SGS XXVI: Sunday, February 26, 2017, Georgia Institute of Technology

Mozghan (Nora) Entekhabi (Wichita State University)
Title: Radial Limits of Bounded Nonparametric Prescribed Mean Curvature Surfaces

Abstract: We show that radial limits of bounded solutions of the equation of prescribed mean curvature over re-entrant corner domains always exist for directions interior to the domain. Under additional conditions we show radial limits also exist for convex corners. No assumptions beyond boundedness are made on the behavior of the traces of solutions on the sides of the domain.

Miyuki Koiso (Kyushu University, Fukuoka)
Title: Stability and bifurcation for surfaces with constant mean curvature

Abstract: A surface with constant mean curvature (CMC surface) is an equilibrium surface of the area functional among surfaces which enclose the same volume (and satisfy given boundary conditions). A CMC surface is said to be stable if the second variation of the area is non-negative for all volume-preserving variations. In this talk we give criteria for stability of CMC surfaces in the three-dimensional euclidean space. We also give a sufficient condition for the existence of smooth bifurcation branches of fixed boundary CMC surfaces, and we discuss stability/ instability issues for the surfaces in bifurcating branches. By applying our theory, we determine the stability/instability of some explicit examples of CMC surfaces.

Vladimir Oliker (Emory University)
Title: Freeform lenses, Jacobian equations, and supporting quadric method(SQM)

Abstract: Design of freeform refractive lenses is known to be a difficult inverse problem. But solutions, if available, can be very useful, especially in devices required to redirect and reshape the radiance of the source into an output irradiance redistributed over a given target according to a prescribed pattern. In this talk I present the results of theoretical and numerical analysis of refractive lenses designed with the Supporting Quadric Method. It is shown that such freeform lenses have a particular simple geometry and qualitatively their diffractive properties are comparable with rotationally symmetric lenses designed with classical methods.

Sungho Park (Hankuk University of Foreign Studies, Seoul)
Title: Circle-foliated minimal and CMC surfaces in S^3

Abstract: We classify circle-foliated minimal surfaces and surfaces of constant mean curvature in $\mathbb S^3$ (locally). In special, we show that there is only one cmc surface foliated by geodesics for each mean curvature.

Yuanzhen Shao (Purdue University)
Title: Degenerate and singular elliptic operators on manifolds with singularities

Abstract: In this talk, we will introduce the concept of manifolds with singularities and study a class of elliptic differential operators that exhibit degenerate or singular behavior near the singularities. Based on this theory, we investigate several linear and nonlinear parabolic equations arising from geometric analysis and PDE. Emphasis will be given to geometric flows with "bad" initial metrics.

Ray Treinen (Texas State University)
Title: Unexpected non-uniqueness of equilibria for the 2D floating ball

Abstract: Abstract: We consider a two-dimensional analogue of the problem of a ball of prescribed density floating on a liquid that partially fills a bounded container. Restricting to the case where the density of the ball is less than that of the liquid, we use a phase plane analysis to show existence of equilibrium configurations. This framework also gives us an approach to studying the uniqueness of the equilibrium configurations, and (surprisingly) there are examples of physical parameters that lead to non-uniqueness.