Research within the Analysis and Partial Differential Equations group concentrates on the nonlinear problems. The focus is on rigorous analysis of mathematical models motivated from applied mathematics involving nonlinear partial differential equations, dynamical systems, evolution equations, geometric measure theory, calculus of variations and their applications in material microstructure, nonlinear elasticity, image processing and financial mathematics.
Current members
- Faculty: Filippo Cagnetti, Miroslav Chlebik, Masoumeh Dashti, Peter Giesl, Gabriel Koch, Michael Melgaard, Ali Taheri, Qi Tang and Arghir Zarnescu.
- Associate Faculty: Peter J. Bushell, David E. Edmunds.
- Postgraduate Students: Abimbola Abolarinwa, Richard Awonusika, Asma Elbirki, Joseph Gavin, Haidar Haidar, James McMichen, Najla Mohammed, Stephen Oshungade, Liu Ping, Nima Shahroozi, Pascal Stiefenhofer, Leila Yadollahi Farsani.
Current research topics
Nonlinear partial differential equations
Nonlinear partial differential equations are one of the key areas of the interaction of mathematics and the sciences. In our group we mostly study qualitative features of equations, such as singularities and oscillations which can manifest themselves as defects, microstructure or blow-up. The opposite phenomena such as regularity, compactness and convergence are of equal interest. In order to extend the mathematical framework to understand these phenomena we pursue a broad spectrum of mathematical ideas. Current research topics include:
- Blow-up phenomena in parabolic problems
- Dynamical systems and evolution equations
- Existence, regularity and singularities for nonlinear elliptic problems
- Critical point theory through topological invariants
- Stochastic integral-differential equations.
Calculus of variations
Many problems in physics, biology, engineering are governed by a maximization or minimization principle (for example for energy or entropy). The corresponding optimality conditions (called Euler-Lagrange equations) usually are Nonlinear Partial Differential Equations and one can apply PDE methods to study the minimizers. Often much more powerful methods are available if one exploits the minimization principle directly. The corresponding branch of mathematics is called the calculus of variations (because originally the Euler-Lagrange equation was derived by studying small variations of supposed minimizer).
Our research address both fundamental problems in the calculus of variations and applications, e.g. in nonlinear elasticity which is closely related to nonconvex minimization problems. Current research topics are:
Lower semicontinuity, relaxation, quasiconvexity and gradient Young measures
- Concentration phenomena in variational problems
- Phase transition problems
- Applications in nonlinear elasticity (existence and regularity)
- Deformations with gradient constraints, differential inclusions.
Geometric measure theory and singular structures
Problems in science often lead to singular or near singular behaviour of the solutions (e.g. boundary layers, transitions zones, shock profiles, fracture surfaces, edges and corners in a digital image, etc.). Therefore a mathematical framework which is closed under scaling operations and natural limits but which provides enough compactness for a good existence theory often requires the inclusion of singular objects. In a second step one can then investigate which precise singularities occur for a concrete problem at hand.
The theory of minimal surfaces can serve as an illustration of this approach. It motivated the development of geometric measure theory leading to the highly flexible generalization of the concept of classical surfaces to the notion of (rectifiable) currents and varifolds. Our research focuses on the following topics:
- Relations between metric and geometry, conditions implying rectifiability
- Fine properties of weakly differentiable (BV, Sobolev) functions
- Singularities of minimal surfaces
- Fractal geometry.
Dynamical systems
Ordinary differential equations model many important processes in biology, physics, economy and other sciences. The theory of Dynamical Systems studies the qualitative properties and long-time behaviour of solutions. We are particularly interested in stability properties of solutions and their basin of attraction. The systems that we consider range from autonomous, non-autonomous, non-smooth and finite-time ordinary differential equations to discrete dynamical systems given by iterations of a map. Current research topics include:
- Basin of attraction of periodic orbits
- Non-autonomous dynamical systems
- Attractivity of solutions over a finite time
- Non-smooth systems
- Numerical methods to determine the basin of attraction.
Financial mathematics
Research in financial mathematics tries to answer questions like 'what factors determine the prices of shares?', 'what role do financial markets play in an economy?', by developing and applying theoretical economic principles to these issues in a rigorous and impartial manner. It requires to develop theoretical models and apply them, in order to gain an understanding of the issues mentioned above as well as most other real world social phenomena.
Our research topics in this area include:
- Risk assessment computation and software making for banking related consultancy companies
- Minimization algorithms for finance industries
- Option prices and related computations.
Inverse problems
Inverse problems in differential equations arise in many application areas such as weather prediction, oceanography, medical tomography and subsurface geophysics, when the object of interest cannot be measured directly. The unknown object in such problems is the input to a known mathematical model and is to be found given (possibly noisy) observations of the solution (output) of this known model.
A fundamental difficulty shared by many inverse problems is that they are ill-posed and some form of regularization is required in order to obtain a reasonable estimate of the unknown in a stable way.
A main research interest in the group is the theory of Bayesian regularization of inverse problems for functions. In this approach to regularization, the prior beliefs about the unknown are described in terms of a probability measure on an appropriate function space, and then the Bayes formula is used to calculate a posterior probability measure. Within this area, the current subjects of research interest include:
- Finding reasonable conditions on the prior for different applications
- Wavelet-based prior measures
- Consistency properties of the posterior measure
Harmonic analysis and applications
Harmonic analysis studies the fine properties of functions, and suitable decompositions of them, with the aim of obtaining an optimal understanding of various interactions between different scales. The harmonic analysis technique have a broad range of applications in areas ranging from signal and image processing to quantum mechanics. Current research topics include:
- Besov and more general function spaces
- Littlewood-Paley applications
- critical embeddings
Applied analysis of physically motivated equations
Most descriptions of physical pheomena involve partial differential equations, often nonlinear. The understanding, from an analytical point of view, of the predictive capacities as well as the limitations of these equations is often a first crucial step in the development and simulations of the numerical solutions of these equations.Current research topics include:
-fluid mechanics problems
-liquid crystal models
-Schroedinger equations