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Monotonicity and compactness methods for nonlinear variational inequalities. (English) Zbl 1192.35083

Chipot, Michel (ed.), Handbook of differential equations: Stationary partial differential equations. Vol. IV. Amsterdam: Elsevier/North Holland (ISBN 978-0-444-53036-3/hbk). Handbook of Differential Equations, 203-298 (2007).
The author studies nonlinear variational inequalities of elliptic type. The approach is based on the theory of nonlinear mappings of the monotone type.
At first an abstract problem is under consideration: \(f\in Au+Bu\), where \(A\) and \(B\) are multivalued mappings from a real Banach space into its dual. The solvability of the problem is reduced to the study of the range of \(A+B\). The notions of monotone, maximal monotone, pseudo-monotone and semi-monotone operators are introduced and applied in the above problem.
Next abstract variational inequalities of the form \(\langle Au-f,u-v\rangle\leq0\) \(\forall v\in K\) are investigated, where \(K\) is a closed convex set in a Banach space.
Finally the above abstract problems are applied to the boundary value problems for elliptic partial differential equations or systems and to variational inequalities.
For the entire collection see [Zbl 1179.35001].

MSC:

35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
35J25 Boundary value problems for second-order elliptic equations
35J57 Boundary value problems for second-order elliptic systems
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