Grant CNCS-UEFISCDI PN-III-P1-1.1-TE-2019-0456

Funding Sources: Consiliul National al Cercetarii Stiintifice & Uniunea Executiva pentru Finantarea Invatamantului Superior a Cercetarii Dezvoltarii si Inovarii

Project code: PN-III-P1-1.1-TE-2019-0456

Contract number: TE 19/2020

Period: 15 September 2020 - 14 September 2022

Title: The Infinity-Laplace Operator

Abstract: In this project we are concerned with the study of some classes of Partial Differential Equations (PDE’s) involving the presence of the Infinity-Laplace operator. There are several research directions undergirding the topic of the present project: the Absolutely Minimizing Lipschitz Extension Problem (AMLE Problem) of Aronsson, the Torsional Creep Problem – studied by Kawohl and Bhattacharya, DiBenedetto and Manfredi, and The ∞-Eigenvalue Problem – investigated by Juutinen, Lindqvist and Manfredi. The motivation to study PDE’s involving the Infinity-Laplace operator partially stems from its usefulness in certain applications such as optimal transportation, image processing or tug-of-war games. Motivated by the classical problems recalled above, we propose the analysis of some related problems which represent the main objectives of our proposal and are expected to complete the existing knowledge to date on the topic: the study of the AMLE problem for non-uniformly elliptic operators; the analysis of some torsional creep problems in Finsler metrics; the investigation of the asymptotic behavior of some new families of eigenvalue problems.

Objectives: a) AMLE Problem for Non-uniformly Elliptic Operators; b) Torsional Creep Problems in Finsler Metrics; c) Asymptotic Behaviour of a Family of Eigenvalue Problems.

TEAM: Denisa Stancu-Dumitru (project leader), Cristian Cazacu, Nicusor Costea, Maria Farcaseanu, Andrei Grecu.

HOST INSTITUTION: Universitatea Politehnica din Bucuresti

PUBLICATIONS 2020:

• Maria Fărcăşeanu, Andrei Grecu, Mihai Mihăilescu and Denisa Stancu-Dumitru, Perturbed eigenvalue problems: an overview, Studia Universitatis Babeș-Bolyai Mathematica 66 (2021), 55-73. DOI: 10.24193/subbmath.2021.1.05; WOS: 000631641400006

• Maria Fărcăşeanu, Mihai Mihăilescu and Denisa Stancu-Dumitru, On a family of torsional creep problems in Finsler metrics, Canadian Journal of Mathematics, accepted. DOI: 10.4153/S0008414X20000681

CONFERENCES 2020:

• International Conference on Applied Mathematics and Numerical Methods, October 29-31, 2020, University of Craiova, Romania (speaker: Denisa Stancu-Dumitru; title of the talk: On a family of torsional creep problems in Finsler metrics; online conference);

• International Conference on Applied Mathematics and Numerical Methods, October 29-31, 2020, University of Craiova, Romania (speaker: Cristian Cazacu; title of the talk: Optimal Hardy-Rellich type inequalities; online conference);

• International Conference on Applied Mathematics and Numerical Methods, October 29-31, 2020, University of Craiova, Romania (speaker: Andrei Grecu; title of the talk: Fractional eigenvalue problems on $\mathbb{R}^N$; online conference);

• International Conference on Applied Mathematics and Numerical Methods, October 29-31, 2020, University of Craiova, Romania (speaker: Nicusor Costea; title of the talk: Existence results for mixed hemivariational-like inequalities; online conference);

• International Conference on Applied Mathematics and Numerical Methods, October 29-31, 2020, University of Craiova, Romania (speaker: Maria Fărcăşeanu; title of the talk: Sharp existence and classification results for nonlinear elliptic equations in with Hardy potential; online conference).

SCIENTIFIC REPORT 2020 (PDF)

Prize:

• 29/12/2020, Maria Fărcăşeanu, "Ad Astra 2020" Award for researchers in the first phase of the carrer, less than 7 years after the PhD defence, in research field of Mathematics and Computer Science, non-Romanian-based institution (Premiu de excelenta in cercetare doctorala in domeniul Matematica si Informatica, categoria cercetatori in prima faza a carierei, sub 7 ani in de la finalizarea tezei de doctorat, cu afiliere in strainatate). Available online at https://www.youtube.com/watch?v=eg__7EqJSFE or https://www.edupedu.ro/cercetatorii-anului-2020-premiati-de-asociatia-ad-astra-lista-castigatorilor/

Papers 2021:

• Andrei Grecu and Denisa Stancu-Dumitru, The asymptotic behavior of solutions to a class of inhomogeneous problems: an Orlicz-Sobolev space approach, Electronic Journal of Qualitative Theory of Differential Equations 38 (2021), 1-20, doi: https://doi.org/10.14232/ejqtde.2021.1.38; WOS: 000641292600001

• Mihai Mihailescu and Denisa Stancu-Dumitru, Torsional creep problems involving Grushin-type operators, Applied Mathematics Letters 121 (2021), no. 107423. https://www.sciencedirect.com/science/article/abs/pii/S0893965921002275 Doi: 10.1016/j.aml.2021.107423, WOS: 000682955100045.

• Cristian Cazacu, Joshua Flynn and Nguyen Lam, Short proofs of refined sharp Caffarelli-Kohn-Nirenberg inequalities, J. Differential Equations 302 (2021), 533-549, https://doi.org/10.1016/j.jde.2021.09.005

• Maria Farcaseanu and Mihai Mihailescu, On the Monotonicity of the Best Constant of Morrey's Inequality in Convex Domains, Proceedings of the American Mathematical Society 150 (2022), 651-660, DOI:10.1090/proc/15826, WOS:000749000100016.

Books 2021:

• Nicusor Costea, Alexandru Kristaly and Csaba Varga, Variational and Monotonicity Methods in Nonsmooth Analysis, Birkhauser Basel, 2021, pages XXI+446, ISBN: 978-3-030-81670-4, e-book ISBN: 978-3-030-81671-1, ISSN: 1660-8046, DOI: 10.1007/978-3-030-81671-1.

CONFERENCES 2021:

• Workshop for Young Researchers in Mathematics – 10th Edition, May 20-21, 2021, IMAR, (speaker: Denisa Stancu-Dumitru; Title of the talk: Torsional creep problems in Finsler metrics; online conference);

• Workshop for Young Researchers in Mathematics – 10th Edition, May 20-21, 2021, IMAR (speaker: Cristian Cazacu; Title of the talk: Second order uncertainty principles: sharp constants and minimizers; online conference);

• Workshop for Young Researchers in Mathematics – 10th Edition, May 20-21, 2021, IMAR (speaker: Andrei Grecu; Title of the talk: The asymptotic behavior of solutions to a class of inhomogeneous problems: an Orlicz-Sobolev space approach; online conference).

• Workshop for Young Researchers in Mathematics – 10th Edition, May 20-21, 2021, IMAR (speaker: Maria Farcaseanu; Title of the talk: Classification of singular solutions to nonlinear elliptic equations with a gradient term; online conference)).

• Mathematics & IT: Research and Education (MITRE-2021), July 1-3, 2021, Moldova State University, Chisinau, Republic of Moldova (speaker: Andrei Grecu; Title of the talk: Two Eigenvalue-type Problems in Exterior Domains; online conference).

• Mathematics & IT: Research and Education (MITRE-2021), July 1-3, 2021, Moldova State University, Chisinau, Republic of Moldova (speaker: Denisa Stancu-Dumitru; Title of the talk: Asymptotic Behaviour of solutions to a class of inhomogeneous problems; online conference).

• 8th International Conference on Mathematics and Informatics, September 9-10, 2021, Universitatea Sapientia, Târgu Mureş/Marosvásárhely, Romania (speaker: Maria Farcaseanu; Title of the talk: Sharp existence and classification results for nonlinear elliptic equations in $mathbb R^Nsetminus{0}$ with Hardy potential; online conference).

• 8th International Conference on Mathematics and Informatics, September 9-10, 2021, Universitatea Sapientia, Târgu Mureş/Marosvásárhely, Romania (speaker: Denisa Stancu-Dumitru; Title of the talk: Torsional creep problems involving Grushin-type operators; online conference).

• 8th International Conference on Mathematics and Informatics, September 9-10, 2021, Universitatea Sapientia, Târgu Mureş/Marosvásárhely, Romania (speaker: Cristian Cazacu; Title of the talk: First and second order Caffarelli-Kohn-Nirenberg type inequalities: sharp constants and minimizers; online conference).

• Third Romanian Itinerant Seminar on Mathematical Analysis and its Applications, “1 Decembrie 1918” University of Alba Iulia, Alba-Iulia, Romania, September 10-12, 2021 (speaker: Denisa Stancu-Dumitru; Title of the talk: Torsional creep problems involving Grushin p-Laplacian).

• Current Trends in Applied Mathematics, November 20, 2021, Octav Mayer Institute of Mathematics of Romanian Academy, Iasi, Romania (speaker: Cristian Cazacu; Title of the talk: Sharp first and second order Caffarelli-Kohn-Nirenberg type inequalities; online conference).

Ph. D DEFENSE 2021:

• On October 15, 2021 Andrei Grecu obtained at University of Craiova the PhD degree with the thesis entitled THE VARIATIONAL ANALYSIS OF SOME CLASSES OF PARTIAL DIFFERENTIAL EQUATIONS.

Prizes/Awards 2021:

• Award for scientific results of papers published in 2021 by CNCS for the paper "Andrei Grecu and Denisa Stancu-Dumitru, The asymptotic behavior of solutions to a class of inhomogeneous problems: an Orlicz-Sobolev space approach, Electronic Journal of Qualitative Theory of Differential Equations, 38 (2021), 1-20, WOS: 000641292600001", PN-III-P1-1.1- PRECISI-2021- 57837, No. 281 (PRECISI 2021-zona rosie) (https://uefiscdi.gov.ro/index.php )

• Award for scientific results of papers published in 2021 by CNCS for the paper "Mihai Mihailescu and Denisa Stancu-Dumitru, Torsional creep problems involving Grushin-type operators, Applied Mathematics Letters 121 (2021) no. 107423, doi: 10.1016/j.aml.2021.107423, WOS: 000682955100045. ", PN-III-P1-1.1- PRECISI-2021- 58075, No. 370 (PRECISI 2021-zona rosie) (https://uefiscdi.gov.ro/index.php )

SCIENTIFIC REPORT 2021 (PDF)

Papers 2022:

• Cristian Enache, Mihai Mihailescu and Denisa Stancu-Dumitru, The Monotonicity of the p-Torsional Rigidity in Convex Domains, Mathematische Zeitschrift 302, Issue 1, (2022), 419-431. https://doi.org/10.1007/s00209-022-03069-6, https://link.springer.com/article/10.1007/s00209-022-03069-6, WOS: 000815441500001.

• Marian Bocea, Mihai Mihailescu and Denisa Stancu-Dumitru, The Monotonicity of the Principal Frequency of the Anisotropic p-Laplacian, Comptes Rendus Mathématique 360 (2022), 993-100, Published online: 2022-09-29, DOI: 10.5802/crmath.348

• Cristian Cazacu, Joshua Flynn and Nguyen Lam, Sharp second order uncertainty principles, Journal of Functional Analysis, Volume 283, Issue 10, 15 November 2022, 109659, http://arxiv.org/abs/2012.12667, https://doi.org/10.1016/j.jfa.2022.109659

• Andrei Grecu and Mihai Mihailescu, 4-Harmonic Functions and Beyond, Le Matematiche 77 (2022), 107-118, doi: 10.4418/2022.77.1.6

• Andrei Grecu, An eigenvalue-type problem involving the $p$-Laplace operator in exterior domain, Quaestiones Mathematicae 45 (7) (2022), 1091-1098, DOI: 10.2989/16073606.2021.1928322, WOS:000661714900001. https://www.tandfonline.com/doi/abs/10.2989/16073606.2021.1928322.

• Andrei Grecu, A perturbed eigenvalue problem in exterior domain, Mathematica Slovaca 72 (4) (2022), 945-958, https://doi.org/10.1515/ms-2022-0065

• Andrei Grecu, The isolation of the first eigenvalue for a Dirichlet eigenvalue problem involving the Finsler $p-$ Laplacian and a nonlocal term, Revista de la Union Matematica Argentina 63, no. 2 (2022), pp. 443–453 (DOI : https://doi.org/10.33044/revuma.2281)

• Nicusor Costea: Nonlinear coupled systems of variational inequalities, Communications in Nonlinear Science and Numerical Simulation, accepted (December 7, 2022) https://arxiv.org/abs/2207.07082

• Yunru Bai, Nicusor Costea and Shengda Zeng: Systems of hemivariational inequalities with nonlinear coupling functions, submitted to Nonlinear Analysis: Real World Applications, https://arxiv.org/abs/2207.07076

Book 2022:

• Denisa Stancu Dumitru, Ecuatii pe fractali (Equations on fractals), Politehnica Press, 2022, 83 pagini, ISBN: 978-606-9608-11-1.

CONFERENCES 2022:

• School on Nonlinear PDEs, Institute od Mathematics, Polish Academy of Sciences, Warsaw, Poland, May 15-21 2022 (speaker: Denisa Stancu-Dumitru; Title of the talk: The asymptotic behavior of a class of phi-harmonic functions in Orlicz-Sobolev spaces);

• School on Nonlinear PDEs, Institute od Mathematics, Polish Academy of Sciences, Warsaw, Poland, May 15-21 2022 (speaker: Maria Farcaseanu; Title of the talk: Isolated singularities for nonlinear elliptic equations with Hardy potential);

• International Conference on Applied Mathematics and Numerical Methods, June 29-July 2, 2022, University of Craiova, Romania (speaker: Denisa Stancu-Dumitru; title of the talk:Torsional creep problems involving Grushin-type operators; online conference);

• International Conference on Applied Mathematics and Numerical Methods, June 29-July 2, 2022, University of Craiova, Romania (speaker: Maria Farcaseanu; title of the talk: Classification of singular solutions to nonlinear elliptic equations with a gradient term; online conference);

• International Conference on Applied Mathematics and Numerical Methods, June 29-July 2, 2022, University of Craiova, Romania (speaker: Andrei Grecu; title of the talk: 4-Harmonic functions and beyond; online conference);

• International Conference on Applied Mathematics and Numerical Methods, June 29-July 2, 2022, University of Craiova, Romania (speaker: Nicusor Costea; title of the talk: Systems of variational inequalities with nonlinear coupling functions; online conference);

• The fourth Romanian Itinerant Seminar on Mathematical Analysis and Applications (RISMAA'22), May 19-21, 2022, University of Brasov, Romania(speaker: Nicusor Costea; title of the talk: On the existence of solutions for coupled systems of variational inequalities);

• Seminar on Nonlinear Operators and Differential Equations, April 14, 2022, Department of Mathematics, Babes-Bolyai University, Cluj-Napoca (speaker: Cristian Cazacu; title of the talk: Sharp Caffarelli-Kohn-Nirenberg type inequalities, seminar online).

• Hausdorff School on Geometric Analysis and Nonlinear Partial Differential Equations, August 08-12, 2022, Hausdorff Center for Mathematics, Bonn, Germany (speaker: Maria Farcaseanu; title of the poster: Isolated singularities for nonlinear elliptic equations with Hardy potential)

• XVème Colloque Franco-Roumain de Mathématiques Appliquées, August 28 - September 03, 2022, Toulouse Institute of Mathematics, Toulouse, France (speaker: Maria Farcaseanu; title of the talk: Classification of singular solutions to nonlinear elliptic equations with a gradient term)

• XVème Colloque Franco-Roumain de Mathématiques Appliquées, August 28 - September 03, 2022, Toulouse Institute of Mathematics, Toulouse, France (speaker: Denisa Stancu-Dumitru; title of the talk: The Monotonicity of the p-Torsional Rigidity in Convex Domains)

Habilitation Thesis DEFENSE 2022:

• February 11, 2022, Denisa Stancu-Dumitru, The Analysis of Some Classes of Nonlinear PDEs, at ``Gheorghe Mihoc – Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy.

Prizes/Awards 2022:

• Award for scientific results of papers published in 2021 by CNCS for the paper " Cristian Cazacu, Joshua Flynn and Nguyen Lam, Short proofs of refined sharp Caffarelli-Kohn-Nirenberg inequalities, J. Differential Equations 302 (2021), 533-549,https://doi.org/10.1016/j.jde.2021.09.005", PN-III-P1-1.1- PRECISI-2021-65865, No. 48 (PRECISI 2021-zona rosie) (https://uefiscdi.gov.ro/index.php )

SCIENTIFIC REPORT 2022 (PDF)