6,410 research outputs found
Loss of control in pattern-directed nucleation: a theoretical study
The properties of template-directed nucleation are studied close to the
transition where full nucleation control is lost and additional nucleation
occurs beyond the pre-patterned regions. First, kinetic Monte Carlo simulations
are performed to obtain information on a microscopic level. Here the
experimentally relevant cases of 1D stripe patterns and 2D square lattice
symmetry are considered. The nucleation properties in the transition region
depend in a complex way on the parameters of the system, i.e. the flux, the
surface diffusion constant, the geometric properties of the pattern and the
desorption rate. Second, the properties of the stationary concentration field
in the fully controlled case are studied to derive the remaining nucleation
probability and thus to characterize the loss of nucleation control. Using the
analytically accessible solution of a model system with purely radial symmetry,
some of the observed properties can be rationalized. A detailed comparison to
the Monte Carlo data is included
Mean curvature flow with free boundary outside a hypersphere
The purpose of this paper is twofold: firstly, to establish sufficient
conditions under which the mean curvature flow supported on a hypersphere with
exterior Dirichlet boundary exists globally in time and converges to a minimal
surface, and secondly, to illustrate the application of Killing vector fields
in the preservation of graphicality for the mean curvature flow with free
boundary. To this end we focus on the mean curvature flow of a topological
annulus with inner boundary meeting a standard n-sphere in \R^{n+1}
perpendicularly and outer boundary fixed to an (n-1)-sphere with radius R>1 at
a fixed height h. We call this the \emph{sphere problem}. Our work is set in
the context of graphical mean curvature flow with either symmetry or mean
concavity/convexity restrictions. For rotationally symmetric initial data we
obtain, depending on the exact configuration of the initial graph, either long
time existence and convergence to a minimal hypersurface with boundary or the
development of a finite-time curvature singularity. With reflectively symmetric
initial data we are able to use Killing vector fields to preserve graphicality
of the flow and uniformly bound the mean curvature pointwise along the flow.
Finally we prove that the mean curvature flow of an initially mean
concave/convex graphical surface exists globally in time and converges to a
piece of a minimal surface.Comment: 23 page
Spectral Theory of Infinite Quantum Graphs
We investigate quantum graphs with infinitely many vertices and edges without
the common restriction on the geometry of the underlying metric graph that
there is a positive lower bound on the lengths of its edges. Our central result
is a close connection between spectral properties of a quantum graph and the
corresponding properties of a certain weighted discrete Laplacian on the
underlying discrete graph. Using this connection together with spectral theory
of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new
results on spectral properties of quantum graphs. Namely, we prove several
self-adjointness results including a Gaffney type theorem. We investigate the
problem of lower semiboundedness, prove several spectral estimates (bounds for
the bottom of spectra and essential spectra of quantum graphs, CLR-type
estimates) and study spectral types.Comment: Dedicated to the memory of M. Z. Solomyak (16.05.1931 - 31.07.2016
Vector-valued optimal Lipschitz extensions
Consider a bounded open set in and a Lipschitz function g from the
boundary of to . Does this function always have a canonical optimal
Lipschitz extension to all of ? We propose a notion of optimal Lipschitz
extension and address existence and uniqueness in some special cases. In the
case , we show that smooth solutions have two phases: in one they are
conformal and in the other they are variants of infinity harmonic functions
called infinity harmonic fans. We also prove existence and uniqueness for the
extension problem on finite graphs.Comment: 24 pages, 10 figure
A symmetrization result for a class of anisotropic elliptic problems
We prove estimates for weak solutions to a class of Dirichlet problems
associated to anisotropic elliptic equations with a zero order term.Comment: arXiv admin note: text overlap with arXiv:1607.0721
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