Rodrigo Bañuelos
Professor of Mathematics
and Department Head

• 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.

1. 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.

2. 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.

3. The third area of my research has been on martingale inequalities and their applications to L^p-estimates for singular integrals of even kernels, and in particular to L^p-estimates for the Beurling-Ahlfors (BA) operator. A celebrated conjecture of T. Iwaniec (1982) asserts that the norm in L^p, 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.

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