Gisèle Ruiz Goldstein
Mathematical Sciences
Professional Development Assignment 2009 Report
In the spring semester of 2009 I had a Professional Development Assignment. This period without
teaching duties allowed me to focus on my research for a long period, and allowed me to concentrate my
efforts on my research at a very important time.
For most of the PDA we remained in Memphis. We had two Ph.D. students (Naima Naheed and Teddy
Clark) in their final semester at University of Memphis, and it would have been a hardship for them if we
were not available in their last semester. Naima was co-advised by Jerry and me; Ted was officially
Jerry’s student, but I worked very closely with him on his thesis as well. We finished papers with both
Naima and Ted during this period. These papers were based on their theses, but the papers extended the
work in their theses in new directions. Both Naima and Ted finished by August 2009 and got tenure-track
jobs in a very difficult job market.
In late 2008 I received an invitation to be a Distinguished Invited Research Professor at the University of
Poitiers in France for 2009. These are very competitive positions, and I was very happy to have received
this honor. We were at the University of Poitiers for the month of June 2009. While there, I was a
member of the Ph.D. Committee for Sami Injieru who received his Ph.D. from the University of Poitiers.
During the period in Poitiers, I began several projects with Alain Miranville, and Jerry and I began joint
projects with Hassan Emmamirad and Arnaud Rougeril. These projects have lead to several important
papers, and the collaborations still continue. Miranville and I studied a problem from materials science;
more specifically we looked at the Cahn-Hilliard equation (which governs binary mixtures) and showed
that the physically correct model is not the traditional model, but rather one which has different behavior
on the surface of the container holding the mixture. In mathematical terms we showed that the traditional
boundary conditions are not physically correct, and we derived the correct ones, and we showed this leads
to a well-posed problem. With Jerry and Hassan we studied the Nobel Prize winning Black-Scholes
equation of mathematical finance, and we showed that solutions of that equation are always chaotic in a
certain sense.
In addition to the new collaborations in Poitiers, I also completed several papers with long time coauthors. With Favini (Bologna), J. Goldstein, Obrecht (Bologna) and Romanelli (Bari), we obtained the
best solution to date of the famous conjecture of Agmon, Douglis and Nirenberg from 1959. This work
was published in Mathematiche Nachscricten. We were very honored when this paper received the
special distinction of Editor’s Choice in that journal. I also completed a paper with G. Coclite and J.
Goldstein on continuous dependence of the solution of nonlinear parabolic problems with dynamic (or
Wentzell) boundary conditions. Several new projects were also largely completed during the period of
this PDA.
I gave several important lectures during this period. I was a Plenary Lecturer at Eighth Mississippi State
Conference on Differential Equations and Computational Simulations, Mississippi State University, May
2009. I am especially proud of this invitation. I am one of two mathematicians from Tennessee ever to be
invited to give a plenary lecture at any of these international conferences. I also gave plenary lectures at
the International Conference on Evolution Equations and Mathematical Models in the Applied Sciences,
University of Taranto, Italy and at the Journée Picto-Charentaise d'EDP, La Rochelle, France. In addition
I gave colloquia at the Université de Poitiers and at the University of Bari.
Below I have listed the papers which were either completed, essentially completed or begun during my
PDA.
1. Elliptic operators with general Wentzell boundary conditions, analytic Semigroups and the Angle
Concavity Theorem, Mathematische Nachrichten, 283 (2010), 504-521 (with A. Favini, J. A. Goldstein,
E. Obrecht and S. Romanelli).
2. Weighted Hardy’s inequality and the Kolmogorov equation perturbed by an inverse square potential,
Applicable Analysis (2012), to appear (with J.A. Goldstein and A. Rhandi).
3. Wellposedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions,
Advances in Differential Equations (2011), 895-916 (with G. Coclite and J.A. Goldstein).
4. A convex minorant problem arising in electron density theory, Communications in Mathematical
Analysis 8 (2010), 1-11 (with J. A. Goldstein and Naima Naheed).
5. A convexified energy functional for the Fermi-Amaldi correction, Discrete and Continuous Dynamical
Systems 28 (2010), 41-65 (with J.A. Goldstein and Naima Naheed).
6. A Cahn-Hilliard model in a domain with non-permeable walls, Physcia D: Nonlinear Phenomena 240
(2011), 754-766 (with A. Miranville and G. Schimperna).
7. The Wentzell telegraph equation: Asymptotics and continuous dependence on the boundary conditions,
Communications in Applied Analysis, 15 (2011), 313-324 (with T. Clarke, J. A. Goldstein and S.
Romanelli).
8. Chaotic solution for the Black-Scholes equation, Proceedings of the American Mathematical Society,
140 (2012), 2043-2052 (with H. Emamirad and J.A. Goldstein).
I sincerely thank the College of Arts and Sciences and the University of Memphis for the Professional
Development Assignment. It was a wonderful opportunity to focus on my research.