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On some Schrödinger-type variational inequalities. (English) Zbl 0892.35090

Abstract Schrödinger-type evolution variational inequalities \[ u'(t) \in K \quad \text{Im}(u'(t) + iA(t)u(t) - f(t), v - u'(t)) \geq 0\quad\text{for a.a. } t \in (0,T), \]
\[ u(0) = u_0 \] are studied. Here \(V \subset H \equiv H^* \subset V^*\) is a standard complex Hilbert space triplet, \((.,.)\) is the antiduality pairing between \(V^*\) and \(V\), \(K\) is a closed convex set in \(V\), \(t \to A(t)\) is a smooth operator function, \(A(t) \in \mathcal L (V,V^*)\) is an Hermitian strictly \(V\)- coercive operator for any \(t \in (0,T)\), \(u_0 \in V\) is given, \(f(t)\) is a \(V^*\)-valued function. The existence and unicity of the strong solution \(u \in W^{1,\infty}(0,T;V)\) is proved. The existence proof is based on a penalty method.
Reviewer: M.Kučera (Praha)

MSC:

35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35Q55 NLS equations (nonlinear Schrödinger equations)
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