In�quations d'evolution paraboliques avec convexes d�pendant du temps. Applications aux in�quations quasi-variationnelles d'evolution

Applications of abstract parabolic quasi-variational inequalities to obstacle problems

“…As far as the existence of a solution of the quasi-variational evolution problem is concerned, we can work in a much larger class of {ϕ t (v; ·)} than that of this paper; see [2] and also [6,9]. Actually our class of {ϕ t (v; ·)} treated in this paper is a special case of that in [2].…”

Nonlinear evolution equations generated by subdifferentials with nonlocal constraints

“…The stationary cases have been dealt with in many papers, for instance, [2,5,10,13,14], but there are not so many papers dealing with the time-evolution problems, because it is not expected for solutions to have much regularity in time. We recall some papers [11,15,16] for time evolution quasi-variational inequalities. In papers [11,16], the so-called monotonicity property of the mapping v → φ s (v; ·) is used as one of key tools in their treatment.…”

“…We recall some papers [11,15,16] for time evolution quasi-variational inequalities. In papers [11,16], the so-called monotonicity property of the mapping v → φ s (v; ·) is used as one of key tools in their treatment. However, the monotonicity property is too restrictive in many important applications, as examples of section 5 suggest.…”

Cauchy Problem for Semilinear Parabolic Equation with Time-Dependent Obstacles: A BSDEs Approach

“…We know from Remark 5.15 that if there is v ∈ W such that h 1 ≤ v ≤ h 2 a.e., then a unique solution of OP(ϕ, f, h 1 , h 2 ) is a weak solution of VI(ϕ, f, h 1 , h 2 ), but in general, the latter problem may have many other solutions (see [24] for relevant examples). To ensure uniqueness of solutions of variational inequalities additional regularity conditions on the data are needed.…”

“…Such a formulation of a solution of the obstacle problem is very useful because instead of variational inequalities we consider variational equalities which provide an additional information on the solution, and what is more important, it guarantees uniqueness of solutions (in the theory of variational inequalities solutions of obstacle problems with timedependent barriers are in general not unique, see e.g. [24]). …”