# Analysis and Partial Differential Equations

### Armin Schikorra (Associate Professor, PhD)

Schikorra's research is focused on the analysis of partial differential equations, often motivated from geometric or topological calculus of variations. The methods involve tools from harmonic analysis, such as commutator estimates and compensation effects.

### Beatrous' Research (Emeritus Professor)

Beatrous' research is primarily in several complex variables and secondarily in harmonic and functional analysis.

### Christopher Lennard (Assoc. Professor,PhD)

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.

### Dehua Wang (Professor, PhD)

Wang's research area is analysis of partial differential equations arising in fluid dynamics, hyperbolic conservation laws, elastodynamics, geometry, kinetic theory, biology, etc. He works on nonlinear PDE problems including transonic flows, free boundary problems, vortex sheets, turbulence, isometric embeddings, boundary layer, regularity, and so on.

### Juan Manfredi (Professor of Mathematics, PhD)

Manfredi works on the p-Laplace equation, including p equals infinity, in Euclidean space and Riemannian and subRiemannian manifolds, and their connections with tug-of-war stochastic games. He studies the regularity of p-harmonic functions, asymptotic mean value properties for p-harmonic functions, and convergence of semi-discrete approximations induced by mean value properties.

### Marta Lewicka (Assoc. Professor, PhD)

Lewicka's research areas are nonlinear analysis, partial differential equations and calculus of variations. She has obtained results on the well-posedness and stability of systems of conservation laws and reaction-diffusion equations. Currently, her research focuses on the mathematical theory of elasticity with connections to Riemannian geometry and with an eye on the applications in morphogenesis of growth.

### Ming Chen (Associate Professor, PhD)

Chen’s main area of research is PDEs with a particular focus on fluid mechanics and interfacial dynamics. Some specific topics include:

- The steady and dynamic aspects of the water wave problem, as well as various nonlinear dispersive equations arising from/related to the full water wave system.
- Stability of traveling waves in abstract Hamiltonian system with symmetry.
- Free boundary problem in compressible elastodynamics.
- Energy balance in incompressible and compressible Navier-Stokes equations.

### Piotr Hajlasz (Professor, PhD)

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.

### Yibiao Pan (Professor, PhD)

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.