Abstract
Abstract This paper considers a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination of partial derivatives of the solutions. The quasi-linear operators are of monotone type, but are not required to be coercive for the existence of weak solutions, which is obtained by a double penalization/regularization for the approximation of the solutions. In the case of time-dependent convex sets that are independent of the solution, we show also the uniqueness and the continuous dependence of the strong solutions of the variational inequalities, extending previous results to a more general framework.
Highlights
While variational inequalities where introduced in 1964, by Fichera and Stampacchia in the framework of minimization problems with obstacle constraints, the first evolutionary variational inequality was solved in the seminal paper of Lions and Stampacchia [24], which was followed by many other works, including the extension to pseudo-monotone operators by Brezis in 1968 [7]
The first physical models with gradient constraints formulated with quasi-variational inequalities of evolution type were proposed by Prighozhin, in [29] and [28], respectively, for the sandpile growth and for the magnetization of type-II superconductors
This last model has motivated a first existence result for stationary problems in [21], including other applications in elastoplasticity and in electrostatics, and, in [31], in the parabolic framework for the p-Laplacian with an implicit gradient constraint, which was later extended to quasi-variational solutions for first-order quasilinear equations in [32], always in the scalar cases
Summary
While variational inequalities where introduced in 1964, by Fichera and Stampacchia in the framework of minimization problems with obstacle constraints, the first evolutionary variational inequality was solved in the seminal paper of Lions and Stampacchia [24], which was followed by many other works, including the extension to pseudo-monotone operators by Brezis in 1968 [7] (see [23, 33]). Weak quasi-variational solutions, which in general are non-unique and do not have the time derivative in the dual space of the solution, are obtained by the passages to the limit of two vanishing parameters, one for an appropriate approximation/penalisation of the constraint on Lu and a second one for a coercive regularisation, as in [32] This method allows the application of the Schauder fixed point theorem to a general regularised two parameters variational equation of the type (1.1) and extends considerably the work [2]. This paper is organized as follows: in Section 2 we state our framework and the main results on the existence of weak quasi-variational solutions and on the well-posedness of the strong variational solutions; in Section 3 we illustrate the nonlocal constraint operator G and the linear partial differential operator L with several examples of applications; Section 4 deals with the approximated problem and a priori estimates; the proof of the existence of the weak quasi-variational solutions is given in Section 5 and, in Section 6 we show the uniqueness and the continuous dependence on the data in the variational inequality case
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