41 research outputs found
Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds I, uniform estimates
We are concerned with solutions to the parabolic Allen-Cahn equation in
Riemannian manifolds. For a general class of initial condition we show non
positivity of the limiting energy discrepancy. This in turn allows to prove
almost monotonicity formula (a weak counterpart of Huisken's monotonicity
formula) which gives a local uniform control of the energy densities at small
scales.
Such results will be used in [40] to extend previous important results from
[31] in Euclidean space, showing convergence of solutions to the parabolic
Allen-Cahn equations to Brakke's motion by mean curvature in space forms
Phase transitions and minimal hypersurfaces in hyperbolic space]
The purpose of this paper is to investigate the Cahn-Hillard approximation
for entire minimal hypersurfaces in the hyperbolic space. Combining comparison
principles with minimization and blow-up arguments, we prove existence results
for entire local minimizers with prescribed behaviour at infinity. Then, we
study the limit as the length scale tends to zero through a
-convergence analysis, obtaining existence of entire minimal
hypersurfaces with prescribed boundary at infinity. In particular, we recover
some existence results proved in M. Anderson and U. Lang using geometric
measure theory
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces
We obtain an improved Sobolev inequality in H^s spaces involving Morrey
norms. This refinement yields a direct proof of the existence of optimizers and
the compactness up to symmetry of optimizing sequences for the usual Sobolev
embedding. More generally, it allows to derive an alternative, more transparent
proof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998]
using the abstract approach of dislocation spaces developed in [K. Tintarev &
K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the
local defect of compactness of the Sobolev embedding in terms of measures in
the spirit of [P. L. Lions, Rev. Mat. Iberoamericana 1985]. As a model
application, we study the asymptotic limit of a family of subcritical problems,
obtaining concentration results for the corresponding optimizers which are well
known when s is an integer ([O. Rey, Manuscripta math. 1989; Z.-C. Han, Ann.
Inst. H. Poincare Anal. Non Lineaire 1991], [K. S. Chou & D. Geng, Differential
Integral Equations 2000]).Comment: 33 page
A Global Compactness type result for Palais-Smale sequences in fractional Sobolev spaces
We extend the Global Compactness result by M. Struwe (Math. Z, 1984) to any
fractional Sobolev spaces for and a bounded domain with smooth boundary. The proof is a simple
direct consequence of the so-called Profile Decomposition of P. Gerard (ESAIM:
Control, Optimisation and Calculus of Variations, 1998).Comment: To appear in Nonlinear Analysis: Theory, Methods & Application
On large deviations of interface motions for statistical mechanics models
We discuss the sharp interface limit of the action functional associated to
either the Glauber dynamics for Ising systems with Kac potentials or the
Glauber+Kawasaki process. The corresponding limiting functionals, for which we
provide explicit formulae of the mobility and transport coefficients, describe
the large deviations asymptotics with respect to the mean curvature flow.Comment: 32 page
TORUS-LIKE SOLUTIONS FOR THE LANDAU-DE GENNES MODEL. PART I: THE LYUKSYUTOV REGIME
We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a corresponding physically relevant norm constraint (Lyuksyutov constraint), we prove full regularity up to the boundary for the (constrained) minimizers. As a consequence, in a relevant range of parameters (which we call Lyuksyutov regime), we show that unconstrained minimizers do not exhibit isotropic melting. In the case of a nematic droplet, the radial hedgehog is even shown to be an unstable equilibrium. Finally, we describe a class of boundary data including radial anchoring for which constrained or unconstrained minimizers are smooth configurations whose biaxiality level sets carry nontrivial topology. Results of this paper will be largely employed and refined in the next of our series. In particular in [16], where we prove that biaxiality level sets are generically finite unions of tori for smooth equilibrium configurations minimizing the energy in a restricted axially symmetric class