46 research outputs found
Existence of Least-perimeter Partitions
We prove the existence of a perimeter-minimizing partition of R^n into
regions of unit volume. We conclude with a short tribute to the late Manuel A.
Fortes.Comment: 5 pages; for submission to Fortes memorial isue of Philosphical
Magazine Letter
Existence of Integral -Varifolds minimizing and , , in Riemannian Manifolds
We prove existence and partial regularity of integral rectifiable
-dimensional varifolds minimizing functionals of the type and
in a given Riemannian -dimensional manifold , , under suitable assumptions on (in the end of the paper we
give many examples of such ambient manifolds). To this aim we introduce the
following new tools: some monotonicity formulas for varifolds in
involving , to avoid degeneracy of the minimizer, and a sort of
isoperimetric inequality to bound the mass in terms of the mentioned
functionals.Comment: 33 pages; this second submission corresponds to the published version
of the paper, minor typos are fixe
Uniaxial versus biaxial character of nematic equilibria in three dimensions
We study global minimizers of the Landau–de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the t→∞ limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829–838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of “strongly biaxial” regions in terms of the parameter t. We further show that global LdG minimizers in the restricted class of uniaxial Q-tensors cannot be stable critical points of the LdG energy in this limit
Publisher Correction: Discovery of rare variants associated with blood pressure regulation through meta-analysis of 1.3 million individuals
Existence of Isoperimetric Sets with Densities \u201cConverging from Below\u201d on
In this paper, we consider the isoperimetric problem in the space RN with a density. Our result states that, if the density f is lower semi-continuous and converges to a limit a> 0 at infinity, with f 64 a far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture of Morgan and Pratelli (Ann Glob Anal Geom 43(4):331\u2013365, 2013. \ua9 2016, Mathematica Josephina, Inc