1,720,986 research outputs found
Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions
Full text linkWe study the higher gradient integrability of distributional solutions u to the equation div (σ∇ u) = 0 in dimension two, in the case when the essential range of σ consists of only two elliptic matrices, i.e., σ∈ σ1, σ2 a.e. in Ω. In Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), for every pair of elliptic matrices σ1 and σ2, exponents pσ1,2,+∞) and qσ1,1,2) have been found so that if u∈W1,qσ1,σ2(Ω) is solution to the elliptic equation then ∇u∈Lweakpσ1,σ2(Ω) and the optimality of the upper exponent pσ1,σ2 has been proved. In this paper we complement the above result by proving the optimality of the lower exponent qσ1,σ2. Precisely, we show that for every arbitrarily small δ, one can find a particular microgeometry, i.e., an arrangement of the sets σ- 1(σ1) and σ- 1(σ2) , for which there exists a solution u to the corresponding elliptic equation such that ∇u∈Lqσ1,σ2-δ, but ∇u∉Lqσ1,σ2. The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case
Derivation of a Rod theory from lattice systems with interactions beyond nearest neighbours
No full textWe study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-tonearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires
A Variational Model for Dislocations at Semi-coherent Interfaces
No full textWe propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations and a bulk elastic energy, spent to decrease the amount of dislocations needed to compensate the lattice misfit. We prove that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is built on some explicit computations done in the framework of the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far-field strain vanishes as the interface size increases
On the effect of interactions beyond nearest neighbours on non-convex lattice systems
No full textWe analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation
Derivation of Linear Elasticity for a General Class of Atomistic Energies
No full textThe purpose of this paper is the derivation, in the framework of Gamma-convergence, of linear elastic continuum theories from a general class of atomistic models, in the regime of small deformations. Existing results are available only in the special case of one-well potentials accounting for very short interactions. We consider here the general case of multiwell potentials accounting for interactions of finite but arbitrarily long range. The extension to this setting requires a novel idea for the proof of the Gamma-convergence which is interesting in its own right and potentially relevant in other applications
Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals
No full textLinear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours
Correction to: Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals (Archive for Rational Mechanics and Analysis, (2018), 230, 1, (1-45), 10.1007/s00205-018-1240-6)
No full textIn the original version there was a mistake in the displacement of the tables.The original version of this article unfortunately contained mistakes
Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions
Full text linkWe propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence
RIGIDITY OF THREE-DIMENSIONAL LATTICES AND DIMENSION REDUCTION IN HETEROGENEOUS NANOWIRES
No full textAbstract. In the context of nanowire heterostructures we perform a discrete to con-tinuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and De-launay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large
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