1,240 research outputs found
Minimizers of Anisotropic Surface Tensions Under Gravity: Higher Dimensions via Symmetrization
We consider a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques, we establish the existence, convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure. In the case of smooth surface tensions, we obtain the uniqueness of minimizers via an ODE characterization.National Science Foundation (U.S.). (Award No. DMS-1204557
Quantitative estimates for bending energies and applications to non-local variational problems
We discuss a variational model, given by a weighted sum of perimeter, bending
and Riesz interaction energies, that could be considered as a toy model for
charged elastic drops. The different contributions have competing preferences
for strongly localized and maximally dispersed structures. We investigate the
energy landscape in dependence of the size of the 'charge', i.e. the weight of
the Riesz interaction energy. In the two-dimensional case we first prove that
for simply connected sets of small elastic energy, the elastic deficit controls
the isoperimetric deficit. Building on this result, we show that for small
charge the only minimizers of the full variational model are either balls or
centered annuli. We complement these statements by a non-existence result for
large charge. In three dimensions, we prove area and diameter bounds for
configurations with small Willmore energy and show that balls are the unique
minimizers of our variational model for sufficiently small charge
On equilibrium shapes of charged flat drops
Equilibrium shapes of two-dimensional charged, perfectly conducting liquid
drops are governed by a geometric variational problem that involves a perimeter
term modeling line tension and a capacitary term modeling Coulombic repulsion.
Here we give a complete explicit solution to this variational problem. Namely,
we show that at fixed total charge a ball of a particular radius is the unique
global minimizer among all sufficiently regular sets in the plane. For sets
whose area is also fixed, we show that balls are the only minimizers if the
charge is less than or equal to a critical charge, while for larger charge
minimizers do not exist. Analogous results hold for drops whose potential,
rather than charge, is fixed
On well-posedness of variational models of charged drops
Electrified liquids are well known to be prone to a variety of interfacial
instabilities that result in the onset of apparent interfacial singularities
and liquid fragmentation. In the case of electrically conducting liquids, one
of the basic models describing the equilibrium interfacial configurations and
the onset of instability assumes the liquid to be equipotential and interprets
those configurations as local minimizers of the energy consisting of the sum of
the surface energy and the electrostatic energy. Here we show that,
surprisingly, this classical geometric variational model is mathematically
ill-posed irrespectively of the degree to which the liquid is electrified.
Specifically, we demonstrate that an isolated spherical droplet is never a
local minimizer, no matter how small is the total charge on the droplet, since
the energy can always be lowered by a smooth, arbitrarily small distortion of
the droplet's surface. This is in sharp contrast with the experimental
observations that a critical amount of charge is needed in order to destabilize
a spherical droplet. We discuss several possible regularization mechanisms for
the considered free boundary problem and argue that well-posedness can be
restored by the inclusion of the entropic effects resulting in finite screening
of free charges.Comment: 18 pages, 2 figure
Equilibrium shapes of charged droplets and related problems: (mostly) a review
We review some recent results on the equilibrium shapes of charged liquid
drops. We show that the natural variational model is ill-posed and how this can
be overcome by either restricting the class of competitors or by adding
penalizations in the functional. The original contribution of this note is
twofold. First, we prove existence of an optimal distribution of charge for a
conducting drop subject to an external electric field. Second, we prove that
there exists no optimal conducting drop in this setting
On an isoperimetric problem with a competing non-local term. II. The general case
This paper is the continuation of [H. Kn\"upfer and C. B. Muratov, Commun.
Pure Appl. Math. (2012, to be published)]. We investigate the classical
isoperimetric problem modified by an addition of a non-local repulsive term
generated by a kernel given by an inverse power of the distance. In this work,
we treat the case of general space dimension. We obtain basic existence results
for minimizers with sufficiently small masses. For certain ranges of the
exponent in the kernel we also obtain non-existence results for sufficiently
large masses, as well as a characterization of minimizers as balls for
sufficiently small masses and low spatial dimensionality. The physically
important special case of three space dimensions and Coulombic repulsion is
included in all the results mentioned above. In particular, our work yields a
negative answer to the question if stable atomic nuclei at arbitrarily high
atomic numbers can exist in the framework of the classical liquid drop model of
nuclear matter. In all cases the minimal energy scales linearly with mass for
large masses, even if the infimum of energy may not be attained
Some minimization problems for planar networks of elastic curve
In this note we announce some results that will appear in [6] (joint work
with also Matteo Novaga) on the minimization of the functional
, where is a network of
three curves with fixed equal angles at the two junctions. The informal
description of the results is accompanied by a partial review of the theory of
elasticae and a diffuse discussion about the onset of interesting variants of
the original problem passing from curves to networks. The considered energy
functional is given by the elastic energy and a term that penalize the
total length of the network. We will show that penalizing the length is
tantamount to fix it. The paper is concluded with the explicit computation of
the penalized elastic energy of the 'Figure Eight', namely the unique closed
elastica with self--intersections.Comment: 24 pages, 7 figure
Existence, regularity and structure of confined elasticae
We consider the problem of minimizing the bending or elastic energy among
Jordan curves confined in a given open set . We prove existence,
regularity and some structural properties of minimizers. In particular, when
is convex we show that a minimizer is necessarily a convex curve. We
also provide an example of a minimizer with self-intersections
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