1,721,015 research outputs found
Variational Approximation of Free-Discontinuity Energies with Linear Growth
No full textWe provide a variational approximation for quasiconvex energies with linear growth, defined on vector valued generalized functions with bounded variation, in the framework of free-discontinuity problems
Local and Nonlocal Continuum Limits of Ising-Type Energies for Spin Systems
Full text linkWe study, through a !-convergence procedure, the discrete to continuum limit of
Ising-type energies of the form F"(u) = −Pi,j c" i,juiuj , where u is a spin variable defined on a
portion of a cubic lattice "Zd 8 ⌦, ⌦ being a regular bounded open set, and valued in {−1, 1}. If the
constants c" i,j are nonnegative and satisfy suitable coercivity and decay assumptions, we show that all possible !-limits of surface scalings of the functionals F" are finite on BV (⌦; {±1}) and of the
formR Su'(x, ⌫u) dH11 d−1. If such decay assumptions are violated, we show that we may approximate
nonlocal functionals of the form
R
Su
'(⌫u) dHd−1+ K(x, y)g(u(x), u(y)) dxdy. We focus on the
approximation of two relevant examples: fractional perimeters and Ohta–Kawasaki-type energies. Eventually, we provide a general criterion for a ferromagnetic behavior of the energies F" even when
the constants c" i,j change sign. If such a criterion is satisfied, the ground states of F" are still the uniform states 1 and −1 and the continuum limit of the scaled energies is an integral surface energy of the form above
Free-discontinuity problems generated by singular perturbation: the N-dimensional case
No full textWe provide an approximation of some free discontinuity problems by local functionals with a singular perturbation of higher order
Variational description of bulk energies for bounded and unbounded spin systems
No full textWe study the asymptotic behaviour, as the mesh size tends to zero, of a class of discrete energies under very general assumptions that cover the case of bounded and unbounded spin systems, leading to variational limits of integral type. The cases of homogenization and of non-pairwise interacting systems (e.g. multiple-exchange spin systems) are also discussed
Variational equivalence between Ginzburg-Landau, XY spin systems and screw dislocations energies
No full textWe introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing ε tends to zero, the relevant energies in these models behave like a free energy in the complex Ginzburg-Landau theory of superconductivity, justifying in a rigorous mathematical language the analogies between screw dislocations in crystals and vortices in superconductors. To this purpose, we introduce a notion of asymptotic variational equivalence between families of functionals in the framework of Γ-convergence. We then prove that, in several scaling regimes, the complex Ginzburg-Landau, the XY spin system and the screw dislocation energy functionals are variationally equivalent. Exploiting such an equivalence between dislocations and vortices, we can show new results concerning the asymptotic behavior of screw dislocations in the | log ε|^2 energetic regime
Free-discontinuity problems via functionals involving the L^1-norm of the gradient and their approximations
No full textWe provide an approximation of free-discontinuity energies by considering a variant of the Ambrosio-Tortorelli construction which allows to obtain as limits more complex surface energies
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-to-nearest 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
Finite-difference approximation of energies in fracture mechanics
No full textWe provide a variational approximation by finite-difference energies of free-discontinuity functionals depending on the symmetrized gradient, which are related to variational models in fracture mechanics for linearly-elastic materials. We perform this approximation in dimension 2 via both discrete and continuous functionals. In the discrete scheme we treat also boundary value problems and we give an extension of the approximation result to dimension 3
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
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