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Some Problems in the Calculus of Variations
Abstract We present some results and open problems in the Calculus of Variations
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
Analytical continuum mechanics \`a la Hamilton-Piola: least action principle for second gradient continua and capillary fluids
In this paper a stationary action principle is proven to hold for capillary
fluids, i.e. fluids for which the deformation energy has the form suggested,
starting from molecular arguments, for instance by Cahn and Hilliard. Remark
that these fluids are sometimes also called Korteweg-de Vries or Cahn-Allen. In
general continua whose deformation energy depend on the second gradient of
placement are called second gradient (or Piola-Toupin or Mindlin or
Green-Rivlin or Germain or second gradient) continua. In the present paper, a
material description for second gradient continua is formulated. A Lagrangian
action is introduced in both material and spatial description and the
corresponding Euler-Lagrange bulk and boundary conditions are found. These
conditions are formulated in terms of an objective deformation energy volume
density in two cases: when this energy is assumed to depend on either C and
grad C or on C^-1 and grad C^-1 ; where C is the Cauchy-Green deformation
tensor. When particularized to energies which characterize fluid materials, the
capillary fluid evolution conditions (see e.g. Casal or Seppecher for an
alternative deduction based on thermodynamic arguments) are recovered. A
version of Bernoulli law valid for capillary fluids is found and, in the
Appendix B, useful kinematic formulas for the present variational formulation
are proposed. Historical comments about Gabrio Piola's contribution to
continuum analytical mechanics are also presented. In this context the reader
is also referred to Capecchi and Ruta.Comment: 52 page
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