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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

An example of non-convex minimization and an application to Newton's problem of the body of least resistance

T. Lachand-Robert, M. A. Peletier (2001)

Annales de l'I.H.P. Analyse non linéaire

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.

Anisotropic symmetrization

Jean Van Schaftingen (2006)

Annales de l'I.H.P. Analyse non linéaire

A-quasiconvexity : weak-star convergence and the gap

Irene Fonseca, Giovanni Leoni, Stefan Müller (2004)

Annales de l'I.H.P. Analyse non linéaire

Big pieces of $C^{1, α}$ -graphs for minimizers of the Mumford-Shah functional

Séverine Rigot (2000)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Biting theorems for jacobians and their applications

K. Zhang (1990)

Annales de l'I.H.P. Analyse non linéaire

Boundary Regularity for Minima of Certain Quadratic Functionals.

J. Jost, M. Meier (1983)

Mathematische Annalen

$C^{1, β}$ -partial regularity of $p$ -harmonic maps at the free boundary

Thomas Müller (2002)

Bollettino dell'Unione Matematica Italiana

We prove the partial $C^{1, β}$ -regolarity up to the free boundary of the $p$ -harmonic maps which minimize the $p$ -energy $\int {|D u|}^{p} d x$ .

C1,... partial regularity of functions minimising quasiconvex integrals.

Nicola Fusco, John Hutchinson (1986)

Manuscripta mathematica

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 $\int_{Ω} f (d ω)$ , where $1 \leq k \leq n$ , $f : Λ^{k} \to ℝ$ 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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