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An approach to minimization under a constraint: the added mass technique. (English) Zbl 1223.49021

The authors study the following class of minimization problems
\[ \text{minimize } J(u)=\int_{\mathbb R^n} j(x,u,|\nabla u|)\,dx \text{ on the functions } u\in H \text{ with } G(u)=\int_{\mathbb R^n} g(u) \,dx=c \]
where \(H\) is a reflexive Banach space and \(J\) are weakly lower semicontinuous. There are presented, through the treatment of some semi-linear or quasi-linear type problems, techniques to show the existence of a minimizer.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35J20 Variational methods for second-order elliptic equations
35J62 Quasilinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:

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