4,250 research outputs found
On the shape of compact hypersurfaces with almost constant mean curvature
The distance of an almost constant mean curvature boundary from a finite
family of disjoint tangent balls with equal radii is quantitatively controlled
in terms of the oscillation of the scalar mean curvature. This result allows
one to quantitatively describe the geometry of volume-constrained stationary
sets in capillarity problems.Comment: 36 pages, 2 figures. In this version we have added an appendix about
almost umbilical surface
On the shape of capillarity droplets in a container
We provide a quantitative description of global minimizers of the Gauss free
energy for a liquid droplet bounded in a container in the small volume regime.Comment: 37 pages, 3 figure
Capillarity problems with nonlocal surface tension energies
We explore the possibility of modifying the classical Gauss free energy
functional used in capillarity theory by considering surface tension energies
of nonlocal type. The corresponding variational principles lead to new
equilibrium conditions which are compared to the mean curvature equation and
Young's law found in classical capillarity theory. As a special case of this
family of problems we recover a nonlocal relative isoperimetric problem of
geometric interest.Comment: 37 pages, 4 figure
Not Always Sparse: Flooding Time in Partially Connected Mobile Ad Hoc Networks
In this paper we study mobile ad hoc wireless networks using the notion of
evolving connectivity graphs. In such systems, the connectivity changes over
time due to the intermittent contacts of mobile terminals. In particular, we
are interested in studying the expected flooding time when full connectivity
cannot be ensured at each point in time. Even in this case, due to finite
contact times durations, connected components may appear in the connectivity
graph. Hence, this represents the intermediate case between extreme cases of
fully mobile ad hoc networks and fully static ad hoc networks. By using a
generalization of edge-Markovian graphs, we extend the existing models based on
sparse scenarios to this intermediate case and calculate the expected flooding
time. We also propose bounds that have reduced computational complexity.
Finally, numerical results validate our models
Asymptotic expansions of the contact angle in nonlocal capillarity problems
We consider a family of nonlocal capillarity models, where surface tension is
modeled by exploiting the family of fractional interaction kernels
, with and the dimension of the ambient space. The
fractional Young's law (contact angle condition) predicted by these models
coincides, in the limit as , with the classical Young's law
determined by the Gauss free energy. Here we refine this asymptotics by showing
that, for close to , the fractional contact angle is always smaller than
its classical counterpart when the relative adhesion coefficient is
negative, and larger if is positive. In addition, we address the
asymptotics of the fractional Young's law in the limit case of
interaction kernels with heavy tails. Interestingly, near , the dependence
of the contact angle from the relative adhesion coefficient becomes linear
- …