Georgia Institute of Technology College of Sciences

The oval problem asks to determine, among all closed loops in${\bf R}^n$ of fixed length, carrying a Schrödinger operator${\bf H}= -\frac{d^2}{ds^2}+\kappa^2$ (with curvature $\kappa$ andarclength $s$), those loops for which the principal eigenvalue of${\bf H}$ is smallest. A 1-parameter family of ovals connecting the circlewith a doubly traversed segment (digon) is conjectured to be the minimizer.Whereas this conjectured solution is an example that proves a lack ofcompactness and coercivity in the problem, it is proved in this talk(via a relaxed variation problem) that a minimizer exists; it is eitherthe digon, or a strictly convex planar analytic curve with positivecurvature. While the Euler-Lagrange equation of the problem appearsdaunting, its asymptotic analysis near a presumptive singularity givesuseful information based on which a strong variation can excludesingular solutions as minimizers.

We prove a quantitative Brunn-Minkowski inequality for sets E and K,one of which, K, is assumed convex, but without assumption on the other set. We are primarily interested in the case in which K is a ball. We use this to prove an estimate on the remainder in the Riesz rearrangement inequality under certain conditions on the three functions involved that are relevant to a problem arising in statistical mechanics: This is joint work with Franceso Maggi.

Following Kato, we define the sum, $H=H_0+V$, of two linear operators, $H_0$ and $V$, in a fixed Hilbert space in terms of its resolvent. In an abstract theorem, we present conditions on $V$ that guarantee $\text{dom}(H_0^{1/2})=\text{dom}(H^{1/2})$ (under certain sectorality assumptions on $H_0$ and $H$). Concrete applications to non-self-adjoint Schr\"{o}dinger-type operators--including additive perturbations of uniformly elliptic divergence form partial differential operators by singular complex potentials on domains--where application of the abstract theorem yields $\text{dom}(H^{1/2})=\text{dom}((H^{\ast})^{1/2})$, will be presented. This is based on joint work with Fritz Gesztesy and Steve Hofmann.

We consider a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature \beta. The system admits the canonical distribution at inverse temperature \beta as the unique equilibrium state. We prove that the any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large system limit, satisfies an effective Boltzmann-type equation. This is joint work with Federico Bonetto and Michael Loss.