57 research outputs found
On some strong ratio limit theorems for heat kernels
We study strong ratio limit properties of the quotients of the heat kernels
of subcritical and critical operators which are defined on a noncompact
Riemannian manifold.Comment: 16 pages. This version coincides with the published one, except for
Remark 4 added after the paper has appeare
On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators
This note contains a representation formula for positive solutions of linear
degenerate second-order equations of the form proved by a functional analytic approach based on Choquet
theory. As a consequence, we obtain Liouville-type theorems and uniqueness
results for the positive Cauchy problem.Comment: The results of the present version recover most of the ones in the
previous version, but, on top of it, this new version contains some further
new and interesting result
Expanding perfect fluid generalizations of the C-metric
We reexamine Petrov type D gravitational fields generated by a perfect fluid
with spatially homogeneous energy density and in which the flow lines form a
timelike non-shearing and non-rotating congruence. It is shown that the
anisotropic such spacetimes, which comprise the vacuum C-metric as a limit
case, can have \emph{non-zero} expansion, contrary to the conclusion in the
original investigation by Barnes (Gen. Rel. Grav. 4, 105 (1973)). This class
consists of cosmological models with generically one and at most two Killing
vectors. We construct their line element and discuss some important properties.
The methods used in this investigation incite to deduce testable criteria
regarding shearfree normality and staticity op Petrov type spacetimes in
general, which we add in an appendix.Comment: 16 pages, extended and amended versio
On the computation of zone and double zone diagrams
Classical objects in computational geometry are defined by explicit
relations. Several years ago the pioneering works of T. Asano, J. Matousek and
T. Tokuyama introduced "implicit computational geometry", in which the
geometric objects are defined by implicit relations involving sets. An
important member in this family is called "a zone diagram". The implicit nature
of zone diagrams implies, as already observed in the original works, that their
computation is a challenging task. In a continuous setting this task has been
addressed (briefly) only by these authors in the Euclidean plane with point
sites. We discuss the possibility to compute zone diagrams in a wide class of
spaces and also shed new light on their computation in the original setting.
The class of spaces, which is introduced here, includes, in particular,
Euclidean spheres and finite dimensional strictly convex normed spaces. Sites
of a general form are allowed and it is shown that a generalization of the
iterative method suggested by Asano, Matousek and Tokuyama converges to a
double zone diagram, another implicit geometric object whose existence is known
in general. Occasionally a zone diagram can be obtained from this procedure.
The actual (approximate) computation of the iterations is based on a simple
algorithm which enables the approximate computation of Voronoi diagrams in a
general setting. Our analysis also yields a few byproducts of independent
interest, such as certain topological properties of Voronoi cells (e.g., that
in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI;
Ref [51] points to a freely available computer application which implements
the algorithms; to appear in Discrete & Computational Geometry (available
online
Singular solutions of fully nonlinear elliptic equations and applications
We study the properties of solutions of fully nonlinear, positively
homogeneous elliptic equations near boundary points of Lipschitz domains at
which the solution may be singular. We show that these equations have two
positive solutions in each cone of , and the solutions are unique
in an appropriate sense. We introduce a new method for analyzing the behavior
of solutions near certain Lipschitz boundary points, which permits us to
classify isolated boundary singularities of solutions which are bounded from
either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as
well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure
Semiclassical stationary states for nonlinear Schroedinger equations with fast decaying potentials
We study the existence of stationnary positive solutions for a class of
nonlinear Schroedinger equations with a nonnegative continuous potential V.
Amongst other results, we prove that if V has a positive local minimum, and if
the exponent of the nonlinearity satisfies N/(N-2)<p<(N+2)/(N-2), then for
small epsilon the problem admits positive solutions which concentrate as
epsilon goes to 0 around the local minimum point of V. The novelty is that no
restriction is imposed on the rate of decay of V. In particular, we cover the
case where V is compactly supported.Comment: 22 page
Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations
Multiple blowing-up and concentrating solutions for Liouville-type equations with singular sources under mixed boundary conditions
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