Quasivariational solutions for first order quasilinear equations with gradient constraint

José Francisco Rodrigues; Lisa Santos

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arXiv · 1201.6132

Quasivariational solutions for first order quasilinear equations with gradient constraint

José Francisco Rodrigues·Lisa Santos

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Abstract

We prove the existence of solutions for an evolution quasi-variational inequality 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 {\em a priori} estimates. We obtain also 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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José Francisco Rodrigues, Lisa Santos. 2012-01-30. Quasivariational solutions for first order quasilinear equations with gradient constraint. https://arxiv.org/abs/1201.6132

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