arXiv subjects
Publications and source records attributed to Giovanni Leoni.
The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order $2$ by $Γ$-convergence of the mass-constrained Cahn-Hilliard functional. This is achieved by introducing a novel rearrangement technique, which works without Dirichlet boundary conditions.
The goal of this paper is the analytical validation of a model of Cermelli and Gurtin for an evolution law for systems of screw dislocations under the assumption of antiplane shear. The motion of the dislocations is restricted to a discrete set of glide directions, which are properties of the material. The evolution law is given by a "maximal dissipation criterion", leading to a system of differential inclusions. Short time existence, uniqueness, cross-slip, and fine cross-slip of solutions are proved.
The asymptotic behavior of an anisotropic Cahn-Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter epsilon that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to |s-1|^b near s=1, with 1<b<2. The first order term in the asymptotic development by Gamma-convergence is well-known, and is related to a suitable anisotropic perimeter of the interface. Here it is shown that, under these assumptions, the second order term is zero, which gives an estimate on the rate of convergence of the minimum values.
Short time existence for a surface diffusion evolution equation with curvature regularization is proved in the context of epitaxially strained three-dimensional films. This is achieved by implementing a minimizing movement scheme, which is hinged on the $H^{-1}$-gradient flow structure underpinning the evolution law. Long-time behavior and Liapunov stability in the case of initial data close to a flat configuration are also addressed.
In this paper, 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 in exam, 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 as the core radius tends to zero and the number of dislocations tends to infinity of this energy (suitably rescaled), takes the form $$ E= \int_{\Om} (W(β^e) + \f (\Curl β^e)) dx, $$ where $β^e$ represents the elastic part of the macroscopic strain, and $\Curl β^e$ represents the geometrically necessary dislocation density. The plastic energy density $\f$ is defined explicitly through an asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed in crystals such as dislocation walls.
The one-dimensional version of the higher order total variation-based model for image restoration proposed by Chan, Marquina, and Mulet in [4] is analyzed. A suitable functional framework in which the minimization problem is well posed is being proposed and it is proved analytically that the higher order regularizing term prevents the occurrence of the staircase effect.
In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.
This note summarizes recent results in which modern techniques of the calculus of variations are used to obtain qualitative features of film-substrate interfaces for a broad class of interfacial energies. In particular, we show that the existence of a critical thickness for incoherency and the formation of interfacial dislocations depend strongly on the convexity and smoothness of the interfacial energy function.