Project code: PN-III-P1-1.1-TE-2016-2233
Contract no.: TE 115/2018
Funding: Ministry of Research and Innovation, CNCS-UEFISCDI, within PNCDI III
Period: 10 October 2018 - 9 October 2020 (extended to 30 November 2020)
Host Institution: University of Bucharest, Research Institute of the University of Bucharest (ICUB)
Team members:
Cristian Cazacu (Project leader)
Abstract: The overall aim of this project is to provide a qualitative analysis of Partial Differential Equations (PDEs) with particular attention on theoretical aspects as well-posedness, asymptotic behavior, control properties for equations with singular/variables coefficients and nonlocal terms, with the auxiliary scope to underline part of them from a numerical analysis point of view. The project is focused on 3 main topics: A). Hardy-type inequalities and singular PDEs, B). Asymptotic properties for evolution equations and C). Control and numerics in PDEs. In A), we will analyze Hardy-type inequalities for Hamiltonian operators of the type H=T+V (where T denotes the kinetic energy and V refers to the potential energy). We will mainly focus on the Hamiltonian operators -Δ+V and (-i∇-A)^2+V, where A is a magnetic field and V is a singular potential. In B), we analyze two types of problems: B1). Local heat-like equations with singular/degenerate potentials; B2). Convection-diffusion equations for nonlocal operators. In C), we are firstly focused on some control problems on networks. Then, we intend to implement numerical schemes to approximate the controls of PDEs under consideration. The discrete control approach in engineering consists in discretizing continuous models and controlling them by using the finite-dimensional control theory.
Objectives: The main goal of this project concerns the study of functional Hardy inequalities and singular equations, of asymptotic properties for evolution equations, as well as of controllability problems for continuous or discrete models. We aim at obtaining relevant scientific results published in internationally recognized journals, disseminating the results within high visibility conferences, organizing workshops and working groups between members of the academic community with expertise in this field.
Estimated results: We expect to develop innovative methods to solve the problems mentioned in the present project. More exactly, efficient approaches to optimal Hardy or Hardy-Rellich constants will be developed with direct consequences and applications to spectral theory of singular operators. A better understanding of both local and nonlocal operators will allow to obtain qualitative properties of their time evolution. We also expect to obtain stabilization or controllability properties for equations on graphs and for continuous or discrete wave equations.
M. Mihăilescu, J. D. Rossi: Monotonicity with respect to p of the first nontrivial eigenvalue of the p-Laplacian with homogeneous Neumann boundary conditions. Commun. Pure Appl. Anal. 19 (2020), no. 9, 4363–4371.
C. Cazacu: A new proof of the Hardy-Rellich inequality in any dimension. Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), no. 6, 2894–2904.
U. Biccari, A. Marica, E. Zuazua: Propagation of one and two-dimensional discrete waves under finite difference approximation. Found. Comput. Math., 20 (2020), 1401–1438.
L. I. Ignat, J. D. Rossi, A. San Antolin: Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length. Ann. Mat. Pura Appl. (4), 200 (2021), no. 3, 1301–1339.
M. Mihăilescu: The spectrum of the mean curvature operator. Proc. Roy. Soc. Edinburgh Sect. A., 151 (2021), no. 2, 451–463.
C. Cazacu: The method of super-solutions in Hardy and Rellich type inequalities in the $L^2$ setting: an overview of well-known results and short proofs. Rev. Roumaine Math. Pures Appl., 66 (2021), no. 3-4, 617–638.
M. Fărcășeanu, A. Grecu, M. Mihăilescu, D. Stancu-Dumitru: An overview on torsional creep problems. Rev. Roumaine Math. Pures Appl., 66 (2021), 3-4, 695–708.
A. Grecu, M. Mihăilescu: Monotonicity of the principal eigenvalue of the p-Laplacian on an annulus. Math. Reports, (Bucur.) 23(73) (2021), no. 1-2, 149–155.
C. Cazacu, L. Ignat, A. Pazoto, J. D. Rossi: A convection-diffusion model on a star shaped graph. NoDEA Nonlinear Differential Equations Appl. 29 (2022), no. 2, Paper no. 17.
C. Cazacu, A. Marica: Hardy inequalities for inverse square potentials with countable number of singularities. Submitted, preprint arXiv:1912.02391.
16/1/2020, C. Cazacu: Hardy inequalities with multipolar potentials, within Workshop "7th Edition on Mathematical Analysis (Septimas Jornadas de Analisis Matematico)", Universidad de Alicante, Spain. Available online at: https://dmat.ua.es/en/activities/7th-edition-mathematical-analysis.html.
16/1/2020, M. Mihăilescu: The Monotonicity of the Principal Eigenvalue of the $p$-Laplace Operator, within Workshop "7th Edition on Mathematical Analysis (Septimas Jornadas de Analisis Matematico)", Universidad de Alicante, Spain. Available online at: https://dmat.ua.es/en/activities/7th-edition-mathematical-analysis.html.
12/10/2020, C. Cazacu: Optimal constants in Hardy and Hardy-Rellich type inequalities, within Online seminar "Geometric and functional inequalities and applications", organized by Joshua Flynn (University of Connecticut, USA), Nguyen Lam (Memorial University of Newfoundland, Canada), Jungang Li (Brown University, USA), Guozhen Lu (University of Connecticut, USA). Available online at: https://researchseminars.org/seminar/GeomInequAndPDEs.
21/10/2020, L. Ignat: Asymptotic behavior of solutions for some local and nonlocal diffusion problems on metric graphs, within the (virtual) seminar organized by COST partener Delio Mugnolo (University of Hagen, Germany). Available online at: https://www.fernuni-hagen.de/analysis/en/research/analysisseminar.shtml
29/10/2020, M. Mihăilescu: The Spectrum of the Relativistic Mean Curvature Operator, within online Workshop "International conference on applied mathematics and numerical methods (ICAMNM) - 3rd edition", Craiova, Romania. Available online at: http://cis01.central.ucv.ro/ICAMNM/?Programme
29/10/2020, L. Ignat: Asymptotic behavior of solutions for local and nonlocal diffusion on metric graphs, within online Workshop "International conference on applied mathematics and numerical methods (ICAMNM) - 3rd edition", Craiova, Romania. Available online at: http://cis01.central.ucv.ro/ICAMNM/?Programme.
15/1/2020-17/1/2020, Workshop 7th Edition on Mathematical Analysis (Septimas Jornadas de Analisis Matematico), organized by L. Ignat, José Carlos Navarro, Julio D. Rossi, Ángel San Antolín, host institution: Universidad de Alicante, Spain. Available online at: https://dmat.ua.es/en/activities/7th-edition-mathematical-analysis.html.
A. Grecu, L. Ignat.: The Schrödinger equation on a star-shaped graph under general coupling conditions. J. Phys. A: Math. Theor. 52 (2019), no. 3, 035202, 26 pp.
12/4/2019, Cristian Cazacu, About some Hardy type inequalities: optimality and minimizers, within ''Italian-Romanian Colloquium on Differential Equations and Applications'', Universita degli Studi di Udine, Italy. Available online at: https://ireda.weebly.com/ .
11/4/2019, Mihai Mihăilescu, The Monotonicity of the Principal Eigenvalue of the p-Laplace Operator and Related Problems, within ''Italian-Romanian Colloquium on Differential Equations and Applications'', Universita degli Studi di Udine, Italy. Available online at: https://ireda.weebly.com/ .
30/5/2019, Cristian Cazacu, Time evolution properties for a convection-diffusion equation on a star-shaped tree, within ''Second Workshop on Analysis, PDEs and Mechanics'', "Gheorghe Mihoc-Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Romania, Available online at: https://sites.google.com/site/marinliviu/2nd-workshop-on-analysis-pdes-and-mechanics---30-may-2019.
30/5/2019, Mihai Mihăilescu, Inhomogeneous torsional creep problems in anisotropic Orlicz Sobolev spaces, within ''Second Workshop on Analysis, PDEs and Mechanics'', "Gheorghe Mihoc-Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Romania, Available online at: https://sites.google.com/site/marinliviu/2nd-workshop-on-analysis-pdes-and-mechanics---30-may-2019.
30/5/2019, Liviu Ignat, The Schr\"{o}dinger equation on a tree: dispersion and Hardy’s uncertainty principle, within ''Second Workshop on Analysis, PDEs and Mechanics'', "Gheorghe Mihoc-Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Romania, Available online at: https://sites.google.com/site/marinliviu/2nd-workshop-on-analysis-pdes-and-mechanics---30-may-2019.
3/6/2019, Cristian Cazacu, Asymptotic properties for a diffusion equation with supercritical convective term on a star-shaped tree, within ''Workshop for Young Researchers in Mathematics-9th Edition'', Faculty of Mathematics and Computer Science, University of Bucharest, Romania. Available online at http://math.univ-ovidius.ro/workshop/2019/WYRM/#/.
4/6/2019, Aurora Marica, Multipliers methods for discrete wave equations, within ''Workshop for Young Researchers in Mathematics-9th Edition'', Faculty of Mathematics and Computer Science, University of Bucharest, Romania. Available online at http://math.univ-ovidius.ro/workshop/2019/WYRM/#/.
29/6/2019, Aurora Marica, Eigenfunction estimates for discrete Laplace operators, within ''The Ninth Congress of Romanian Mathematicians'', Galați, Romania. Available online at: https://sites.google.com/view/congmatro9/home.
30/6/2019, Cristian Cazacu, Functional inequalities of Hardy type, within ''The Ninth Congress of Romanian Mathematicians'', Galați, Romania. Available online at: https://sites.google.com/view/congmatro9/home.
30/6/2019, Liviu Ignat, The Schrodinger equation on a tree: dispersion and Hardy’s uncertainty principle, within ''The Ninth Congress of Romanian Mathematicians'', Galați, Romania. Available online at: https://sites.google.com/view/congmatro9/home.
18/7/2019, Liviu Ignat, Null controllability of the Kuramoto-Sivashinsky equation on a star-shaped tree, within "9th International Congress on Industrial and Applied Mathematics", Valencia, Spain. Available online at https://iciam2019.org/.
5/10/2019, Liviu Ignat, The Schrodinger equation on a tree: dispersion and Hardy’s uncertainty principle, IMAR70, Bucharest, Romania. Available online at: https://sites.google.com/view/imar70/accueil.
9/7/2019-21/7/2019, Liviu Ignat, University of Alicante, Spain (Invited by Prof. Angel San Antolin)
15/08/2019-17/08/2019, Cristian Cazacu - Department of Applied Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania (Invited by Prof. Radu Precup)
28/08/2019-07/09/2019, Cristian Cazacu - Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague, Czech Republic (Invited by Prof. David Krejčiřík)
16/09/2019-20/09/2019, Cristian Cazacu - Faculty of Mathematics, Al. I. Cuza University of Iasi, Romania (Invited by Prof. Ionut Munteanu)
30/5/2019-5/6/2019, Aingeru Fernandez Bertolin, Universidad del Pais Vasco, Bilbao, Spain.
5/10/2019-12/10/2019, Diana Stan, University of Cantabria, Santander, Spain.
18/11/2019-30/11/2019, Alejandro Garriz Molina, Universidad Autonoma de Madrid, Spain.
3/6/2019-4/6/2019, Workshop for Young Researchers in Mathematics -9th Edition, organized by Cristian Cazacu, Cristina Anghel, Iulian Cîmpean, Constantin Costara, Nicușor Costea, Anca Măcinic, Dumitru Stamate, host institution: Faculty of Mathematics and Computer Science, University of Bucharest, Romania. Available online at http://math.univ-ovidius.ro/workshop/2019/WYRM/#/.
1. 16/08/2019, Cristian Cazacu, "Rada Mihalcea" Award for Young Researchers in Science and Engineering (second prize), Cluj-Napoca, 2019. Available online at https://primariaclujnapoca .ro/evenimente/tinerii-cercetatori-premiul-pentru-stiinta-si-inginerie/#c4aa8775190ead201.
9/11/2018, Cristian Cazacu, About a control problem on star-shaped trees and a functional inequality for vector fields, within “Workshop de analiză, ecuații diferențiale și mecanică”, University of Bucharest, Romania. Available online at: https://sites.google.com/site/cristianmihaicazacu/mini-workshop-icub-fmi
9/11/2018, Liviu Ignat, Asymptotic behaviour for nonlocal diffusion-convection equations, within “Workshop de analiză, ecuații diferențiale și mecanică”, University of Bucharest, Romania. Available online at: https://sites.google.com/site/cristianmihaicazacu/mini-workshop-icub-fmi
22/11/2018, Liviu Ignat, The Schrödinger equation on a tree, as part of the research visit at Universidad del Pais Vasco (University of the Basque Country), Bilbao, Spain. Available online at: https://sites.google.com/view/apdebilbao/seminars
21/11/2018, Liviu Ignat, The Schrödinger equation on a tree, at Universidad de Cantabria, Santander, Spain.
9/11/2018, Workshop de analiză, ecuații diferențiale și mecanică, organized by Cristian Cazacu and Liviu Marin, host institutions: the Research Institute of the University of Bucharest (ICUB) & Faculty of Mathematics and Computer Science, University of Bucharest, Romania. Available online at: http://fmi.unibuc.ro/ro/pdf/2018/conferinte/Mini-workshop_9noi2018.pdf
13/12/2018-14/12/2018, Atelier de travail en Equations aux Dérivées Partielles, organized by Liviu Ignat, Mihai Mihăilescu, Andreea Grecu, Cristian Cazacu, host institution: Institute of Mathematics of the Romanian Academy, Bucharest, Romania. Available online at: http://www.imar.ro/~purice/Inst/2018/afis%20Happy%20PDE-2018.pdf
1. 17/11/2018-28/11/2018, Liviu Ignat, Universidad del Pais Vasco (University of the Basque Country), Bilbao, Spain.
1. 13/12/2019, Cristian Cazacu, "Dimitrie Pompeiu" Prize of the Romanian Academy, awarded in 2018 (for 2016).