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Une méthode variationnelle pour une classe d’équations non linéaires dans les espaces de Hilbert. (A variational method for a class of nonlinear equations in Hilbert spaces). (French) Zbl 0594.47051

Multivalued equations of the form \(Au+\partial \beta (u)\ni f\) in which A is a linear, symmetric and positive definite operator in a Hilbert space are considered. The concept of classical solution \(u_ c\) is introduced and a uniqueness result is proved.
Since in general the equation \(Au+\partial \beta (u)\ni f\) has no classical solution, the concept of generalized solution in the Sobolev sense \(u_ S\) is introduced, using a variational method. The connection between the solutions \(u_ c\) and \(u_ S\) is given and also the dependence of the generalized solution in the Sobolev sense with respect to the element f is studied.
Finally some classical results in the variational method for linear and positive definite operators are generalized.

MSC:

47J05 Equations involving nonlinear operators (general)
47B25 Linear symmetric and selfadjoint operators (unbounded)
35A15 Variational methods applied to PDEs
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