752 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
On a diffusion model with absorption and production
We discuss the structure of radial solutions of some superlinear elliptic
equations which model diffusion phenomena when both absorption and production
are present. We focus our attention on solutions defined in R (regular) or in R
\ {0} (singular) which are infinitesimal at infinity, discussing also their
asymptotic behavior. The phenomena we find are present only if absorption and
production coexist, i.e., if the reaction term changes sign. Our results are
then generalized to include the case where Hardy potentials are considered
Lyapunov-type Inequalities for Partial Differential Equations
In this work we present a Lyapunov inequality for linear and quasilinear
elliptic differential operators in dimensional domains . We also
consider singular and degenerate elliptic problems with coefficients
involving the Laplace operator with zero Dirichlet boundary condition.
As an application of the inequalities obtained, we derive lower bounds for
the first eigenvalue of the Laplacian, and compare them with the usual ones
in the literature
Existence and classification of nonoscillatory solutions of two dimensional time scale systems
During the past years, there has been an increasing interest in studying oscillation and nonoscillation criteria for dynamic equations and systems on time scales that harmonize the oscillation and nonoscillation theory for the continuous and discrete cases in order to combine them in one comprehensive theory and eliminate obscurity from both.
We not only classify nonoscillatory solutions of dynamic equations and systems on time scales but also guarantee the (non)existence of such solutions by using the Knaster fixed point theorem, Schauder - Tychonoff fixed point theorem, and Schauder fixed point theorem. The approach is based on the sign of nonoscillatory solutions. A short introduction to the time scale calculus is given as well.
Examples are significant in order to see if nonoscillatory solutions exist or not. Therefore, we give several examples in order to highlight our main results for the set of real numbers R, the set of integers Z, and qN0 = {1, q, q2, q3, ...}, q \u3e 1, which are the most well-known time scales --Abstract, page iv
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