APPLIED ANALYSIS - IACM

Applied Analysis

ABOUT R&D ACTIVITIES R&D PROGRAMS PUBLICATIONS PEOPLE CONTACT US

ABOUT

The field of Applied Analysis brings together several mathematical topics of great interest and aims at investigating, among others, partial differential equations, probability theory, stochastic partial differential equations, infinite dynamical systems of ordinary differential equations and stochastic analysis. More specifically, the group studies rigorously the dynamics of interfaces for certain reaction-diffusion equations both deterministic and stochastic, sharp interface limits and conservation laws under mean curvature flow. The physical background is diverse, and is stemming mainly from problems of phase transitions, and mathematical physics. Morevover, the group studies large-scale systems of interacting particles and their hydrodynamic limits to model and to rigorously analyze complex systems arising in engineering sciences and physics. The group works towards novel and rigorous results and particular attention is paid to the training of graduate students and postdocs.

RESEARCH AND DEVELOPMENT ACTIVITIES

Partial differential equations and stochastic differential equations: The interaction between the theory of partial differential equations and probability is very fruitful for research in mathematics, both because of the intrinsic mathematical challenges presented, and because of the range of applications. The study of evolution equations with a stochastic forcing has seen major advances in the recent years. Applications include a variety of important reaction-diffusion equations arising in mathematical physics, but the potential for development of this theory in other applications, such as mathematical finance, mathematical biology and others, is enormous. Research in this direction aims at developing innovative analysis suitable for a rigorous mathematical study of the effects of thermal fluctuations. These include among others the study of stochastic reaction-diffusion equations, motion by mean curvature with random forcing, Malliavin calculus, applications to finance and others.

Integrable Systems: We have analyzed so-called "completely integrable" infinite dimensional Hamiltonian systems, in particular partial differential equations and nonlinear lattices (large systems of ODEs). We have been particularly interested in asymptotic problems like the investigation of long time asymptotics, semiclassical asymptotics, zero dispersion limits and continuum limits of solutions of initial and initial-boundary value problems for nonlinear dispersive partial differential equations and nonlinear lattices, including difficult problems involving instabilities. We have used and extended techniques from PDE theory, complex analysis, harmonic analysis, potential theory and algebraic geometry. Along the way, we have made contributions to the analysis of Riemann-Hilbert factorization problems on the complex plane or a hyperelliptic Riemann surface and the related theory of variational problems for Green potentials with harmonic external fields. In a sense we have worked on a "nonlinear microlocal analysis" that generalizes the classical theory of stationary phase and steepest descent.

Large Scale Stochastic Dynamics: Interacting particle systems are models with many degrees of freedom whose stochastic dynamics is prescribed at a microscopic scale. Such models have been very successful in modelling and rigorously analyzing complex systems motivated by engineering sciences, physics, and finance. The aim is to understand and to derive quantitative effective descriptions of phenomena that emerge on the macroscopic scale, and to relate macroscopic observables of these systems to the parameters of their microscopic interactions.

Education and Training: The group contributes to the education and training of undergraduate, graduate and post-graduate students as well as of PhD candidates and Postdoctoral researchers.

Applied Analysis

RESEARCH AND DEVELOPMENT PROGRAMS

A. ONGOING PROJECTS

Title: IPMAFIN: Interfacial phenomena in stochastic reaction-diffusion systems & applications to mathematical finance
Funding Source and funding scheme: Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "Basic Research Financing Action (Horizontal support of all Sciences), Subaction II, Greece 2.0
Duration: 2023-2025

B. COMPLETED PROJECTS

Title: iSTAMP: Innovative Stochastic Dynamics, Analysis and Simulations for Phase Transitions
Funding Source and funding scheme: Hellenic Foundation for Research and Innovation (H.F.R.I) under the "First Call for H.F.R.I. Research Projects to support Faculty members and Researchers"
Duration: 2020-2023

PUBLICATIONS

2025
2024
- ND Alikakos, Z Geng (2024) On the Triple Junction Problem without Symmetry Hypotheses, Archive for Rational Mechanics and Analysis 248 (2), 24.
- ND Alikakos, M Nikolouzos, AN Yannacopoulos (2024) Variational maximum principle for elliptic systems involving the fractional Laplacian, São Paulo Journal of Mathematical Sciences, 1-20.
- DC Antonopoulou, D Dimitriou, G Karali, K Tzirakis (2024) Malliavin Calculus for the one-dimensional Stochastic Stefan Problem, arXiv preprint arXiv:2407.20389
- G Barbatis, KT Gkikas, A Tertikas (2024) Heat and Martin kernel estimates for Schrödinger operators with critical Hardy potentials, Mathematische Annalen 389 (3), 2123-2192.
- G Barbatis, A Tertikas (2024) Sobolev improvements on sharp Rellich inequalities, Journal of Spectral Theory 14 (2024), 641–663, DOI 10.4171/JST/508
- J.L. Bona, A. Chatziafratis, H. Chen, S. Kamvissis (2024) The linear BBM-equation on the half-line, revisited, Letters in Mathematical Physics, Vol. 114, 91, https://doi.org/10.1007/s11005-024-01820-0
- A Chatziafratis, S Kamvissis (2024) Continuous dependence on data for linear evolution PDEs on the quarter-plane, arXiv preprint arXiv:2403.14323
- arXiv:2401.07642
2023
- arXiv:2302.09841
- https://doi.org/10.1007/s00208-023-02693-9
- G Barbatis, A Tertikas (2023) Sobolev improvements on sharp Rellich inequalities, arXiv preprint arXiv:2312.00433
- Filippas, S., Tersenov, A. (2023) Estimates on the velocity of a rigid body moving in a fluid, arXiv preprint arXiv:2304.11887
- S Fujiié, N Hatzizisis, S Kamvissis (2023) Semiclassical WKB problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential, Journal of Differential Equations, Vol. 360, pp. 90-150. (html file)
- N Hatzizisis, S Kamvissis (2023) Semiclassical WKB problem for the non-self-adjoint Dirac operator with a multi-humped decaying potential, Asymptotic Analysis, vol. 137, no. 3-4, pp. 177-243.
2022
- https://doi.org/10.1111/sapm.12542
- S Fujiie, N Hatzizisis, S Kamvissis (2022) Semiclassical Wkb Problem for the Non-Self-Adjoint Dirac Operator Withan Analytic Rapidly Oscillating Potential, Available at SSRN 4108409 (html)
2021
- Alikakos N., Fusco G. (2021) Sharp lower bounds for vector Allen-Cahn energy and qualitative properties of minimizes under no symmetry hypotheses, arXiv preprint arXiv:2110.00388
- Filippas, S., Tersenov, A. (2021) On Vector Fields Describing the 2d Motion of a Rigid Body in a Viscous Fluid and Applications, J. Math. Fluid Mech. 23, 5 . https://doi.org/10.1007/s00021-020-00528-0
- arXiv:2108.11077
- R Kousovista, G Karali, K Vlasopoulou, V Karalis (2021) Validation of population pharmacokinetic models: a comparison of internal and external validation approaches for hydrochlorothiazide, Xenobiotica 51 (12), 1372-1388.
2020
- DC Antonopoulou, M Bitsaki, G Karali (2020) The multi-dimensional Stochastic Stefan Financial Model for a portfolio of assets, arXiv preprint arXiv:2012.13432
- Cheliotis, D., Loulakis, M., Kontoyiannis, I., Toumpis, S. (2020) ‘A Simple Network of Nodes Moving on the Circle, Random Struct Alg 2020. DOI:10.1002/rsa.20932
- Fujiie, S., Kamvissis, S., (2020) Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential, Journal of Mathematical Physics, v.61, n.1, 011510.
- Haskovec, J. Markou, I., (2020) Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Models 13, 795—813.
- Haskovec, J., Markou, I., (2020) Exponential asymptotic flocking in the Cucker-Smale model with distributed delays, Math. Biosci. Eng., 17, 5651—5671.
2019
- Loulakis, M., Kossioris G.T., Souganidis, P.E., (2019) The Deterministic and Stochastic Shallow Lake Problem. In: Friz P., König W., Mukherjee C., Olla S. (eds) Probability and Analysis in Interacting Physical Systems, in honor of S.R.S. Varadhan 75th birthday, Springer 2019, 49-74. DOI:10.1007/978-3-030-15338-0_3210
2018
- Alikakos, N., Fusco, G., Smyrnelis, P., (2018) Elliptic systems of phase transition type, Monograph, PNLDE, Volume 91, 2018, Birkhauser, 343 pages.
- Alikakos, N., Fusco, G., (2018) Almost entire solutions of the Burgers equation, Electron.J. Differ. Eq., no 53, pp 1-6.
- Barbatis, G., Filippas, S., Tertikas, A. (2018) ‘Sharp Hardy and Hardy–Sobolev inequalities with point singularities on the boundary’, Journal des Mathematiques Pures et Appliquees, 2018, 117, pp. 146–184.
2017
2016
- Alikakos, N.D.; Fusco, G. (2016), Asymptotic behavior and rigidity results for symmetric solutions of the elliptic system Δu=Wu(u), Annali della Scuola Normale Superiore di Pisa. XV issue special (2016), pp. 809-836.
- Loulakis, M., (2016). Introduction to Mathematical Finance, Hellenic Academic Libraries available at http://hdl.handle.net/11419/3481
2015
- Alikakos, ND.; Fusco, G. (2015), A maximum principle for systems with variational structure and an application to standing waves, Journal of the European Mathematical Society 17, No. 7 (2015), pp. 1547-1567.
- DC Antonopoulou, PW Bates, D Blömker, GD Karali (2015) Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation, arXiv preprint arXiv:1501.05288
- Antonopoulou, D., Karali, G., Plexousakis, M., Zouraris, G., (2015). Crank-Nicolson finite element discretizations for a 2d linear Schrodinger-type equation posed in a noncylindrical domain, Mathematics of Computation, 84, no 294, 1571-1598.
- G Barbatis, A Tertikas (2015) On the Hardy constant of some non-convex planar domains, Geometric Methods in PDE’s, 15-41.
- J Dolbeault, MJ Esteban, S Filippas, A Tertikas (2015) Rigidity results with applications to best constants and symmetry of Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities, Calculus of Variations and Partial Differential Equations 54, 2465-2481.
- Karali, G., Sourdis, C., (2015). The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit, Archive for Rational Mechanics and Analysis, 217, no 2, 439-523.
- Stai, E., Loulakis, M., Papavassiliou, S., (2015). Cross-Layer Design of Wireless Multihop Networks over Stochastic Channels with Time-Varying Statistics, IEEE Trans. Wireless Commun. 14, no. 12, 6967-6980.
2014
- https://core.ac.uk/download/pdf/20257788.pdf
- Dellis, A.T., Loulakis, M., Kominis, I.K., (2014).Spin-noise correlations and spin-noise exchange driven by low-field spin-exchange collisions, Phys Rev A, 90 (3), 032905.
- Spectral theory and differential equations 233, 95-115.
- arXiv:1409.4519
- SK Giannopoulou, GN Makrakis (2014) Uniformization of WKB functions by Wigner transform Applicable Analysis 93 (3), 624-645.
- arXiv:1402.6194
2013
- Alikakos, ND., (2013), On the structure of phase transition maps for three or more coexisting phases. In Geometric partial differential equations M. Novaga and G. Orlandi eds. Publications of the Scuola Normale Superiore, CRM Series, Birkhauser.
- Armendariz, I., Grosskinsky, S. and Loulakis, M., (2013). Zero Range Condensation at Criticality, Stoch. Proc. and Appl. 123, no. 9, 3466-3496.
- Kossioris, G.T.; Zouraris, G.E. (2013). Finite element approximations for a linear fourth-order parabolic SPDE in two and three space dimensions with additive space-time white noise. Appl. Numer. Math. 67, 243–261.

PEOPLE

RESEARCHERS

CONTACT US

For any information regarding the group please contact:

Applied Analysis Group,
Institute of Applied and Computational Mathematics,
Foundation for Research and Technology - Hellas
Nikolaou Plastira 100, Vassilika Vouton,
GR 700 13 Heraklion, Crete
GREECE

Tel: +30 2810 391800
E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it. (Mrs. Maria Papadaki)

Tel.: +30 2810 391805
E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it. (Mrs. Yiota Rigopoulou)