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Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs. (English) Zbl 1340.34216
Summary: Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize it as a parabolic inclusion by using the Lagrange multiplier and the nonlinear maximal monotone operator associated with the time differential under time-dependent double obstacles.

MSC:
34G25 Evolution inclusions
35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
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