Local and Nonlocal Continuum Limits of Ising-Type Energies for Spin Systems

We 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

Monotonicity formulas for obstacle problems with Lipschitz coefficients

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Hölder continuous linear term.With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli (J Fourier Anal Appl 4(4–5), 383–402, 1998), Monneau (J Geom Anal 13(2), 359–389, 2003), and Weiss (Invent Math 138(1), 23–50, 1999)

The role of intrinsic distances in the relaxation of L∞-functionals

We consider a supremal functional of the form $$F(u)= supess_{x in Omega}f(x,Du(x))$$ where $Omegasubseteq R^N$ is a regular bounded open set, $uin wi$ and $f$ is a Borel function. Assuming that the intrinsic distances $d^{lambda}_F(x,y):= sup Big{ u(x) - u(y): , F(u)leq lambda Big}$ are locally equivalent to the euclidean one for every lambda>inf_{wi} F, we give a description of the sublevel sets of the weak$^*$-lower semicontinuous envelope of $F$ in terms of the sub-level sets of the difference quotient functionals $R_{d^lambda_F}(u):=sup_{x ot =y} rac{u(x)-u(y)}{d^lambda_F(x,y)}.$ As a consequence we prove that the relaxed functional of positive $1$-homogeneous supremal functionals coincides with $R_{d^1_F}$. Moreover, for a more general supremal functional $F$ (a priori non coercive), we prove that the sublevel sets of its relaxed functionals with respect to the weak$^*$ topology, the weak$^*$ convergence and the uniform convergence are convex. The proof of these results relies both on a deep analysis of the intrinsic distances associated to $F$ and on a careful use of variational tools such as $Gamma$-convergence

Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems

In this paper we provide rigorous statements and proofs for the asymptotic analysis of discrete energies defined on a two-dimensional triangular lattice allowing for fracture in presence of a microscopic impenetrability constraint. As the lattice parameter goes to 0, we prove that any limit deformation with finite energy is piecewise rigid and we prove a general lower bound with a suitable Griffith-fracture energy density which reflects the anisotropies of the underlying triangular lattice. For such a continuum energy we also provide a class of (piecewise rigid) deformations satisfying "opening-crack" conditions on which the lower bound is sharp. Relying on these results, some consequences have been already presented in the companion paper [A. Braides et al., J. Mech. Phys. Solids 96 (2016) 235-251] to validate models in Computational Mechanics in the small-deformation regime

Relaxation of free-discontinuity energies with obstacles

Given a Borel function ψ defined on a bounded open set Ω with Lipschitz boundary and $\varphi\in L^1(\partial\Omega,{\mathcal H}^{n-1})$, we prove an explicit representation formula for the L1 lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint $u^+\ge\psi$ ${\mathcal H}^{n-1}$ a.e. on Ω and the Dirichlet boundary condition $u=\varphi$ on $\partial\Omega$