668 research outputs found
Spindle Starshaped Sets
In this paper, spindle starshaped sets are introduced and investigated, which
apart from normalization form an everywhere dense subfamily within the family
of starshaped sets. We focus on proving spindle starshaped analogues of recent
theorems of Bobylev, Breen, Toranzos, and Zamfirescu on starshaped sets.
Finally, we consider the problem of guarding treasures in an art gallery (in
the traditional linear way as well as via spindles).Comment: 16 pages, 2 figure
Initial trace of solutions of Hamilton-Jacobi parabolic equation with absorption
Here we study the initial trace problem for the nonnegative solutions of the
equation in
where
and or is a smooth bounded domain of
and on We can
define the trace at as a nonnegative Borel measure where is the closed set where it is infinite, and is a
Radon measure on We show that the trace is a
Radon measure when For and any given Borel
measure, we show the existence of a minimal solution, and a maximal one on
conditions on When and
is an open subset of the existence extends to any
when and any when . In
particular there exists a self-similar nonradial solution with trace
with a growth rate of order as for fixed
Moreover we show that the solutions with trace in
may present near a growth rate of order
in and of order on $\partial
\omega.
Starshapedeness for fully-nonlinear equations in Carnot groups
In this paper we establish the starshapedness of the level sets of the
capacitary potential of a large class of fully-nonlinear equations for
condensers in Carnot groups, once a natural notion of starshapedness has been
introduced. Our main result is Theorem 1.2 below.Comment: Accepted for publication in the Journal of the London Mathematical
Societ
Geometric-type Sobolev inequalities and applications to the regularity of minimizers
The purpose of this paper is twofold. We first prove a weighted Sobolev
inequality and part of a weighted Morrey's inequality, where the weights are a
power of the mean curvature of the level sets of the function appearing in the
inequalities. Then, as main application of our inequalities, we establish new
and estimates for semi-stable solutions of in
a bounded domain of . These estimates lead to an
bound for the extremal solution of when and the domain is convex. We recall that extremal
solutions are known to be bounded in convex domains if , and that
their boundedness is expected ---but still unkwown--- for .Comment: 20 pages; 1 figur
Topology of geometric joins
We consider the geometric join of a family of subsets of the Euclidean space.
This is a construction frequently used in the (colorful) Carath\'eodory and
Tverberg theorems, and their relatives. We conjecture that when the family has
at least sets, where is the dimension of the space, then the
geometric join is contractible. We are able to prove this when equals
and , while for larger we show that the geometric join is contractible
provided the number of sets is quadratic in . We also consider a matroid
generalization of geometric joins and provide similar bounds in this case
Separable solutions of quasilinear Lane-Emden equations
For and , we prove the existence of solutions of
-\Gd_pu=\ge u^q in a cone , with vertex 0 and opening , vanishing on
\prt C_S, under the form u(x)=|x|^\gb\gw(\frac{x}{|x|}). The problem
reduces to a quasilinear elliptic equation on and existence is based upon
degree theory and homotopy methods. We also obtain a non-existence result in
some critical case by an integral type identity.Comment: To appear in Journal of the European Mathematical Societ
Solutions of some nonlinear parabolic equations with initial blow-up
We study the existence and uniqueness of solutions of () in where
is a domain with a compact boundary, subject to the conditions on
and the initial condition . By means of Brezis' theory of maximal monotone operators in
Hilbert spaces, we construct a minimal solution when , whatever is the
regularity of the boundary of the domain. When satisfies the
parabolic Wiener criterion and is continuous, we construct a maximal
solution and prove that it is the unique solution which blows-up at
Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles
The problem of deciding if a given triangulation of a sphere can be realized
as the boundary sphere of a simplicial, convex polytope is known as the
"Simplicial Steinitz problem". It is known by an indirect and non-constructive
argument that a vast majority of Bier spheres are non-polytopal. Contrary to
that, we demonstrate that the Bier spheres associated to threshold simplicial
complexes are all polytopal. Moreover, we show that all Bier spheres are
starshaped. We also establish a connection between Bier spheres and
Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the
KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to
the Bier sphere of the associated simplicial complex of "short sets"
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