- Office: MATH 830
- Phone: 765-494-1908
- Email: banuelos@math.purdue.edu
RESEARCH INTERESTS
My research is at the interface of probability, harmonic analysis,
partial differential equations and spectral theory. As reflected by
my list of
publications, I have devoted
considerable time in the last few years to the following three areas of
research.
- The first concerns the study of the "fine" spectral theoretic
properties of symmetric Levy processes. These stochastic processes
are natural extensions of Brownian motion. But despite the extensive literature on
their potential theory, many of the basic spectral
theoretic properties for which there is now a vast knowledge for the case
of Brownian motion, as for example geometric and analytic properties of
eigenvalues, eigenfunctions, heat kernels, and more, remain completely
open. This is the case even in the simplest geometric setting of the
interval (-1, 1) in R. Indeed, my interest on this topic started with a
question raised to me by Davar Khoshnevisan of the University of
Utah: What is the lowest
eigenvalue for the symmetric stable processes in (-1, 1)?
I found it difficult to believe that such a simple question had not been answered.
Several years later, and after
writing several papers on the spectral theory of symmetric
stable processes, I have learned to appreciate this "simple" problem.
- My second area of research in recent years has centered on efforts
to gain a better understanding of the celebrated "hot-spots" conjecture. This
conjecture, made in 1974 by J. Rauch, asserts that the maximum (and
minimum) of the first non-constant Neumann eigenfunction for a smooth
bounded domain in Rn is attained on the boundary and only on
the
boundary of the domain. Without any assumptions on the domain
the conjecture is false. However, it is widely believe to be true
for convex domains but this remains open even
in two dimensions, and even for arbitrary triangles. With some understanding of the geometry of the
nodal curve of the eigenfunction, the conjecture reduces to a maximum
principle for a mixed (Dirichlet-Neumann) boundary value problem. One
can then relate this problem to properties of Brownian motion in the domain
with killing and reflection. The Brownian motion approach to the problem was initiated in the
article title "On the 'hot spots' conjecture of J.Rauch," co-author with K.Burdzy
of the University of Washington, Seattle, and published in 1999.
- The third area of my research has been on martingale inequalities and their applications to Lp-estimates for singular integrals of even kernels, and in particular to Lp-estimates for the Beurling-Ahlfors (BA) operator. A celebrated conjecture of T. Iwaniec (1982) asserts that the norm in Lp, 1<p<infinity, of the BA operator is p*-1 where p* is the maximum of p and its conjugate exponent. There are many interesting consequences of this conjecture to quasiconformal mappings and to regularity results for solutions of certain nonlinear PDE's. The BA operator, like many other classical Calderón-Zygmund singular integrals, can be represented by certain Itô stochastic integrals. From here the powerful sharp martingale inequality techniques of D.L. Burkholder can be brought to bear to to make significant progress on the conjecture.