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In this note we prove compactness for the Cahn–Hilliard functional without assuming coercivity of the multi-well potential.
Vengono studiate proprietà di semicontinuità per integrali multipli quando soddisfa a condizioni di semicontinuità nelle variabili e può non essere soggetta a ipotesi di coercitività, e le successioni ammissibili in convergono fortemente in .
We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space and in the space of functions of bounded
variation.
We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations.
We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem.
The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the -limit of this
energy (suitably rescaled),...
Integral representation of relaxed energies and of
-limits of functionals
are obtained when sequences of fields may develop oscillations and are
constrained to satisfy
a system of first order linear partial differential equations. This
framework includes the
treatement of divergence-free fields, Maxwell's equations in
micromagnetics, and curl-free
fields. In the latter case classical relaxation theorems in , are
recovered.
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