455 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
Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-Order PDEs
Very singular self-similar solutions of semilinear odd-order PDEs are studied
on the basis of a Hermitian-type spectral theory for linear rescaled odd-order
operators.Comment: 49 pages, 12 Figure
A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart
We establish the nonexistence of nontrivial ancient solutions to the
nonlinear heat equation which are smaller in absolute
value than the self-similar radial singular steady state, provided that the
exponent is strictly between Serrin's exponent and that of Joseph and
Lundgren. This result was previously established by Fila and Yanagida [Tohoku
Math. J. (2011)] by using forward self-similar solutions as barriers. In
contrast, we apply a sweeping argument with a family of time independent weak
supersolutions. Our approach naturally lends itself to yield an analogous
Liouville type result for the steady state problem in higher dimensions. In
fact, in the case of the critical Sobolev exponent we show the validity of our
results for solutions that are smaller in absolute value than a 'Delaunay'-type
singular solution.Comment: In this third version, we clarified that the approach of Fila and
Yanagida [Tohoku Math. J. (2011)] also works in the subcritical regim
Hyperboloidal layers for hyperbolic equations on unbounded domains
We show how to solve hyperbolic equations numerically on unbounded domains by
compactification, thereby avoiding the introduction of an artificial outer
boundary. The essential ingredient is a suitable transformation of the time
coordinate in combination with spatial compactification. We construct a new
layer method based on this idea, called the hyperboloidal layer. The method is
demonstrated on numerical tests including the one dimensional Maxwell equations
using finite differences and the three dimensional wave equation with and
without nonlinear source terms using spectral techniques.Comment: 23 pages, 23 figure
Entire large solutions for semilinear elliptic equations
We analyze the semilinear elliptic equation , in
, with a particular emphasis put on the qualitative
study of entire large solutions, that is, solutions such that
. Assuming that satisfies the
Keller-Osserman growth assumption and that decays at infinity in a
suitable sense, we prove the existence of entire large solutions. We then
discuss the more delicate questions of asymptotic behavior at infinity,
uniqueness and symmetry of solutions.Comment: Journal of Differential Equations 2012, 28 page
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