Please visit www.ms.uky.edu/~hislop/PDEseminarF2015 for the title and abstract and for other Analysis and PDE Seminar events.
Title: Large-Scale Boundary Regularity Estimates in Periodic Homogenization
Abstract: In this talk I will discuss some of recent progress on boundary regularity at large scale in homogenization of second-order elliptic systems in divergence form with rapidly oscillating periodic coecients. Here we are interested in the estimates, due to homogenization, for operators with bounded measurable coecients. These large-scale estimates, combined with the small-scale estimates obtained by rescaling, lead to the sharp regularity estimates at all scales. In particular, the cases of the Rellich estimates in Lipschitz domains and the Lipschitz estimates in C1; domains will be presented.
TITLE: The Robin Laplacian, Faber-Krahn inequalities, and mean curvature
Abstract: Available at http://www.ms.uky.edu/~hislop/PDEseminarS2015
Title: Oscillatory Riemann Hilbert
Abstract: Please see the seminar web page at http://www.ms.uky.edu/~hislop/PDEseminarS2015
TITLE: Scattering Resonances on Hyperbolic Manifolds as a Model of Chaotic Scattering
ABSTRACT: Scattering resonances represent "almost standing waves" in a scattering system which have a nite lifetime as measured by energy decay in a nte region. In this survey talk well review the basics of scattering theory on geometrically nite, real hyperbolic manifolds and their role as models of open chaotic systems. As such they have attracted the interest of both mathematicians and physicists. Work to be discussed includes the work of Patterson and Perry and papers by Borthwick, Guillope-Zworski, Guillarmou, Naud, and others. Real hyperbolic manifolds provide a useful "laboratory" for scattering because their symmetries allow for the use of powerful methods from the theory of automorphic functions, dynamics, and the theory of Fuchsian groups. We'll discuss the connection between scattering resonances and Helberg's zeta function for a hyperbolic surface, and in turn the connection between Selberg's zeta function and the Ruelle zeta function from dynamical systems. Through this connection one can uncover close relationships between the Hausdor dimension of the trapped set for geodesic ow on the one hand, and the distribution of scattering resonances on the other.
Title: Gradient estimates for the Stokes semigroup subject Neumann boundary conditions in bounded convex domains.
Abstract available at http://www.ms.uky.edu/~hislop/PDEseminarS2015
Abstract available at:
Title: Fluid PDEs with partial or fractional dissipation.
Title: Boundary integral equations of coupled thermoelastodynamics
Title: Using the method of layer potentials to solve a mixed boundary value problem
Abstract: Following the exposition by William McLean in his book Strongly Elliptic Systems and Boundary Integral Equations, we use the method of layer potentials to show that on a bounded Lipschitz domain, the mixed problem for Laplace’s equation is equivalent to a 2 × 2 system of boundary integral equations.
