Analysis and Partial Differential Equations
Professors Beatrous, Jiang, Hajlasz, Lennard, Lewicka, Manfredi, Pakzad, Pan and Rabier
The research of the analysis group covers functional analysis, harmonic analysis, several complex variables, partial differential equations, and analysis on metric and Carnot-Caratheodory spaces. Some of the topics are described in more detail below.
Beatrous' research is primarily in several complex variables and secondarily in harmonic and functional analysis.
Hajlasz’s research is focused on the theory of Sobolev spaces with applications to various areas like the theory of quasiconformal mappings, calculus of variations, regularity of nonlinear elliptic PDEs, and Carnot-Caratheodory spaces. He is particularly interested in analysis on metric spaces, including the theory of Sobolev spaces on metric spaces. Recently, Hajlasz has been interested in the theory of Sobolev mappings between manifolds and metric spaces. This includes questions about density of smooth and Lipschitz mappings with connection topology of spaces and also regularity theory for p-harmonic mappings between manifolds.
Lennard's research interests include these topics.
- Banach space geometry and metric fixed point theory. He works mainly with Paddy Dowling and Barry Turett, trying to understand which Banach spaces support fixed-point-free non-expansive mappings on small sets (e.g., those that are weakly compact and convex).
- Convergence properties in Banach spaces. The uniform Kadec-Klee property is an analogue of uniform convexity that many classical nonreflexive spaces enjoy.
- Banach and Hilbert frames. Frames are non-linearly independent analogues of bases in Banach spaces that have many applications (e.g., in signal processing).
- Roundness and metric type. The notions of roundness, generalized roundness, and metric type are related to the isometric embedding of metric spaces into Hilbert and Banach spaces and to the classification of Banach spaces via uniform homeomorphisms.
Manfredi and his graduate students Robert Berry and Alexander Sviridov work on the p-Laplace equation, including p equals infinity, in Euclidean space and Carnot groups, and their connection with the Monge-Kantorovich mass transfer problem. They study the regularity of p-harmonic functions in the Heisenberg group, the connection between the Neuman problem for the infinite Laplacian and mass transport, and fine properties of p-harmonic measure. More recently, Manfredi has developed an interest in the theory of stochastic games in connection with their applications to nonlinear elliptic equations.
Pakzad's research concerns nonconvex calculus of variations and geometric analysis. He has worked on harmonic mappings into sphere, on Sobolev spaces between manifolds and Sobolev isometric immersions. This includes study of regularity and rigidity properties of isometric immersions from a an analytical perspective. Most recently he works on problems in nonlinear theory of elasticity and shell and plate theories, and also on metric-driven shape formation and non-Euclidean elasticity. Study of geometric PDEs from various new perspectives is a major ingredient in advancing these projects.
Pan's main research area is harmonic analysis. Currently his research is focused on problems related to singular and oscillatory integrals and their behavior on L^p and Hardy spaces.
Rabier's interests are in functional analysis, PDEs, ODEs, and numerical analysis (in that order).