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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 Melgaard, Mariapia 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

 

2023

Taheri, Ali and Vahidifar, Vahideh (2023) Nonlinear Analysis: Theory, Methods and Applications, 232. pp. 1-37. ISSN 0362-546X

Taheri, Ali and Vahidifar, Vahideh (2023) Mathematika: a journal of pure and applied mathematics, 69 (3). pp. 751-779. ISSN 0025-5793

Taheri, Ali and Vahideh, Vahidifar (2023) Advances in Nonlinear Analysis, 12 (1). pp. 1-16. ISSN 2191-950X

Giesl, Peter (2023) Journal of Computational Dynamics. pp. 1-23. ISSN 2158-2505

2022

Louzeiro, Mauricio, Kawan, Christoph, Hafstein, Sigurdur, Giesl, Peter and Yuan, Jinyun (2022) SIAM Journal on Applied Dynamical Systems, 21 (4). pp. 2297-2696. ISSN 1536-0040

Argáez, Carlos, Giesl, Peter and Freyr Hafstein, Sigurdur (2022) Complexity, 2022. pp. 1-17. ISSN 1076-2787

Mohammed, Najla and Giesl, Peter (2022) Journal of Computational Dynamics. pp. 1-25. ISSN 2158-2491

Taheri, Ali and Vahidifar, Vahideh (2022) Nonlinear Analysis Theory Methods and Applications, 221. a112889. ISSN 0362-546X

Giesl, Peter, Hafstein, Sigurdur and Kawan, Christoph (2022) Journal of Computational Dynamics. pp. 1-47. ISSN 2158-2491

Abolarinwa, Abimbola and Taheri, Ali (2022) Complex Variables and Elliptic Equations: an international journal, 67 (6). pp. 1379-1392. ISSN 1747-6933

Cagnetti, Filippo, Dal Maso, Gianni, Scardia, Lucia and Zeppieri, Caterina Ida (2022) Annals of PDE, 8 (1). a8 1-89. ISSN 2524-5317

Melgaard, Michael, Yang, Minbo and Zhou, Xianmei (2022) Pacific Journal of Mathematics, 317 (1). pp. 153-186. ISSN 0030-8730

2021

Taheri, Ali (2021) Potential Analysis. ISSN 0926-2601

Taheri, Ali (2021) Advances in Calculus of Variations. pp. 1-17. ISSN 1864-8258

Abolarinwa, Abimbola and Taheri, Ali (2021) Chaos, Solitons & Fractals, 142. a110329 1-14. ISSN 0960-0779

2020

Morrison, George and Taheri, Ali (2020) Advances in Operator Theory, 6. a17 1-28. ISSN 2662-2009

Campos Cordero, Judith and Koumatos, Konstantinos (2020) ESAIM: Control, Optimisation and Calculus of Variations, 26. a49. ISSN 1292-8119

Day, Stuart and Taheri, Ali (2020) The Journal of Analysis, 28 (2). pp. 431-460. ISSN 0971-3611

Koumatos, Konstantinos and Spirito, Stefano (2020) Communications on Pure and Applied Mathematics. ISSN 0010-3640

Morrison, George and Taheri, Ali (2020) Differential Equations and Applications, 12 (1). pp. 47-67. ISSN 1847-120X

2019

Morrison, George and Taheri, Ali (2019) Topological Methods in Nonlinear Analysis, 54 (2A). pp. 833-862. ISSN 1230-3429

Giesl, Peter and Mohammed, Najla (2019) Discrete and Continuous Dynamical Systems - Series B, 24 (9). pp. 4955-4981. ISSN 1531-3492

Giesl, Peter (2019) 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) 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) Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire, 36 (4). pp. 1035-1079. ISSN 0294-1449

Albertsson, Sigurdur, Hafsteom, Sigurdur, Giesl, Peter and Gudmundsson, Skuli (2019) Journal of Computational and Applied Mathematics. ISSN 0377-0427

Koumatos, Konstantinos and Spirito, Stefano (2019) Communications on Pure and Applied Mathematics, 72 (6). pp. 1288-1320. ISSN 0010-3640

Giesl, Peter (2019) Discrete and Continuous Dynamical Systems - Series A, 39 (9). pp. 5339-5363. ISSN 1078-0947

Taheri, Ali and Morris, Charles (2019) Journal of Mathematical Analysis and Applications, 473 (18). pp. 1-26. ISSN 0022-247X

Stiefenhofer, Pascal, Giesl, Peter and Wagner, Heiko (2019) Nonlinear Analysis and Differential Equations, 7 (1). pp. 17-32. ISSN 2367-4814

Bjoernsson, Hjortur, Hafstein, Sigurdur, Giesl, Peter, Scalas, Enrico and Gudmundsson, Skuli (2019) 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) Schizophrenia Bulletin, 45 (Supp 1). S5-S23. ISSN 0586-7614

Bond, Stuart and Taheri, Ali (2019) Advances in Operator Theory, 4 (1). pp. 226-250. ISSN 2538-225X

Giesl, Peter (2019) ZAMM, 99 (4). e201800141 1-18. ISSN 0044-2267

Giesl, Peter and Wendland, Holger (2019) Discrete and Continuous Dynamical Systems - Series B, 24 (8). pp. 3843-3863. ISSN 1531-3492

2018

Morris, Charles and Taheri, Ali (2018) Journal of Elasticity, 133 (2). pp. 201-222. ISSN 0374-3535

Giesl, Peter and Wendland, Holger (2018) SIAM Journal on Numerical Analysis, 56 (6). pp. 3386-3406. ISSN 0036-1429

Deribe, Kebede (2018) Transactions of The Royal Society of Tropical Medicine and Hygiene, 112 (11). pp. 473-475. ISSN 0035-9203

Argaez, C and Melgaard, M (2018) Applicable Analysis, 97 (14). pp. 2377-2403. ISSN 0003-6811

Awonusika, Richard Olu and Taheri, Ali (2018) Canadian Mathematical Bulletin, 61. pp. 473-482. ISSN 0008-4395

Taheri, Ali and Morrison, George (2018) Boundary Value Problems, 2018 (130). pp. 1-24. ISSN 1687-2762

Morrison, George and Taheri, Ali (2018) Nonlinear Analysis Theory Methods & Applications, 173. pp. 209-219. ISSN 0362-546X

Agapiou, Sergios, Burger, Martin, Dashti, Masoumeh and Helin, Tapio (2018) Inverse Problems, 34 (4). pp. 1-37. ISSN 0266-5611

Morris, Charles and Taheri, Ali (2018) Nonlinear Differential Equations and Applications, 25 (2). pp. 1-35. ISSN 1021-9722

Morris, Charles and Taheri, Ali (2018) Journal of Fourier Analysis and Applications, 24 (1). pp. 184-209. ISSN 1069-5869

Hafstein, Sigurdur, Gudmundsson, Skuli, Giesl, Peter and Scalas, Enrico (2018) Discrete and Continuous Dynamical Systems - Series B, 23 (2). pp. 939-956. ISSN 1531-3492

2017

Fanzon, Silvio, Palombaro, Mariapia and Ponsiglione, Marcello (2017) 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) Artificial Cells, Nanomedicine, and Biotechnology. pp. 1-14. ISSN 2169-1401

Fanzon, Silvio and Palombaro, Mariapia (2017) Calculus of Variations and Partial Differential Equations, 56 (137). ISSN 0944-2669

Guo, Zhenyu, Melgaard, Michael and Zou, Wenming (2017) Electronic Journal of Differential Equations, 2017 (199). pp. 1-18. ISSN 1072-6691

Argaez, Carlos, Hafstein, Sigurdur and Giesl, Peter (2017) 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) Journal of Alzheimer's Disease, 58 (4). pp. 993-1002. ISSN 1387-2877

Awonusika, Richard Olu and Taheri, Ali (2017) 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) Malaria Journal, 16. a229. ISSN 1475-2875

Awonusika, Richard Olu and Taheri, Ali (2017) Results in Mathematics, 72 (3). pp. 1359-1367. ISSN 1422-6383

Giesl, Peter and McMichen, James (2017) Journal of Computational Dynamics, 3 (2). pp. 191-210. ISSN 2158-2491

Alicandro, Roberto, Lazzaroni, Giuliano and Palombaro, Mariapia (2017) Calculus of Variations and Partial Differential Equations, 56 (2). a42. ISSN 0944-2669

Morris, C and Taheri, A (2017) Nonlinear Analysis: Theory, Methods and Applications, 152. pp. 250-275. ISSN 0362-546X

Koumatos, K and Muehlemann, A (2017) Acta Crystallographica Section A: Foundations and Advances, A73. pp. 115-123. ISSN 2053-2733

Artina, Marco, Cagnetti, Filippo, Fornasier, Massimo and Solombrino, Francesco (2017) Mathematical Models and Methods in Applied Sciences, 27 (2). pp. 231-290. ISSN 0218-2025

Kungsman, J and Melgaard, Michael (2017) Journal of Operator Theory, 77 (1). pp. 133-147. ISSN 1841-7744

Argáez, Carlos, Giesl, Peter and Hafstein, Sigurdur (2017) 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) Mathematical Modelling of Natural Phenomena, 11 (5). pp. 4-32. ISSN 0973-5348

Argaez, C and Melgaard, M (2016) Methods and Applications of Analysis, 23 (3). pp. 269-292. ISSN 1073-2772

Agostiniani, Virginia, DeSimone, Antonio and Koumatos, Konstantinos (2016) Journal of Elasticity, 127 (1). pp. 1-24. ISSN 0374-3535

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

Koumatos, Konstantinos and Muehlemann, Anton (2016) Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472 (2188). ISSN 1364-5021

Ball, J M and Koumatos, K (2016) Archive for Rational Mechanics and Analysis, 219 (1). pp. 89-157. ISSN 0003-9527

2015

Koumatos, Konstantinos and Muehlemann, Anton (2015) MATEC Web of Conferences, 33. 07003. ISSN 2261-236X

Koumatos, Konstantinos, Rindler, Filip and Wiedemann, Emil (2015) SIAM Journal on Mathematical Analysis, 47 (2). pp. 1169-1195. ISSN 0036-1410

2014

Agostiniani, V, Blass, T and Koumatos, K (2014) Mathematical Models and Methods in Applied Sciences, 25 (1). pp. 1-38. ISSN 0218-2025

Ball, J M and Koumatos, K (2014) 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) 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) Analysis and Applications, 11 (2). pp. 124-150. ISSN 0219-5305

2011

Taheri, Ali (2011) Nonlinear Differential Equations and Applications, 19 (1). pp. 79-95. ISSN 1021-9722

2009

Ball, J M, Koumatos, K and Seiner, H (2009) 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