate systems of variational inequalities with natural growth (1992) Comment Math. Univ. Carolinae, 33, pp. 427-435Fuchs, M., Smoothness for systems of degenerate variational inequalities with natural growth (1992) Comment. Math. Univ. Carolinae, 33, pp. 33-41Fuchs, M., On the existence of weak solutions for degenerate systems of variational inequalities with critical growth (1994) Comment. Math. Univ. Carolinae, 35, pp. 445-449Gilbarg, D., Trudinger, N., (1977) Elliptic Partial Differential Equations of Second Order, , Springer-VerlagHeinonen, J., Kilpelainen, T., Martio, O., (1993) Nonlinear Potential Theory of Degenerate Elliptic Equations, , Oxford Math. Monographs, Clarendon PressHildebrandt, S., Widman, O., Variational inequalities for vector-valued functions (1979) J. Reine Angew. Math., 309, pp. 181-220Kinderlehrer, D., Stampacchia, G., (1990) An Introduction to Variational Inequalities, , Academic PressLi, G., Martio, O., Stability in obstacle problems (1994) Math. Scand., 75, pp. 87-100Park, D., (1979) Classical Dynamics and Its Quantum Analogues, , Lecture Notes in Physics, Springer-VerlagRoyden, H.L., (1968) Real Analysis, 2 nd Edition, , The Macmillan CompanyStein, E., Weiss, G., (1971) Introduction to Fourier Analysis on Euclidean Spaces, , Princeton University PressTroianello, G., (1987) Elliptic Differential Equations and Obstacle Problems, , Univ. Series in Math. Plenum Pres