2,617 research outputs found
Functional Inequalities: New Perspectives and New Applications
This book is not meant to be another compendium of select inequalities, nor
does it claim to contain the latest or the slickest ways of proving them. This
project is rather an attempt at describing how most functional inequalities are
not merely the byproduct of ingenious guess work by a few wizards among us, but
are often manifestations of certain natural mathematical structures and
physical phenomena. Our main goal here is to show how this point of view leads
to "systematic" approaches for not just proving the most basic functional
inequalities, but also for understanding and improving them, and for devising
new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a
pre-publication pdf cop
Existence, stability and oscillation properties of slow decay positive solutions of supercritical elliptic equations with Hardy potential
We prove the existence of a family of slow decay positive solutions of a
supercritical elliptic equation with Hardy potential in the entire space and
study stability and oscillation properties of these solutions. We also
establish the existence of a continuum of stable slow decay positive solutions
for the relevant exterior Dirichlet problem
"Boundary blowup" type sub-solutions to semilinear elliptic equations with Hardy potential
Semilinear elliptic equations which give rise to solutions blowing up at the
boundary are perturbed by a Hardy potential. The size of this potential effects
the existence of a certain type of solutions (large solutions): if the
potential is too small, then no large solution exists. The presence of the
Hardy potential requires a new definition of large solutions, following the
pattern of the associated linear problem. Nonexistence and existence results
for different types of solutions will be given. Our considerations are based on
a Phragmen-Lindelof type theorem which enables us to classify the solutions and
sub-solutions according to their behavior near the boundary. Nonexistence
follows from this principle together with the Keller-Osserman upper bound. The
existence proofs rely on sub- and super-solution techniques and on estimates
for the Hardy constant derived in Marcus, Mizel and Pinchover.Comment: 23 pages, 3 figure
Sobolev inequalities for the Hardy-Schr\"odinger operator: Extremals and critical dimensions
In this expository paper, we consider the Hardy-Schr\"odinger operator
on a smooth domain \Omega of R^n with 0\in\bar{\Omega},
and describe how the location of the singularity 0, be it in the interior of
\Omega or on its boundary, affects its analytical properties. We compare the
two settings by considering the optimal Hardy, Sobolev, and the
Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites:
for all ,
where \gamma <n^2/4, s\in [0,2) and p:=2(n-s)/(n-2). We address questions
regarding the explicit values of the optimal constant C, as well as the
existence of non-trivial extremals attached to these inequalities. Scale
invariance properties lead to situations where the best constants do not depend
on the domain and are not attainable. We consider two different approaches to
"break the homogeneity" of the problem:
One approach was initiated by Brezis-Nirenberg and by Janelli. It is suitable
for the case where 0 is in the interior of \Omega, and consists of considering
lower order perturbations of the critical nonlinearity. The other approach was
initiated by Ghoussoub-Kang , C.S. Lin et al. and Ghoussoub-Robert. It consists
of considering domains where the singularity is on the boundary.
Both of these approaches are rich in structure and in challenging problems.
If 0\in \Omega, a negative linear perturbation suffices for higher dimensions,
while a positive "Hardy-singular interior mass" is required in lower
dimensions. If the singularity is on the boundary, then the local geometry
around 0 plays a crucial role in high dimensions, while a positive
"Hardy-singular boundary mass" is needed for the lower dimensions.Comment: Expository paper. 48 page
Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains
We study the existence and nonexistence of positive (super-)solutions to a
singular semilinear elliptic equation in cone--like domains of (),
for the full range of parameters and . We provide a
complete characterization of the set of such that the
equation has no positive (super-)solutions, depending on the values of
and the principle Dirichlet eigenvalue of the cross--section of the cone.
The proofs are based on the explicit construction of appropriate barriers and
involve the analysis of asymptotic behavior of super-harmonic functions
associated to the Laplace operator with critical potentials,
Phragmen--Lindel\"of type comparison arguments and an improved version of
Hardy's inequality in cone--like domains.Comment: 30 pages, 1 figur
On semilinear elliptic equations with borderline Hardy potentials
In this paper we study the asymptotic behavior of solutions to an elliptic
equation near the singularity of an inverse square potential with a coefficient
related to the best constant for the Hardy inequality. Due to the presence of a
borderline Hardy potential, a proper variational setting has to be introduced
in order to provide a weak formulation of the equation. An Almgren-type
monotonicity formula is used to determine the exact asymptotic behavior of
solutions
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