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Periodic solutions of elliptic-parabolic variational inequalities with time-dependent constraints. (English) Zbl 1110.35006

The authors study periodic solutions of the quasilinear elliptic-parabolic variational inequality of the form \[ \begin{aligned} &u(t)\in K(t),\;t>0,\\ & (b(u)_t,u-v)+\int_\Omega a(x,b(u),\nabla u).\nabla (u-v)\,dx \leq (f,u-v)\;\forall v\in K(t),\;t>0,\\ & b(u(0,.))=b_0\;\text{ in}\;\Omega.\end{aligned} \] For all \(t\in \mathbb{R}^+, K(t)\) is a nonempty, closed and convex set in \(H^1(\Omega)\) satisfying the periodicity condition with period \(T_0>0: K(t+T_0)=K(t)\;\forall t\in\mathbb{R}_+.\) The existence of periodic solutions is proved. The applications of the general results to the periodic interior time-dependent double obstacle problem and to the problem with Signorini-Dirichlet-Neumann type mixed boundary conditions is presented.

MSC:

35B10 Periodic solutions to PDEs
35M10 PDEs of mixed type
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
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