Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint |
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Authors: | José Francisco Rodrigues Lisa Santos |
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Affiliation: | (1) CONICET, Rosario, Argentina;(2) Centro At?mico Bariloche, TEMADI, Av. Bustillo 9500, Bariloche, 8400, Argentina;(3) Depto. de Matem?tica, Universidad Austral, Paraguay 1950, Rosario, S2000FZF, Argentina;(4) Facultad de Ingenier?a, Universidad Nacional de Salta, Buenos Aires 144, Salta, 4400, Argentina |
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Abstract: | We prove the existence of solutions for a quasi-variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. We also obtain the existence of stationary solutions by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results. |
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