We study the asymptotic behavior, as mesh size $\varepsilon$ tends to zero, of a general class discrete energies defined on functions $u:\alpha\in\varepsilon\mathbb Z^N\cap\ \Omega\mapsto u(\alpha)\in{{\mathbb R}^d}$ form \begin{eqnarray*} F_{\varepsilon}(u)=\sum\limits_{ \begin{array}{ll} {\scriptstyle \alpha, \beta \in \varepsilon\mathbb Z^N} \\ \scriptstyle \symbol{91}\alpha ,\beta \symbol{93} \subset\Omega \end{array} }\hskip-0.3cm g_{\varepsilon}(\alpha,\beta,u(\alpha)-u(\beta))...

Abstract We give a detailed description of the geometry single‐droplet patterns in nonlocal isoperimetric problem. In particular, we focus on sharp interface limit Ohta‐Kawasaki free energy for diblock copolymers, regarded as paradigm energies with short‐ and long‐range interactions. Exploiting fine properties regularity theory minimal surfaces, extend previous partial results different directions robust tools geometric analysis more complex patterns. © 2012 Wiley Periodicals inc.

We provide a variational description of nearest-neighbours andnext-to-nearest neighbours binary lattice systems. By studying the$\Gamma$-limit proper scaling the energies thesystems, we highlight phase and anti-phase boundary phenomena andshow how they depend on geometry lattice.

We investigate the flat flow solution for surface diffusion equation via discrete minimizing movements scheme proposed by Cahn and Taylor. prove that in dimension three converges to unique smooth of equation, provided initial set is sufficiently regular.

We analyze the variational limit of one-dimensional next-to-nearest neighbours (NNN) discrete systems as lattice size tends to zero when energy densities are multiwell or Lennard–Jones type. Properly scaling energies, we study several phenomena formation boundary layers and phase transitions. also presence local patterns anti-phase transitions in asymptotic behaviour ground states NNN model subject Dirichlet conditions. use this information prove a localization fracture result case type potentials.

We introduce and discuss discrete two-dimensional models for XY spin systems screw dislocations in crystals. prove that, as the lattice spacing $\e$ tends to zero, relevant energies these behave like a free energy complex Ginzburg-Landau theory of superconductivity, justifying rigorous mathematical language analogies between crystals vortices superconductors. To this purpose, we notion asymptotic variational equivalence families functionals framework $\Gamma$-convergence. then several...

We prove some results in the context of isoperimetric inequalities with quantitative terms. In 2 -dimensional case, our main contribution is a method for determining optimal coefficients c_{1},\dots,c_{m} inequality \delta P(E) \geq \sum_{k=1}^{m}c_{k}\alpha (E)^{k} + o(\alpha (E)^{m}) , valid each Borel set E positive and finite area, \alpha (E) being, respectively, \textit{isoperimetric deficit} \textit{Fraenkel asymmetry} . n dimensions, besides proving existence regularity properties...

We study the discrete-to-continuum limit of ferromagnetic spin systems when lattice spacing tends to zero. assume that atoms are part a (maybe) non-periodic close flat set in lower dimensional space, typically plate three dimensions. Scaling particle positions by small parameter $\varepsilon>0$ we perform $\Gamma$-convergence analysis properly rescaled interfacial-type energies. show that, up subsequences, energies converge surface integral defined on partitions space. In second paper...

We consider a class of spin-type discrete systems and analyze their continuum limit as the lattice spacing goes to zero. Under standard coerciveness growth assumptions together with an additional head-to-tail symmetry condition, we observe that this can be conveniently written functional in space $Q$-tensors. further characterize energy density several cases (both two three dimensions). In planar case also develop second-order theory derive gradient or concentration-type models according...

We consider a variational model for two interacting species (or phases), subject to cross and self attractive forces. show existence several qualitative properties of minimizers. Depending on the strengths forces, different behaviors are possible: phase mixing or separation with nested disjoint phases. In case Coulomb interaction we characterize ground state configurations.

We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study geometric variational problems stratified singular sets.We then apply this isoperimetric for planar bubble clusters.In setting we develop an improved convergence theorem, showing that sequence almost-minimizing clusters converging L 1 limit cluster has actually converge strong C 1,α -sense.Applications result classification and qualitative...

We study the discrete-to-continuum variational limit of $J_{1}$-$J_{3}$ spin model on square lattice in vicinity ferromagnet/helimagnet transition point as spacing vanishes. Carrying out $\Gamma$-convergence analysis proper scalings energy, we prove emergence and characterize geometric rigidity chirality phase transitions.

Abstract We study the discrete-to-continuum variational limit of antiferromagnetic XY model on two-dimensional triangular lattice. The system is fully frustrated and displays two families ground states distinguished by chirality spin field. compute $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Γ</mml:mi></mml:math> -limit energy in a regime which detects transitions one-dimensional interfaces between admissible phases.

We study the asymptotic behaviour of a general class discrete energies defined on functions form , as mesh size ε goes to 0. prove that under assumptions cover case bounded and unbounded spin systems in thermodynamic limit, variational limit Eε has . The cases homogenization non-pairwise interacting (e.g. multiple-exchange systems) are also discussed.