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Abstract variational problems with volume constraints

Marc Oliver Rieger (2004)

ESAIM: Control, Optimisation and Calculus of Variations

Existence results for a class of one-dimensional abstract variational problems with volume constraints are established. The main assumptions on their energy are additivity, translation invariance and solvability of a transition problem. These general results yield existence results for nonconvex problems. A counterexample shows that a naive extension to higher dimensional situations in general fails.

Abstract variational problems with volume constraints

Marc Oliver Rieger (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Existence results for a class of one-dimensional abstract variational problems with volume constraints are established. The main assumptions on their energy are additivity, translation invariance and solvability of a transition problem. These general results yield existence results for nonconvex problems. A counterexample shows that a naive extension to higher dimensional situations in general fails.

An existence theorem for extended mildly nonlinear complementarity problem in semi-inner product spaces

M. S. Khan (1995)

Commentationes Mathematicae Universitatis Carolinae

We prove a result for the existence and uniqueness of the solution for a class of mildly nonlinear complementarity problem in a uniformly convex and strongly smooth Banach space equipped with a semi-inner product. We also get an extension of a nonlinear complementarity problem over an infinite dimensional space. Our last results deal with the existence of a solution of mildly nonlinear complementarity problem in a reflexive Banach space.

Calculus of variations with differential forms

Saugata Bandyopadhyay, Bernard Dacorogna, Swarnendu Sil (2015)

Journal of the European Mathematical Society

We study integrals of the form Ω f d ω , where 1 k n , f : Λ k is continuous and ω is a k - 1 -form. We introduce the appropriate notions of convexity, namely ext. one convexity, ext. quasiconvexity and ext. polyconvexity. We study their relations, give several examples and counterexamples. We finally conclude with an application to a minimization problem.

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