 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 blowup. 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:
 Blowup phenomena in parabolic problems
 Dynamical systems and evolution equations
 Existence, regularity and singularities for nonlinear elliptic problems
 Critical point theory through topological invariants
 Stochastic integraldifferential 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 EulerLagrange 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 EulerLagrange 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.
 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 longtime 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, nonautonomous, nonsmooth and finitetime 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
 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 illposed 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
 Waveletbased 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
 LittlewoodPaley 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 simulation of their numerical solutions.
Current research topics include:
 fluid mechanics problems (Gabriel Koch)
 Schroedinger equations (Gabriel Koch, Michael Melgaard)
 Homogenisation theory (Mariapia Palombaro)
 Propagation of waves in periodic media (Mariapia Palombaro)
 Analysis of Nonlinear PDEs arising for manybody models in Quantum Chemistry (Michael Melgaard)
 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 scalarvalued and matrixvalued 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
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. 15361564. ISSN 15360040
Cagnetti, Filippo, Dal Maso, Gianni, Scardia, Lucia and Ida Zeppieri, Caterina (2019) Stochastic homogenisation of freediscontinuity problems. Archive for Rational Mechanics and Analysis, 233 (240). pp. 935974. ISSN 00039527
Cagnetti, Filippo, Dal Maso, Gianni, Scardia, Lucia and Ida Zeppieri, Caterina (2019) ΓConvergence of free discontinuity problems. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 36 (4). pp. 10351079. ISSN 02941449
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 03770427
Day, Stuart and Taheri, Ali (2019) Stability and local minimality of spherical harmonic twists u = Q(x)xx −1 , positivity of second variations and conjugate points on SO(n). The Journal of Analysis. ISSN 09713611
Koumatos, Konstantinos and Spirito, Stefano (2019) Quasiconvex elastodynamics: weakstrong uniqueness for measurevalued solutions. Communications on Pure and Applied Mathematics, 72 (6). pp. 12881320. ISSN 00103640
Giesl, Peter (2019) Converse theorem on a global contraction metric for a periodic orbit. Discrete and Continuous Dynamical Systems  Series A, 39 (9). pp. 53395363. ISSN 10780947
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. 126. ISSN 0022247X
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. 1732. ISSN 23674814
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 12303429 (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 nonsmooth domains. ESAIM: Control, Optimisation and Calculus of Variations. ISSN 12928119 (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. 42474269. ISSN 15313492
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 BartelsVelthuis, Agna A (2019) Hallucinations in children and adolescents: an updated review and practical recommendations for clinicians. Schizophrenia Bulletin, 45 (Supp 1). S5S23. ISSN 05867614
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. 226250. ISSN 2538225X
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 15313492
Giesl, Peter (2019) Construction of FinslerLyapunov functions with meshless collocation. ZAMM, 99 (4). e201800141 118. ISSN 00442267
Giesl, Peter and Wendland, Holger (2019) Construction of a contraction metric by meshless collocation. Discrete and Continuous Dynamical Systems  Series B, 24 (8). pp. 38433863. ISSN 15313492
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. 201222. ISSN 03743535
Deribe, Kebede (2018) [Editorial] Podoconiosis today: challenges and opportunities. Transactions of The Royal Society of Tropical Medicine and Hygiene, 112 (11). pp. 473475. ISSN 00359203
Giesl, Peter and Wendland, Holger (2018) Kernelbased discretisation for solving matrixvalued PDEs. SIAM Journal on Numerical Analysis, 56 (6). pp. 33863406. ISSN 00361429
Argaez, C and Melgaard, M (2018) Ground state solutions to Hartree–Fock equations with magnetic fields. Applicable Analysis, 97 (14). pp. 23772403. ISSN 00036811
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. 124. ISSN 16872762
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. 209219. ISSN 0362546X
Agapiou, Sergios, Burger, Martin, Dashti, Masoumeh and Helin, Tapio (2018) Sparsitypromoting and edgepreserving maximum a posteriori estimators in nonparametric Bayesian inverse problems. Inverse Problems, 34 (4). pp. 137. ISSN 02665611
Morris, Charles and Taheri, Ali (2018) Annular rearrangements, incompressible axisymmetric whirls and L1local minimisers of the distortion energy. Nonlinear Differential Equations and Applications, 25 (2). pp. 135. ISSN 10219722
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. 184209. ISSN 10695869
Hafstein, Sigurdur, Gudmundsson, Skuli, Giesl, Peter and Scalas, Enrico (2018) Lyapunov function computation for autonomous linear stochastic differential equations using sumofsquares programming. Discrete and Continuous Dynamical Systems  Series B, 23 (2). pp. 939956. ISSN 15313492
2017
Fanzon, Silvio, Palombaro, Mariapia and Ponsiglione, Marcello (2017) A variational model for dislocations at semicoherent interfaces. Journal of Nonlinear Science, 27 (5). pp. 14351461. ISSN 09388974
Mohammadi, Samaneh, ZakeriMilani, Parvin, Golkar, Nasim, Farkhani, Samad Mussa, Shirani, Ali, Shahbazi Mojarrad, Javid, Nokhodchi, Ali and Valizadeh, Hadi (2017) Synthesis and cellular characterization of various nanoassemblies of cell penetrating peptideepirubicinpolyglutamate conjugates for the enhancement of antitumor activity. Artificial Cells, Nanomedicine, and Biotechnology. pp. 114. ISSN 21691401
Fanzon, Silvio and Palombaro, Mariapia (2017) Optimal lower exponent for the higher gradient integrability of solutions to twophase elliptic equations in two dimensions. Calculus of Variations and Partial Differential Equations, 56 (137). ISSN 09442669
Guo, Zhenyu, Melgaard, Michael and Zou, Wenming (2017) Schrödinger equations with magnetic fields and HardySobolev critical exponents. Electronic Journal of Differential Equations, 2017 (199). pp. 118. ISSN 10726691
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. 323330. 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. 9931002. ISSN 13872877
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. 139166. ISSN 09713611
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 14752875
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. 13591367. ISSN 14226383
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. 191210. ISSN 21582491
Alicandro, Roberto, Lazzaroni, Giuliano and Palombaro, Mariapia (2017) On the effect of interactions beyond nearest neighbours on nonconvex lattice systems. Calculus of Variations and Partial Differential Equations, 56 (2). a42. ISSN 09442669
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. 250275. ISSN 0362546X
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. 115123. ISSN 20532733
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. 231290. ISSN 02182025
Kungsman, J and Melgaard, Michael (2017) Poisson wave trace formula for perturbed Dirac operators. Journal of Operator Theory, 77 (1). pp. 133147. ISSN 18417744
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. 134144. Madrid, Spain. ISBN 9789897582653
2016
Madzvamuse, A and Chung, A H (2016) Analysis and simulations of coupled bulksurface reactiondiffusion systems on exponentially evolving volumes. Mathematical Modelling of Natural Phenomena, 11 (5). pp. 432. ISSN 09735348
Argaez, C and Melgaard, M (2016) Minimizers for openshell, spinpolarised Kohn–Sham equations for nonrelativistic and quasirelativistic molecular systems. Methods and Applications of Analysis, 23 (3). pp. 269292. ISSN 10732772
Agostiniani, Virginia, DeSimone, Antonio and Koumatos, Konstantinos (2016) Shape programming for narrow ribbons of nematic elastomers. Journal of Elasticity, 127 (1). pp. 124. ISSN 03743535
Koumatos, Konstantinos, Rindler, Filip and Wiedemann, Emil (2016) Orientationpreserving Young measures. Quarterly Journal of Mathematics, 67 (3). pp. 439466. ISSN 00335606
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 13645021
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. 89157. ISSN 00039527
2015
Koumatos, Konstantinos and Muehlemann, Anton (2015) The morphology of lath martensite: a new perspective. MATEC Web of Conferences, 33. 07003. ISSN 2261236X
Koumatos, Konstantinos, Rindler, Filip and Wiedemann, Emil (2015) Differential inclusions and young measures involving prescribed Jacobians. SIAM Journal on Mathematical Analysis, 47 (2). pp. 11691195. ISSN 00361410
2014
Agostiniani, V, Blass, T and Koumatos, K (2014) From nonlinear to linearized elasticity via Γconvergence: the case of multiwell energies satisfying weak coercivity conditions. Mathematical Models and Methods in Applied Sciences, 25 (1). pp. 138. ISSN 02182025
Ball, J M and Koumatos, K (2014) An investigation of nonplanar austenite–martensite interfaces. Mathematical Models and Methods in Applied Sciences, 24 (10). pp. 19371956. ISSN 02182025
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) s37s42. Elsevier ISSN 09258388
ShahrokhiDehkordi, 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. 124150. ISSN 02195305
2011
Taheri, Ali (2011) Spherical twists, stationary loops and harmonic maps from generalised annuli into spheres. Nonlinear Differential Equations and Applications, 19 (1). pp. 7995. ISSN 10219722
2009
Ball, J M, Koumatos, K and Seiner, H (2009) An analysis of nonclassical AusteniteMartensite interfaces in CuAlNi. Published in: Olson, G B, Lieberman, D S and Saxena, A, (eds.) Proceedings of the International Conference on Martensitic Transformations, June 29July 05, 2008: ICOMAT : Santa Fe, New Mexico, USA, 2008. John Wiley & Sons ISBN 9781118820438