Γ\Gamma-convergence analysis of a generalized XYXY model: fractional vortices and string defects

We propose and analyze a generalized two dimensional $XY$ model, whose interaction potential has $n$ weighted wells, describing corresponding symmetries of the system. As the lattice spacing vanishes, we derive by $\Gamma$-convergence the discrete-to-continuum limit of this model. In the energy regime we deal with, the asymptotic ground states exhibit fractional vortices, connected by string defects. The $\Gamma$-limit takes into account both contributions, through a renormalized energy, depending on the configuration of fractional vortices, and a surface energy, proportional to the length of the strings. Our model describes in a simple way several topological singularities arising in Physics and Materials Science. Among them, disclinations and string defects in liquid crystals, fractional vortices and domain walls in micromagnetics, partial dislocations and stacking faults in crystal plasticity

Ground states of a two phase model with cross and self attractive interactions

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

Minimising movements for the motion of discrete screw dislocations along glide directions

In [3] a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in [8] and predicts motion along the glide directions of the crystal. In this paper, we provide rigorous proofs of the results in [3], and in particular of the passage from the discrete to the continuous dynamics. The proofs are based on $\Gamma$-convergence techniques

Γ\Gamma-convergence of the Heitmann-Radin sticky disc energy to the crystalline perimeter

We consider low energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle, i.e., the so called kissing number. For configurations whose energy scales like the perimeter, we prove a compactness result which shows the emergence of polycrystalline structures: The empirical measure converges to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function. Whenever the limit configuration is a single crystal, i.e., it has constant orientation, we show that the $\Gamma$-limit is the anisotropic perimeter, corresponding to the Finsler metric determined by the orientation of the single crystal

Two slope functions minimizing fractional seminorms and applications to misfit dislocations

We consider periodic piecewise affine functions, defined on the real line, with two given slopes and prescribed length scale of the regions where the slope is negative. We prove that, in such a class, the minimizers of $s$-fractional Gagliardo seminorm densities, with $0<s<1$, are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals where the slope is constant vanishes. Our results, for $s=\frac 1 2$, have relevant applications to the van der Merwe theory of misfit dislocations at semi-coherent straight interfaces. We consider two elastic materials having different elastic coefficients and casting parallel lattices having different spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semi-coherent limit as the ratio between the two lattice spacings tends to one

Insufficient control of blood pressure and incident diabetes

OBJECTIVE: Incidence of type 2 diabetes might be associated with preexisting hypertension. There is no information on whether incident diabetes is predicted by blood pressure control. We evaluated the hazard of diabetes in relation to blood pressure control in treated hypertensive patients. RESEARCH DESIGN AND METHODS: Nondiabetic, otherwise healthy, hypertensive patients (N = 1,754, mean +/- SD age 52 +/- 11 years, 43% women) participated in a network over 3.4 +/- 1 years of follow-up. Blood pressure was considered uncontrolled if systolic was >or=140 mmHg and/or diastolic was >or=90 mmHg at the last outpatient visit. Diabetes was defined according to American Diabetes Association guidelines. RESULTS: Uncontrolled blood pressure despite antihypertensive treatment was found in 712 patients (41%). At baseline, patients with uncontrolledblood pressure were slightly younger than patients with controlled blood pressure (51 +/- 11 vs. 53 +/- 12 years, P < 0.001), with no differences in sex distribution, BMI, duration of hypertension, baseline blood pressure, fasting glucose, serum creatinine and potassium, lipid profile, or prevalence of metabolic syndrome. During follow-up, 109 subjects developed diabetes. Incidence of diabetes was significantly higher in patients with uncontrolled (8%) than in those with controlled blood pressure (4%, odds ratio 2.08, P < 0.0001). In Cox regression analysis controlling for baseline systolic blood pressure and BMI, family history of diabetes, and physical activity, uncontrolled blood pressure doubled the risk of incident diabetes (hazard ratio [HR] 2.10, P < 0.001), independently of significant effects of age (HR 1.02 per year, P = 0.03) and baseline fasting glucose (HR 1.10 per mg/dl, P < 0.001). CONCLUSIONS: In a large sample of treated nondiabetic hypertensive subjects, uncontrolled blood pressure is associated with twofold increased risk of incident diabetes independently of age, BMI, baseline blood pressure, or fasting glucose