117 research outputs found
The fractional Laplacian in power-weighted spaces: integration-by-parts formulas and self-adjointness
We consider the fractional Laplacian operator (let ) on Euclidean space and investigate the validity of the classical
integration-by-parts formula that connects the scalar
product between a function and its fractional Laplacian to the nonlocal norm of
the fractional Sobolev space . More precisely, we
focus on functions belonging to some weighted space whose fractional
Laplacian belongs to another weighted space: we prove and disprove the
validity of the integration-by-parts formula depending on the behaviour of the
weight at infinity. The latter is assumed to be like a power both
near the origin and at infinity (the two powers being possibly different). Our
results have direct consequences for the self-adjointness of the linear
operator formally given by . The generality of the
techniques developed allows us to deal with weighted spaces as well
Porous medium equations on manifolds with critical negative curvature: unbounded initial data
We investigate existence and uniqueness of solutions of the Cauchy problem
for the porous medium equation on a class of Cartan-Hadamard manifolds. We
suppose that the radial Ricci curvature, which is everywhere nonpositive as
well as sectional curvatures, can diverge negatively at infinity with an at
most quadratic rate: in this sense it is referred to as critical. The main
novelty with respect to previous results is that, under such hypotheses, we are
able to deal with unbounded initial data and solutions. Moreover, by requiring
a matching bound from above on sectional curvatures, we can also prove a
blow-up theorem in a suitable weighted space, for initial data that grow
sufficiently fast at infinity
Radial Fast Diffusion on the Hyperbolic Space
We consider radial solutions to the fast diffusion equation
on the hyperbolic space for , ,
. By radial we mean solutions depending only on the
geodesic distance from a given point . We investigate
their fine asymptotics near the extinction time in terms of a separable
solution of the form , where
is the unique positive energy solution, radial w.r.t. , to for a suitable , a semilinear elliptic problem thoroughly
studied in \cite{MS08}, \cite{BGGV}. We show that converges to in relative error, in the sense that as . In particular the solution is
bounded above and below, near the extinction time , by multiples of
.Comment: To appear in Proc. London Math. So
Smoothing effects for the filtration equation with different powers
We study the nonlinear diffusion equation on general
Euclidean domains, with homogeneous Neumann boundary conditions. We assume that
is bounded from below by for small
and by for large , the two exponents being
possibly different and larger than one. The equality case corresponds to the
well-known porous medium equation. We establish sharp short- and long-time - smoothing estimates: similar issues have widely been
investigated in the literature in the last few years, but the Neumann problem
with different powers had not been addressed yet. This work extends some
previous results in many directions
Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods
This paper is the second part of the study. In Part~I, self-similar solutions
of a weighted fast diffusion equation (WFD) were related to optimal functions
in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied
to radially symmetric functions. For these inequalities, the linear instability
(symmetry breaking) of the optimal radial solutions relies on the spectral
properties of the linearized evolution operator. Symmetry breaking in (CKN) was
also related to large-time asymptotics of (WFD), at formal level. A first
purpose of Part~II is to give a rigorous justification of this point, that is,
to determine the asymptotic rates of convergence of the solutions to (WFD) in
the symmetry range of (CKN) as well as in the symmetry breaking range, and even
in regimes beyond the supercritical exponent in (CKN). Global rates of
convergence with respect to a free energy (or entropy) functional are also
investigated, as well as uniform convergence to self-similar solutions in the
strong sense of the relative error. Differences with large-time asymptotics of
fast diffusion equations without weights will be emphasized
Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities
In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation
inequalities (CKN), with two radial power law weights and exponents in a
subcritical range. We address the question of symmetry breaking: are the
optimal functions radially symmetric, or not ? Our intuition comes from a
weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy
- entropy production inequality which governs the intermediate asymptotics is
indeed equivalent to (CKN), and the self-similar profiles are optimal for
(CKN). We establish an explicit symmetry breaking condition by proving the
linear instability of the radial optimal functions for (CKN). Symmetry breaking
in (CKN) also has consequences on entropy - entropy production inequalities and
on the intermediate asymptotics for (WFD). Even when no symmetry holds in
(CKN), asymptotic rates of convergence of the solutions to (WFD) are determined
by a weighted Hardy-Poincar{\'e} inequality which is interpreted as a
linearized entropy - entropy production inequality. All our results rely on the
study of the bottom of the spectrum of the linearized diffusion operator around
the self-similar profiles, which is equivalent to the linearization of (CKN)
around the radial optimal functions, and on variational methods. Consequences
for the (WFD) flow will be studied in Part II of this work
Conditions at infinity for the inhomogeneous filtration equation
We investigate existence and uniqueness of solutions to the filtration
equation with an inhomogeneous density in , approaching at
infinity a given continuous datum of Dirichlet type.Comment: To appear in Annales de l'Institut Henri Poincar\'e (C) Analyse Non
Lin\'eair
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