Research Grant PN-III-P1-1.1-TE-2021-1539 

Research Grant CNCS-UEFISCDI

''Sharp analysis of partial differential operators''

Project code: PN-III-P1-1.1-TE-2021-1539

Contract no.: TE 146/2022

Funding: Ministry of Research, Innovation and Digitization, CNCS - UEFISCDI, within PNCDI III

Period: 1 June 2022 - 31 May 2024 

Host Institution: University of Bucharest,  Research Institute of the University of Bucharest (ICUB)

Team members:

• Cristian Cazacu (Project leader)

• Liviu Ignat (Researcher)

• Denisa Stancu Dumitru (Researcher, Involved after 1 October 2022)

• Teodor Rugină (PhD Student)

• Dragoș Manea (PhD Student, Involved after 5 January 2023)

Abstract: This project proposal is mainly dedicated to the analysis of the sharp constants and the existence/nonexistence of minimizers in functional inequalities involving partial differential operators of first or second order such as the free/magnetic gradient and the Laplacian applied to either scalar or vector fields. The study of the present project proposal is threefold: a) Hardy inequality (HI) and the magnetic p-Laplacian; b) Sharp Caffarelli-Kohn-Nirenberg (CKN) inequalities; c) Stability of Hardy and CKN inequalities. In a) we distinguish two main problems, namely a free magnetic multipolar HI in L^p setting and a magnetic L^p HI with one singular potential; in b) we also analyze two situations related to a first order CKN inequality for curl-free and divergence-free vector fields and a second order CKN for scalar fields; in c) we develop stability estimates for the part of the inequalities studied in a) and b).

Objectives:   Hardy inequalities (HI) with trivial and nontrivial magnetic fields or HI with multipolar potentials and Caffarelli-Kohn-Nirenberg (CKN) inequalities will be analyzed with the scope of applying them directly in the study of qualitative and quantitative properties of PDEs. A better understanding of spectral properties of magnetic or singular operators and the study of evolution equations on various structures will allow to develop well-posedness, regularity, asymptotic, stabilization or controllability properties of such systems.  

Estimated results:  We expect to discover new efficient methods and techniques or to improve methods already known in the literature to solve the mathematical problems proposed in this project such as the development of optimal functional inequalities (especially of Hardy or CKN type) with direct applications in the sharp/qualitative analysis of differential operators (local or non-local) to obtain asymptotic properties, stability results  for evolution equations associated to the operators under consideration. We hope to provide as many deliverables as possible, among which we list i) to publish research papers in prestigious international ISI journals - especially in the first 25% according to the Article Influence Score classification- or monographs ii) to disseminate the obtained results by delivering talks at international congresses, workshops and schools iii) to attend international conferences, workshops or  schools iii) to provide a webpage of the project; iv) to organize (at least one) international workshop dedicated to young researchers; v) to invite experienced researchers to disseminate their results in our institution.

Obtained results

The results obtained during the period of  the project implementation mostly fullfil all the estimated results we expected and hoped to obtain at the beginning of the funding period.  The most important scientific contributions refer to improved L^p magnetic Hardy inequalities, sharp weighted CKN inequalities for vector fields,  sharp stability constant in the Heisenberg Uncertainty Principle,  asymptotic behavior of solutions for diffusion equations with either local or nolocal terms,  minimization problems for Rayleigh-type quotients.         

2024

Publications

1. F. Della Pietra, G. Fantuzzi, L. I. Ignat, A. L.  Masiello, G. Paoli and E. Zuazua, Finite element approximation of the Hardy constant. J. Convex Anal. 31 (2024), no. 2, 497-523.

2.  C. Cazacu, D. Krejčiřík, N. Lam and A. Laptev, Hardy inequalities for magnetic p-Laplacians. Nonlinearity 37 (2024), no. 3, 035004 (27 pp). 

3. J. Endal, L. I. Ignat and F. Quirós, Heat equations with fast convection: Source-type solutions and large-time behaviour. Discrete Contin. Dyn. Syst. Ser. S 17 (2024), no. 4, 1497–1512.

4. C. Cazacu, J. Flynn, N. Lam and G. Lu, Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities.  J. Math. Pures Appl., 182 (2024) 253-284.

5. C. Cazacu and T. Rugină, Best constants in bipolar $L^p$ Hardy-type Inequalities. J. Math. Anal. Appl. 530 (2024) 127635. 

6. T. Crin-Barat and D. Manea, Asymptotic-preserving finite difference method for partially dissipative hyperbolic systems. Preprint, arXiv:2404.06380.

7. M. M. González, L. I. Ignat, D. Manea and S. Moroianu, Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space. Nonlinear anal. TMA, minor revision, preprint arXiv:2302.02624.  

8. M. Mihăilescu, D. Stancu-Dumitru and  A.-M. Teca,  On the  infimum of a Rayleigh-type quotient involving a variable exponent which depends on test functions. Submitted.

9. C. Cazacu, I. Fidel, Weighted Hardy-Rellich type inequalities: improved best constants and symmetry breaking. To appear in Rev. Roumaine Math. Pures Appl. Preprint arXiv:2406.15792.

Talks

1. 23/05/2024, T. Rugină: Hardy-Rellich Inequalities in domains of $\mathbb{R}^N$ and applications, "Workshop for Young Researchers in Mathematics"-13th edition, Iași, Romania. Available online at: https://fmi.univ-ovidius.ro/cercetare/wyrm2024/. Invited speaker.  

2. 23/05/2024, D. Manea: Atmospherical frontogenesis in vortex liquid mixing and optimal placement, "Workshop for Young Researchers in Mathematics"-13th edition, Iași, Romania. Available online at: https://fmi.univ-ovidius.ro/cercetare/wyrm2024/. Invited speaker. 

3. 23/05/2024, L. Ignat: Approximations of the best constants, "Workshop for Young Researchers in Mathematics"-13th edition, Iași, Romania. Available online at: https://fmi.univ-ovidius.ro/cercetare/wyrm2024/. Invited speaker. 

4. 25/04/2024, C. Cazacu: Magnetic Hardy inequalities in L^p setting, Analysis and Operator Theory Seminar, The Ohio State University, USA (online seminar). Available online at https://u.osu.edu/aots/ . Invited speaker. 

5. 09/02/2024, D. Manea:  A Payne–Whitham model of urban traffic networks in the presence of traffic lights, Final Conference of the Action Mathematical Models for Interacting Dynamics on Networks, University of Minho, Braga, Portugal. Available online at https://w3.math.uminho.pt/FinalConfMatDynNet. Invited speaker.

6. 23/01/2024, T. Rugină: Hardy-Rellich Inequalities in domains of $\mathbb{R}^N$ and applications,  Workshop Analysis and Potential, on the Anniversary of Gheorghe Bucur, Institute of Mathematics of the Romanian Academy, Bucharest, Romania. Available online at http://www.imar.ro/organization/activities/ archive/conferences_arh_conf_2024_s.php. Invited speaker. 

Research visits and conference attendances 

1. 27/05/2024-31/05/2024, C. Cazacu, Modern Methods in Nonlinear Elliptic and Parabolic PDE, Plovdiv, Bulgaria. Available online at https://sites.google.com /view/pdes-in-plovdiv/ .

2. 23/05/2024-24/05/2024, C. Cazacu, "Workshop for Young Researchers in Mathematics"-13th edition, Iași, Romania. Available online at: https://fmi.univ-ovidius.ro/cercetare/wyrm2024/.

3.  03/04/2024-07/04/2024, L. Ignat, research visit at Sapientia Hungarian University of Transylvania, Târgu Mureș, Romania.  Invited by Prof. Csaba Farkas 

4.  19/03/2024-24/03/2024, L. Ignat, research visit at University of Craiova, Romania. Invited by Prof. Mihai Mihăilescu

Organized workshop

1. 29/05/2024, Workshop on PDEs - Research in Pairs in Bucharest, Romania. Available online at http://www.imar.ro/~imar/2024/Conferinte/Workshop_on_PDEs .pdf . Organized by L. Ignat. 

Scientific report 2024 (RO)

2023

Publications

1. C. Cazacu, J. Flynn and N. Lam, Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives.  Calc. Var Partial Differential Equations 62 (2023), no. 4, Paper No. 118, 26 pp.

2. J. Endal, L. I. Ignat and F. Quirós, Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection, J. Math. Pures Appl. (9) 179 (2023), 277–336.

3. M. Mihăilescu and  D. Stancu-Dumitru,  Monotonicity with respect to $p$ of the best constants associated with Sobolev immersions of type $W_0^{1,p (\Omega)\hookrightarrow L^q(\Omega)$ when $q\in\{1,p,\infty\}$, Studia Universitatis Babeş-Bolyai Mathematica 68 (2023), no. 1, 109-123. 

4. C. Cazacu, L. Ignat and D. Manea, The Hardy inequality and large time behaviour of the heat equation on $\mathbb{R}^{N-k}\times (0,\infty)^k$. J. Differential Equations 375 (2023) 682–705.  

5.  M. Farcaseanu, M. Mihailescu and Denisa Stancu-Dumitru: A minimization problem related to the principal frequency of the p-Bilaplacian with coupled Dirichlet-Neumann conditions. Electron. J. Qual. Theory Differ. Equ. (2023) No. 51, 1-9. 

Talks

1. 09/10/2023,  L. Ignat: Asymptotic behavior of solutions for some diffusion problems on metric graphs, PDE SEMINAR, UAM - ICMAT, Madrid, Spain. Available online at  https://sites.google.com/view/seminarioedpsuam-icmat. Invited speaker. 

2. 16/09/2023, C. Cazacu: Sharp uncertainty principles and stability results, The 30th Conference on Applied and Industrial Mathematics CAIM, 14-17 Sept. 2023,  Iasi, Romania.  Available online at https://caim.romai.ro/Caim23/caim_23.htm. Plenary speaker.

3. 20/09/2023,  L. Ignat:  Asymptotic behavior of solutions for some diffusion problems on metric graphs, NCSU Differential Equations/Nonlinear Analysis Seminars,  NC state university, SUA (online seminar), https://wp.math.ncsu.edu/nat/ncsu-differential-equations-nonlinear-analysis-seminar-schedule-fall-2023/. Invited speaker.

4. 30/06/2023, C. Cazacu: Recent results on Hardy inequalities for magnetic p-Laplacians, The Tenth Congress of Romanian Mathematicians, Piteşti, Romania.   Available online at http://www.imar.ro/~congmatro10/program. Invited speaker. 

5. 01/07/2023, D. Manea:  Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space, The Tenth Congress of Romanian Mathematicians, Piteşti, Romania.  Available online at http://www.imar.ro/~congmatro10/program. Invited speaker. 

6. 01/07/2023, L. Ignat:  Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection, The Tenth Congress of Romanian Mathematicians, Piteşti, Romania.  Available online at http://www.imar.ro/~congmatro10/program. Invited speaker. 

7. 04/07/2023, D. Stancu-Dumitru: The Monotonicity of the principal frequency of the anisotropic p-Laplacian, The Tenth Congress of Romanian Mathematicians, Piteşti, Romania.   Available online at http://www.imar.ro/~congmatro10/program. Invited speaker.  

8.  13/06/2023,  L. Ignat: Asymptotic behaviour of solutions for some diffusion problems on a metric graphs, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Praga, Republica Cehă. Available online at https://www.fjfi.cvut.cz/en/26-cz/aktuality/8324-mafia-seminar-13-6-12pm-liviu-ignat. Invited speaker.

9. 26/05/2023, L. Ignat: Asymptotic behavior of solutions for some diffusion problems on metric graph, Fifth Romanian Itinerant Seminar on Mathematical Analysis and its Applications, Craiova,  Romania.  Available online at http://rismaa.ucv.ro/. Plenary speaker.

10. 27/05/2023, C. Cazacu: The Hardy inequality and asymptotic behaviour of the heat equation on $\rr^{N-k}\times (0,\infty)^k$, Fifth Romanian Itinerant Seminar on Mathematical Analysis and its Applications, Craiova,  Romania.  Available online at http://rismaa.ucv.ro/. Invited speaker.

11. 27/05/2023, D. Stancu-Dumitru: A minimization problem related to the principal frequency of the p-Bilaplacian with coupled Dirichlet-Neumann boundary conditions, Fifth Romanian Itinerant Seminar on Mathematical Analysis and its Applications”, Craiova,  Romania.  Available online at http://rismaa.ucv.ro/. Invited speaker.

12. 18/05/2023, C. Cazacu: Stability results for uncertainty principles, "Workshop for Young Researchers in Mathematics"-12th edition, Iași, Romania. Available online at: https://fmi.univ-ovidius.ro/cercetare/wyrm2023/. Invited speaker. 

13. 18/05/2023, T. Rugină: H∞-control problem for parabolic systems with singular inverse-square potential and convection, "Workshop for Young Researchers in Mathematics"-12th edition, Iași, Romania. Available online at: https://fmi.univ-ovidius.ro/cercetare/wyrm2023/. Invited speaker. 

14. 11/05/2023,  L. Ignat: Asymptotic behavior of solutions for some diffusion problems on metric graphs, Mini-workshop: Analysis, Numerics and Control,  FAU, Erlangen, Germany. Available online at https://dcn.nat.fau.eu/wp-content/uploads/FAUDCNAvHworkshop_lIgnat_slides_11may2023.pdf

15. 10/03/2023, L. Ignat: Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection, Workshop ""Nonlocal Equations : Analysis and Numerics", Bielefeld, Germany. Available online at "https://www.sfb1283.uni-bielefeld.de/Talks/filter?event_id=556". Invited speaker.    

16. 4/03/2023, D. Stancu-Dumitru: The Monotonicity of the Principal Frequency of the Anisotropic p-Laplacian, “UAE Math Day”, Zayed University Dubai, Dubai, United Arab Emirates. Available online at: https://www.zu.ac.ae/main/en/uae-mathday/UAE_Math_Day_Program. Invited speaker.  

17. 19/01/2023, C. Cazacu: Uncertainty principles: sharp constants, extremal functions and stability results, "Numerical and theoretical advances in quantum mechanics", Université Paul Sabatier, Toulouse, France.   Available online at: https://indico.math.cnrs.fr/event/7593/timetable/#20230119. Invited speaker.

18. 19/01/2023, L. Ignat: Asymptotic behavior of solutions for some diffusion problems on metric graphs, "Numerical and theoretical advances in quantum mechanics", Université Paul Sabatier, Toulouse, France.   Available online at: https://indico.math.cnrs.fr/event/7593/timetable/#20230119. Invited speaker.

Poster

1. 17/04/2023, T. Rugină: H∞-control problem for parabolic equations with singular inverse-square potentials with convection, Workshop "Geometric Aspects of Evolution and Control", FernUniversität in Hagen, Germany. Available online at: https://www.fernunihagen.de/analysis/forschung/veranstaltungen/workshop_2023

Research visits and conference attendances 

1. 09/11/2013-12/11/2023, T. Rugină, International Conference on Applied and Pure Mathematics (ICAPM) - 7th Edition 9-12 Nov 2023, Iași, Romania. Available online at http://math.etti.tuiasi.ro/apm2023/

2. 02/10-04/11/2023, L. Ignat, research visit at ICMAT &  Universidad Autonoma de Madrid, Spain. Invited by Prof. Fernando Quiros.

3. 30/06/2013-05/07/2023, T. Rugină, attendance at The Tenth Congress of Romanian Mathematicians,  June 30 to July 5, 2023, Pitești, Romania. Available online at http://www.imar.ro/~congmatro10/

4. 11/06/2023-19/06/2023, L. Ignat, reseach visit at Faculty of Nuclear Sciences and Technical Engineering, Czech Technical University, Prague, Czech Republic. Invited by Prof. David Krejcirik.

5. 12/06/2023-16/06/2023, D. Stancu-Dumitru,  research visit and participant at Calculus of Variations & Geometric Measure Theory, Conference in honor of Luigi Ambrosio on the occasion of his 60th birthday, University of Pisa, Italy. Available online at  https://la2023.cs.dm.unipi.it/home-page.

6. 12/06/2023-16/06/2023, C. Cazacu,  research visit and participant at Calculus of Variations & Geometric Measure Theory, Conference in honor of Luigi Ambrosio on the occasion of his 60th birthday, University of Pisa, Italy. Available online at  https://la2023.cs.dm.unipi.it/home-page. 

7. 2/03/2023-9/03/2023, D. Stancu-Dumitru, research visit at Department of Mathematics and Statistics, American University of Sharjah, United Arab Emirates. Invited by Prof. Cristian Enache. 

8. 15/01/2023-26/01/2023, C. Cazacu, research visit at Université Paul Sabatier, Toulouse, France. Invited by Prof. Claudia Negulescu. 

9. 15/01/2023-26/01/2023, L. Ignat, research visit at Université Paul Sabatier, Toulouse, France. Invited by Prof. Claudia Negulescu.

Organized workshop

1. 26/06/2023-27/06/2023, Workshop on PDEs - Research in Pairs in Bucharest, Romania. Available online at http://www.imar.ro/~imar/RIPB/Workshop-on-PDEs-Research-in-Pairs-June2023.pdf. Organized by L. Ignat. 

Prize

1. 30/06/2023, C. Cazacu, "Nicolae Dinculeanu" Prize at the 10th Congress of the Romanian Mathematicians, awarded by the Mathematical Sciences Section of the Romanian Academy. 

Scientific report 2023 (RO)

2022

Talks

1. 22/11/2022, C. Cazacu: The Hardy inequality and large time behaviour of the heat equation on $\rr^{N-k}\times (0,\infty)^k$,  "MAFIA-Methods of Algebra and Functional Analysis In Applications" seminar, Faculty of Nuclear Sciences and Technical Engineering, Czech Technical University, Prague, Czech Republic. Available online at: http://mafia.fjfi.cvut.cz/index.php?str=seminar. Invited speaker.

2. 7/11/2022, L. Ignat: Asymptotic behavior of solutions for some diffusion problems on metric graphs,  The DNA Seminar, NTNU – Norwegian University of Science and Technology, Trondheim, Norway. Available online at: https://wiki.math.ntnu.no/seminar/dna. Invited speaker.  

3. 28/10/2022, L. Ignat: Large-time asymptotics of some numerical schemes for Burgers' like equations, "Numerical Analysis, Numerical Modeling, Approximation Theory", online conference, Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania. Available online at:  https://ictp.acad.ro/nanmat-2022-abstracts/. Invited speaker. 

4. 29/08/2022, C. Cazacu: Sharp weighted first and second order Caffarelli-Kohn-Nirenberg type inequalities in the $L^2$ setting, "XVème Colloque Franco-Roumain de Mathématiques Appliquées", Université Paul Sabatier, Toulouse, France. Available online at: https://15colfrro.sciencesconf.org/. Invited speaker.

5. 31/08/2022, L. Ignat: Asymptotic behavior of solutions for some diusion problems on metric graphs, "XVème Colloque Franco-Roumain de Mathématiques Appliquées", Université Paul Sabatier, Toulouse, France. Available online at: https://15colfrro.sciencesconf.org/. Plenary speaker.

6. 29/06/2022, C. Cazacu: Hardy inequalities for magnetic p-Laplacians, within online Workshop "International conference on applied mathematics and numerical methods (ICAMNM) - 4rd edition", Craiova, Romania.  Available online at: https://cis01.central.ucv.ro/ICAMNM/?Programme.  Invited speaker.

Research visits

1. 18/11/2022-26/11/2022, C. Cazacu, reseach visit at Faculty of Nuclear Sciences and Technical Engineering, Czech Technical University, Prague, Czech Republic. Invited by Prof. David Krejcirik. 

Scientific report 2022 (RO)