Analysis Seminar

ABSTRACT: Bochner’s tube theorem states that a holomorphic function of several complex variables defined on an open tube can be extended to the convex hull of the tube. We will describe some recent CR generalizations of this result.

Abstract: Fluid flows in the presence of free surfaces occur in a great many situations in nature; examples include waves on the ocean and the flow of groundwater. In this talk, I will discuss my contributions to the understanding of the systems of nonlinear partial differential equations which model such phenomena. The most important step in these results is making a suitable formulation of the problem. Influenced by the computational work of Hou, Lowengrub, and Shelley, we formulate the problems in natural, geometric variables. I will discuss my proofs (most of which are joint with Nader Masmoudi) of existence of solutions to the initial value problems for vortex sheets and water waves. I will also discuss computational results, including work with Jon Wilkening on the computation of special solutions, especially time- periodic interfacial flows

Abstract: In this talk we study a reflector system which consists of a point light source, a reflecting surface and an object to be illuminated. Due to its practical applications in optics, electro-magnetics, and acoustic, it has been extensively studied during the last half century. This problem involves a fully nonlinear partial differential equation of Monge-Ampere type, subject to a nonlinear second boundary condition. In the far field case, it is related to the reflector antenna design problem and optimal transportation problem. Therefore, the regularity results of optimal transportation can be applied. However, in the general case, the reflector problem is not an optimal transportation problem and the regularity is an extremely complicated issue. In this talk, we give necessary and sufficient conditions for the global regularity and briefly discuss their connection with the Ma-Trudinger-Wang condition in optimal transportation. This is a joint work with Neil Trudinger.

Abstract: The problem of designing optical systems that contain free- form surfaces (i.e. not rotationally symmetric) is a challenging one, even in the case of designing a single surface. Part of the reason for this is that solu- tions do not always exist. Here we present a method for the coupled design of two free-formreflective surfaces (i.e. mirrors) which will have a prescribed distortion. One should think for example of a child’s periscope with curved mirrors so as to give a wider field of view. The method is motivated by viewing the problem in the language of distributions from differential geom- etry and makes use of the Cartan Kaehler theorem from exterior differential systems for proof of existence. The method can also be described using tra- ditional vectors and matrices, which we do. We give example applications to the design of a mirror pair that increases the field of view of an observer, a similar mirror pair that also rotate the observers view, and a pair of mirrors that give the observer a traditional panoramic strip view of the scene.

Abstract: We discuss viscosity solutions $u\in C(\overline{\Omega})$ to the Dirichlet problem \[ (DP)\qquad \begin{cases} \Delta_{\infty}u=f(x,u), & x\in\Omega,\\ u=b, & x\in\partial\Omega. \end{cases} \] Here $\Omega\subset\mathbb{R}^{n}$ is a bounded domain, $b\in\partial\Omega$, $f$ is continuous on $\Omega\times\mathbb{R}$ and $\Delta_{\infty}u$ is the infinity Laplacian, a highly degenerate elliptic operator, given by $\Delta_{\infty}u=\langle D^{2}uDu,Du\rangle$. General conditions on $f$ under which the above Dirichlet problem admits a solution in $C(\overline{\Omega})$ for any bounded domain $\Omega$ will be given. In contrast, we will identify a class of inhomogeneous terms $f$ for which Problem (DP) has no solution in $C(\overline{\Omega})$ provided $\Omega$ contains a large ball. Some Comparison principles and Harnack inequalities will also be discussed. Finally we will mention several open problems. This talk is based on a joint work with Tilak Bhattacharya.

Abstract: Quasiconformal mappings $u:\Omega\to\Omega'$ between open domains in $\mathbb{R}^{n}$, are $W^{1,n}$ homeomorphisms whose dilation $K = |du|/(\det du)^{1/n}$ is in $L^{\infty}$. A classical problem in geometric function theory consists in finding QC minimizers for the dilation within a given homotopy class or with prescribed boundary data. In a joint work with A. Raich we study $C^{2}$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a ‘localized’ form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.