8,663 research outputs found
Analysis of local minima for constrained minimization problems
We consider vectorial problems in the calculus of variations with an
additional pointwise constraint. Our admissible mappings satisfy , where
is a manifold embedded in Euclidean space. The main results of the paper all
formulate necessary or sufficient conditions for a given mapping to
be a weak or strong local minimizer. Our methods involve using projection
mappings in order to build on existing, unconstrained, local minimizer results.
We apply our results to a liquid crystal variational problem to quantify the
stability of the unwound cholesteric state under frustrated boundary
conditions
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
Majorisation with applications to the calculus of variations
This paper explores some connections between rank one convexity,
multiplicative quasiconvexity and Schur convexity. Theorem 5.1 gives simple
necessary and sufficient conditions for an isotropic objective function to be
rank one convex on the set of matrices with positive determinant. Theorem 6.2
describes a class of possible non-polyconvex but multiplicative quasiconvex
isotropic functions. This class is not contained in a well known theorem of
Ball (6.3 in this paper) which gives sufficient conditions for an isotropic and
objective function to be polyconvex. We show here that there is a new way to
prove directly the quasiconvexity (in the multiplicative form). Relevance of
Schur convexity for the description of rank one convex hulls is explained.Comment: 13 page
The inverse problem for Lagrangian systems with certain non-conservative forces
We discuss two generalizations of the inverse problem of the calculus of
variations, one in which a given mechanical system can be brought into the form
of Lagrangian equations with non-conservative forces of a generalized Rayleigh
dissipation type, the other leading to Lagrangian equations with so-called
gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free
conditions for the existence of a suitable non-singular multiplier matrix,
which will lead to an equivalent representation of a given system of
second-order equations as one of these Lagrangian systems with non-conservative
forces.Comment: 28 page
A Unifying Variational Perspective on Some Fundamental Information Theoretic Inequalities
This paper proposes a unifying variational approach for proving and extending
some fundamental information theoretic inequalities. Fundamental information
theory results such as maximization of differential entropy, minimization of
Fisher information (Cram\'er-Rao inequality), worst additive noise lemma,
entropy power inequality (EPI), and extremal entropy inequality (EEI) are
interpreted as functional problems and proved within the framework of calculus
of variations. Several applications and possible extensions of the proposed
results are briefly mentioned
On the generalized Helmholtz conditions for Lagrangian systems with dissipative forces
In two recent papers necessary and sufficient conditions for a given system
of second-order ordinary differential equations to be of Lagrangian form with
additional dissipative forces were derived. We point out that these conditions
are not independent and prove a stronger result accordingly.Comment: 10 pages, accepted for publ in Z. Angew. Math. Mech
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