948 research outputs found
Exterior convexity and classical calculus of variations
We study the relation between various notions of exterior convexity
introduced in Bandyopadhyay-Dacorogna-Sil \cite{BDS1} with the classical
notions of rank one convexity, quasiconvexity and polyconvexity. To this end,
we introduce a projection map, which generalizes the alternating projection for
two-tensors in a new way and study the algebraic properties of this map. We
conclude with a few simple consequences of this relation which yields new
proofs for some of the results discussed in Bandyopadhyay-Dacorogna-Sil
\cite{BDS1}.Comment: The original publication is available at www.esaim-cocv.org
https://www.esaim-cocv.org/articles/cocv/abs/2016/02/cocv150007/cocv150007.htm
Convexity of certain integrals of the calculus of variations
In this paper we study the convexity of the integral over the space . We isolate a necessary condition on f and we find necessary and sufficient conditions in the case where f(x, u, u′) = a(u)u′2n or g(u) + h(u′
Notions of affinity in calculus of variations with differential forms
Ext-int.\ one affine functions are functions affine in the direction of
one-divisible exterior forms, with respect to exterior product in one variable
and with respect to interior product in the other. The purpose of this article
is to prove a characterization theorem for this class of functions, which plays
an important role in the calculus of variations for differential forms
Direct approach to the problem of strong local minima in Calculus of Variations
The paper introduces a general strategy for identifying strong local
minimizers of variational functionals. It is based on the idea that any
variation of the integral functional can be evaluated directly in terms of the
appropriate parameterized measures. We demonstrate our approach on a problem of
W^{1,infinity} weak-* local minima--a slight weakening of the classical notion
of strong local minima. We obtain the first quasiconvexity-based set of
sufficient conditions for W^{1,infinity} weak-* local minima.Comment: 26 pages, no figure
Potentials for -quasiconvexity
We show that each constant rank operator admits an exact
potential in frequency space. We use this fact to show that the
notion of -quasiconvexity can be tested against compactly
supported fields. We also show that -free Young measures are
generated by sequences , modulo shifts by the barycentre.Comment: 15 pages; to appear in Calculus of Variations and Partial
Differential Equation
- …