In this paper we derive explicit formulas for disarrangement densities of submacroscopic separations, switches, and interpenetrations in the context of firstorder structured deformations. Our derivation employs relaxation within one mathematical setting for structured deformations of a specific, purely interfacial density, and the formula we obtain agrees with one obtained earlier in a different setting for structured deformations. Coincidentally, our derivation provides an alternative method for obtaining the earlier result, and we establish new explicit formulas for other measures of disarrangements that are significant in applications.
We study necessary and sufficient conditions for the existence of so-where E ⊆ Λ k+1 (R n ) is a given set. Special attention is given to the case of the curl (i.e. k = 1), particularly in dimension 3. Some applications to the calculus of variations are also stated.
ABSTRACT. Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral representation for a relaxed energy functional in the setting of second-order structured deformations. Our derivation covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position); finally, we provide explicit formulas for bulk relaxed energies as well as anticipated applications.
Abstract. The asymptotic behavior of a family of singular perturbations of a non-convex second order functional of the typeis studied through Γ-convergence techniques as a variational model to address two-phase transition problems.
We look for necessary and sufficient conditions for the existence of solutions to the minimisation problemwhere the boundary data u ξ 0 satisfies curlu ξ 0 (x) = ξ 0 , for ξ 0 a given vector in R 3 .
In this paper we investigate the possibility of obtaining a measure representation for functionals arising in the context of optimal design problems under non-standard growth conditions and perimeter penalization. Applications to modelling of strings are also provided.
MSC (2010): 49J45, 74K10
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