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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 $$u\in W^{k,1}\left(\mathrm{\Omega };{\mathbb{R}}^d\right)\mapsto {\int }_{\mathrm{\Omega }}f\left(x,u\left(x\right),\mathrm{\dots }{\nabla }^{k}u\left(x\right)\right)dx$$ quando $f$ soddisfa a condizioni di semicontinuità nelle variabili $\left(x,u,\mathrm{\dots },{\nabla }^{k-1}u\left(x\right)\right)$ e può non essere soggetta a ipotesi di coercitività, e le successioni ammissibili in $W^{k,1}\left(\mathrm{\Omega };{\mathbb{R}}^d\right)$ convergono fortemente in $W^{k-1,1}\left(\mathrm{\Omega };{\mathbb{R}}^d\right)$.

We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space $W_{\text{loc}}^{1,1}({ℝ}^N;{ℝ}^d)$ and in the space $B{V}_{\text{loc}}({ℝ}^N;{ℝ}^d)$ 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 $\Gamma$-limit of this energy (suitably rescaled),...

Integral representation of relaxed energies and of -limits of functionals $$(u,v)\mapsto {\int }_{\Omega }f(x,u\left(x\right),v\left(x\right))dx$$ 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.