Analysis and Partial Differential Equations

Research

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.

Gabriel Koch, Michael Melgaard

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 addresses 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 below.

Michael MelgaardMariapia Palombaro

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.

Mariapia Palombaro

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 and equilibria
  • Numerical methods to determine the basin of attraction
  • Lyapunov functions and complete Lyapunov functions
  • Contraction metrics

Peter Giesl

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

Gabriel Koch

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 simulation of their numerical solutions.

Current research topics include:

 

Spectral and scattering properties of differential operators

Spectral and scattering properties of differential operators is one of the most successful and most fascinating areas of modern mathematical physics.

It includes the qualitative and quantitative study of the possible energies and their associated states for particular physical systems (e.g., in quantum mechanics, in electromagnetic theory etc).

Research includes:

  • Phase space bounds and eigenvalue asymptotics for Schroedinger operators and Dirac operators
  • Spectral and scattering properties at thresholds for scalar-valued and matrix-valued Schroedinger operators
  • Spectral and scattering properties for Schroedinger and Dirac operators with magnetic fields
  • Spectral and scattering properties for quantum wires
  • Existence and approximation of resonances for Schroedinger and Dirac operators

 Michael Melgaard

 

2019

Giesl, Peter (2019) Computation of a contraction metric for a periodic orbit using meshfree collocation. SIAM Journal on Applied Dynamical Systems, 18 (3). pp. 1536-1564. ISSN 1536-0040

Cagnetti, Filippo, Dal Maso, Gianni, Scardia, Lucia and Ida Zeppieri, Caterina (2019) Stochastic homogenisation of free-discontinuity problems. Archive for Rational Mechanics and Analysis, 233 (240). pp. 935-974. ISSN 0003-9527

Cagnetti, Filippo, Dal Maso, Gianni, Scardia, Lucia and Ida Zeppieri, Caterina (2019) Γ-Convergence of free discontinuity problems. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 36 (4). pp. 1035-1079. ISSN 0294-1449

Albertsson, Sigurdur, Hafsteom, Sigurdur, Giesl, Peter and Gudmundsson, Skuli (2019) Simplicial complex with approximate rotational symmetry: a general class of simplicial complexes. Journal of Computational and Applied Mathematics. ISSN 0377-0427

Day, Stuart and Taheri, Ali (2019) Stability and local minimality of spherical harmonic twists u = Q(|x|)x|x| −1 , positivity of second variations and conjugate points on SO(n). The Journal of Analysis. ISSN 0971-3611

Koumatos, Konstantinos and Spirito, Stefano (2019) Quasiconvex elastodynamics: weak-strong uniqueness for measure-valued solutions. Communications on Pure and Applied Mathematics, 72 (6). pp. 1288-1320. ISSN 0010-3640

Giesl, Peter (2019) Converse theorem on a global contraction metric for a periodic orbit. Discrete and Continuous Dynamical Systems - Series A, 39 (9). pp. 5339-5363. ISSN 1078-0947

Taheri, Ali and Morris, Charles (2019) On the uniqueness and monotonicity of energy minimisers in the homotopy classes of incompressible mappings and related problems. Journal of Mathematical Analysis and Applications, 473 (18). pp. 1-26. ISSN 0022-247X

Stiefenhofer, Pascal, Giesl, Peter and Wagner, Heiko (2019) A system of inverted nonsmooth pendula: modelling an elderly person stepping over an obstacle. Nonlinear Analysis and Differential Equations, 7 (1). pp. 17-32. ISSN 2367-4814

Morrison, George and Taheri, Ali (2019) Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation L [u] = ∇P. Topological Methods in Nonlinear Analysis. ISSN ISSN 1230-3429 (Accepted)

Campos Cordero, Judith and Koumatos, Konstantinos (2019) Necessary and sufficient conditions for the strong local minimality of C^1 extremals on a class of non-smooth domains. ESAIM: Control, Optimisation and Calculus of Variations. ISSN 1292-8119 (Accepted)

Bjoernsson, Hjortur, Hafstein, Sigurdur, Giesl, Peter, Scalas, Enrico and Gudmundsson, Skuli (2019) Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete and Continuous Dynamical Systems - Series B, 24 (8). pp. 4247-4269. ISSN 1531-3492

Maijer, Kim, Hayward, Mark, Fernyhough, Charles, Calkins, Monica E, Debbané, Martin, Jardri, Renaud, Kelleher, Ian, Raballo, Andrea, Rammou, Aikaterini, Scott, James G, Shinn, Ann K, Steenhuis, Laura A, Wolf, Daniel H and Bartels-Velthuis, Agna A (2019) Hallucinations in children and adolescents: an updated review and practical recommendations for clinicians. Schizophrenia Bulletin, 45 (Supp 1). S5-S23. ISSN 0586-7614

Bond, Stuart and Taheri, Ali (2019) Operators of Laplace transform type and a new class of hypergeometric coefficients. Advances in Operator Theory, 4 (1). pp. 226-250. ISSN 2538-225X

Giesl, Peter and Mohammed, Najla (2019) Verification estimates for the construction of Lyapunov functions using meshfree collocation. Discrete and Continuous Dynamical Systems - Series B, 22. ISSN 1531-3492

Giesl, Peter (2019) Construction of Finsler-Lyapunov functions with meshless collocation. ZAMM, 99 (4). e201800141 1-18. ISSN 0044-2267

Giesl, Peter and Wendland, Holger (2019) Construction of a contraction metric by meshless collocation. Discrete and Continuous Dynamical Systems - Series B, 24 (8). pp. 3843-3863. ISSN 1531-3492

2018

Morris, Charles and Taheri, Ali (2018) Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy. Journal of Elasticity, 133 (2). pp. 201-222. ISSN 0374-3535

Deribe, Kebede (2018) [Editorial] Podoconiosis today: challenges and opportunities. Transactions of The Royal Society of Tropical Medicine and Hygiene, 112 (11). pp. 473-475. ISSN 0035-9203

Giesl, Peter and Wendland, Holger (2018) Kernel-based discretisation for solving matrix-valued PDEs. SIAM Journal on Numerical Analysis, 56 (6). pp. 3386-3406. ISSN 0036-1429

Argaez, C and Melgaard, M (2018) Ground state solutions to Hartree–Fock equations with magnetic fields. Applicable Analysis, 97 (14). pp. 2377-2403. ISSN 0003-6811

Taheri, Ali and Morrison, George (2018) On a class of stationary loops on SO(n) and the existence of multiple twisting solutions to a nonlinear elliptic system subject to a hard incompressibility constraint. Boundary Value Problems, 2018 (130). pp. 1-24. ISSN 1687-2762

Morrison, George and Taheri, Ali (2018) An infinite scale of incompressible twisting solutions to the nonlinear elliptic system L [u; A, B] = ∇P and the discriminant ∆(h, g). Nonlinear Analysis Theory Methods & Applications, 173. pp. 209-219. ISSN 0362-546X

Agapiou, Sergios, Burger, Martin, Dashti, Masoumeh and Helin, Tapio (2018) Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems. Inverse Problems, 34 (4). pp. 1-37. ISSN 0266-5611

Morris, Charles and Taheri, Ali (2018) Annular rearrangements, incompressible axi-symmetric whirls and L1-local minimisers of the distortion energy. Nonlinear Differential Equations and Applications, 25 (2). pp. 1-35. ISSN 1021-9722

Morris, Charles and Taheri, Ali (2018) On Weyl’s asymptotics and remainder term for the orthogonal and unitary groups. Journal of Fourier Analysis and Applications, 24 (1). pp. 184-209. ISSN 1069-5869

Hafstein, Sigurdur, Gudmundsson, Skuli, Giesl, Peter and Scalas, Enrico (2018) Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete and Continuous Dynamical Systems - Series B, 23 (2). pp. 939-956. ISSN 1531-3492

2017

Fanzon, Silvio, Palombaro, Mariapia and Ponsiglione, Marcello (2017) A variational model for dislocations at semi-coherent interfaces. Journal of Nonlinear Science, 27 (5). pp. 1435-1461. ISSN 0938-8974

Mohammadi, Samaneh, Zakeri-Milani, Parvin, Golkar, Nasim, Farkhani, Samad Mussa, Shirani, Ali, Shahbazi Mojarrad, Javid, Nokhodchi, Ali and Valizadeh, Hadi (2017) Synthesis and cellular characterization of various nano-assemblies of cell penetrating peptide-epirubicin-polyglutamate conjugates for the enhancement of antitumor activity. Artificial Cells, Nanomedicine, and Biotechnology. pp. 1-14. ISSN 2169-1401

Fanzon, Silvio and Palombaro, Mariapia (2017) Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions. Calculus of Variations and Partial Differential Equations, 56 (137). ISSN 0944-2669

Guo, Zhenyu, Melgaard, Michael and Zou, Wenming (2017) Schrödinger equations with magnetic fields and Hardy-Sobolev critical exponents. Electronic Journal of Differential Equations, 2017 (199). pp. 1-18. ISSN 1072-6691

Argaez, Carlos, Hafstein, Sigurdur and Giesl, Peter (2017) Wendland functions a C++ code to compute them. SIMULTECH 2017, Madrid, Spain, 26/07/17 - 28/07/17. Published in: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, 2017, Madrid, Spain. 323-330. SCITEPRESS, Madrid, Spain. ISBN 9789897582653

Macedo, Arthur Cassa, Balouch, Sara and Tabet, Naji (2017) Is sleep disruption a risk factor for Alzheimer’s disease? Journal of Alzheimer's Disease, 58 (4). pp. 993-1002. ISSN 1387-2877

Awonusika, Richard Olu and Taheri, Ali (2017) On Jacobi polynomials (P (α,β) k : α, β > −1) and Maclaurin spectral functions on rank one symmetric spaces. Journal of Analysis, 25 (1). pp. 139-166. ISSN 0971-3611

Abdelrazig, Salah, Ortori, Catherine A, Davey, Gail, Deressa, Wakgari, Dhaba, Mulleta, Barret, David A, Amberbir, Alemayhu and Fogarty, Andrew W (2017) A metabolomic analytical approach permits identification of urinary biomarkers for Plasmodium falciparum infection. Malaria Journal, 16. a229. ISSN 1475-2875

Awonusika, Richard Olu and Taheri, Ali (2017) On Gegenbauer polynomials and coefficients cℓj(ν) (1≤j≤ℓ, ν>−1/2). Results in Mathematics, 72 (3). pp. 1359-1367. ISSN 1422-6383

Giesl, Peter and McMichen, James (2017) Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 3 (2). pp. 191-210. ISSN 2158-2491

Alicandro, Roberto, Lazzaroni, Giuliano and Palombaro, Mariapia (2017) On the effect of interactions beyond nearest neighbours on non-convex lattice systems. Calculus of Variations and Partial Differential Equations, 56 (2). a42. ISSN 0944-2669

Morris, C and Taheri, A (2017) Twist maps as energy minimisers in homotopy classes: Symmetrisation and the coarea formula. Nonlinear Analysis: Theory, Methods and Applications, 152. pp. 250-275. ISSN 0362-546X

Koumatos, K and Muehlemann, A (2017) A theoretical investigation of orientation relationships and transformation strains in steels. Acta Crystallographica Section A: Foundations and Advances, A73. pp. 115-123. ISSN 2053-2733

Artina, Marco, Cagnetti, Filippo, Fornasier, Massimo and Solombrino, Francesco (2017) Linearly constrained evolutions of critical points and an application to cohesive fractures. Mathematical Models and Methods in Applied Sciences, 27 (2). pp. 231-290. ISSN 0218-2025

Kungsman, J and Melgaard, Michael (2017) Poisson wave trace formula for perturbed Dirac operators. Journal of Operator Theory, 77 (1). pp. 133-147. ISSN 1841-7744

Argáez, Carlos, Giesl, Peter and Hafstein, Sigurdur (2017) Analysing dynamical systems towards computing complete Lyapunov functions. SIMULTECH 2017, Madrid, Spain, 26/07/17 - 28/07/17. Published in: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, 2017, Madrid, Spain. 134-144. Madrid, Spain. ISBN 9789897582653

2016

Madzvamuse, A and Chung, A H (2016) Analysis and simulations of coupled bulk-surface reaction-diffusion systems on exponentially evolving volumes. Mathematical Modelling of Natural Phenomena, 11 (5). pp. 4-32. ISSN 0973-5348

Argaez, C and Melgaard, M (2016) Minimizers for open-shell, spin-polarised Kohn–Sham equations for non-relativistic and quasi-relativistic molecular systems. Methods and Applications of Analysis, 23 (3). pp. 269-292. ISSN 1073-2772

Agostiniani, Virginia, DeSimone, Antonio and Koumatos, Konstantinos (2016) Shape programming for narrow ribbons of nematic elastomers. Journal of Elasticity, 127 (1). pp. 1-24. ISSN 0374-3535

Koumatos, Konstantinos, Rindler, Filip and Wiedemann, Emil (2016) Orientation-preserving Young measures. Quarterly Journal of Mathematics, 67 (3). pp. 439-466. ISSN 0033-5606

Koumatos, Konstantinos and Muehlemann, Anton (2016) Optimality of general lattice transformations with applications to the Bain strain in steel. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472 (2188). ISSN 1364-5021

Ball, J M and Koumatos, K (2016) Quasiconvexity at the boundary and the nucleation of austenite. Archive for Rational Mechanics and Analysis, 219 (1). pp. 89-157. ISSN 0003-9527

2015

Koumatos, Konstantinos and Muehlemann, Anton (2015) The morphology of lath martensite: a new perspective. MATEC Web of Conferences, 33. 07003. ISSN 2261-236X

Koumatos, Konstantinos, Rindler, Filip and Wiedemann, Emil (2015) Differential inclusions and young measures involving prescribed Jacobians. SIAM Journal on Mathematical Analysis, 47 (2). pp. 1169-1195. ISSN 0036-1410

2014

Agostiniani, V, Blass, T and Koumatos, K (2014) From nonlinear to linearized elasticity via Γ-convergence: the case of multiwell energies satisfying weak coercivity conditions. Mathematical Models and Methods in Applied Sciences, 25 (1). pp. 1-38. ISSN 0218-2025

Ball, J M and Koumatos, K (2014) An investigation of non-planar austenite–martensite interfaces. Mathematical Models and Methods in Applied Sciences, 24 (10). pp. 1937-1956. ISSN 0218-2025

2013

Ball, J M, Koumatos, K and Seiner, H (2013) Nucleation of austenite in mechanically stabilized martensite by localized heating. ICOMAT 2011, Osaka, Japan, 4 to 9 September 2011. Published in: Miyazaki, Shuichi, Kakeshita, Tomoyuki and Nishida, Minorui, (eds.) Journal of Alloys and Compounds. 557 (supp1) s37-s42. Elsevier ISSN 0925-8388

Shahrokhi-Dehkordi, Mohammad and Taheri, Ali (2013) Generalised twists as elastic energy extremals on annuli, Squaternions and lifting twist loops to the spinor groups. Analysis and Applications, 11 (2). pp. 124-150. ISSN 0219-5305

2011

Taheri, Ali (2011) Spherical twists, stationary loops and harmonic maps from generalised annuli into spheres. Nonlinear Differential Equations and Applications, 19 (1). pp. 79-95. ISSN 1021-9722

2009

Ball, J M, Koumatos, K and Seiner, H (2009) An analysis of non-classical Austenite-Martensite interfaces in CuAlNi. Published in: Olson, G B, Lieberman, D S and Saxena, A, (eds.) Proceedings of the International Conference on Martensitic Transformations, June 29-July 05, 2008: ICOMAT : Santa Fe, New Mexico, USA, 2008. John Wiley & Sons ISBN 9781118820438