Eigenvalues on curved spaces
Summaries 2021-2023
The main objective of the present project is to prove new geometric inequalities for eigenvalues describing nonlinear phenomena that connect physical and geometrical quantities.
Task I. Qualitative properties of eigenvalues: from comparisons to Brunn-Minkowski inequalities.
Problem 1. Provide sharp spectral gap estimates on (non-)compact Finsler manifolds.
Problem 2. Provide spectral gap estimates in Heisenberg groups.
Problem 3. Prove Brunn-Minkowski-type inequalities for first eigenvalues on curved spaces.
Task II. Lord Rayleigh's conjectures: from clamped plates to saturation phenomena.
Problem 4. Solve Lord Rayleigh's conjecture for clamped plates in positively curved spaces.
Problem 5. Characterize the saturation phenomena on positively/negatively curved spaces.
Summary 2023
Scientific papers
Balogh Z, Kristály A, Tripaldi F, Sharp log-Sobolev inequalities in CD(0,N) spaces with applications, J. FUNCT. ANAL., 286 (2024), no. 2, paper no. 110217 [pdf]
Kajántó S, Rellich inequalities via Riccati pairs on model space forms. J. MATH. ANAL. APPL., 531 (2024), no. 2, paper no. 127870 [pdf]
Farkas C, Mezei I I, Nagy, Zs T, Multiple solution for a fourth-order nonlinear eigenvalue problem with singular and sublinear potential, STUD. UNIV. BABEȘ-BOLYAI MATH., 68 (2023), no.1, 139–149.
Farkas C, Critical elliptic equations on non-compact Finsler manifolds, accepted, PUBL. MATH. DEBRECEN, 2023 [pdf]
Kajántó S, Kristály A, Saturation phenomena of a nonlocal eigenvalue problem: the Riemannian case, accepted, OPTIMIZATION, 2023 [pdf]
Disseminations, talks, visits
Kristály A:
Hungarian Day of Science in Transylvania, MTNE, 2023, Cluj-Napoca, 16-17 November 2023, Title of the talk: Lord Rayleigh's conjecture positively curved spaces
International Conference on Variational Analysis and Optimization with Applications, Aligarh Muslim University, Aligarh, India, 23-25 September 2023. Presented: Optimal mass transport approach to sharp geometric inequalities on curved structures
9th International Conference on Mathematics and Informatics, Sapientia Hungarian University of Transylvania, Târgu-Mureş/Marosvásárhely, Romania, 7-8 September, 2023. Presented: Clamped plates on curved spaces
20th EUROpt Workshop: continuous optimization working group is coming home, Corvinus University of Budapest, Hungary, 23-25 August 2023. Presented: Sharp isoperimetric and Sobolev inequalities on CD(0,N) spaces: an optimal mass transport approach
Hungarian Optimization Conference, Corvinus University of Budapest, Hungary, 21-23 June 2023. Presented: Optimal mass transport with applications to sharp isoperimetric and functional inequalities
Colloquium on Finsler Geometry and its Applications, University of Debrecen, Hungary, 12-16 June 2023. Presented: Sharp geometric and functional inequalities on Finsler manifolds
Partial Differential Equations Seminar, University of Oxford, Mathematical Institute, UK, 15 May 2023. Presented: Lord Rayleigh’s conjecture for clamped plates in curved spaces
Visit/Scientific collaboration: University of Oxford, 15-20 May, 2023 (Collaboration with Z. Balogh, A. Mondino).
Visit/Scientific collaboration: University of Bern, 17-31 October, 2023 (Collaboration with Z. Balogh).
Farkas C:
Hungarian Day of Science in Transylvania, MTNE, 2023, Cluj-Napoca, 16-17 November 2023, Title of the talk: Spectral gaps via convexity-based techniques on Riemannian manifolds
International Conference on Variational Analysis and Optimization with Applications, Aligarh Muslim University, Aligarh, India, 23-25 September 2023. Presented: Sharp spectral gaps on Riemannian manifolds
9th International Conference on Mathematics and Informatics, Sapientia Hungarian University of Transylvania, Târgu-Mureş/Marosvásárhely, Romania, 7-8 September, 2023. Presented: Sharp spectral gaps and Hardy-Rellich inequalities via convexity-based techniques
20th EUROpt Workshop: continuous optimization working group is coming home, Corvinus University of Budapest, Hungary, 23-25 August 2023. Presented: Lower semicontinuity of Kirchhoff-type energy functionals and spectral gaps on non-euclidean structures
Colloquium on Finsler Geometry and its Applications, University of Debrecen, Hungary, 12-16 June 2023. Presented: Critical elliptic equations on non-compact Finsler manifolds.
17th International Symposium on Applied Computational Intelligence and Informatics (SACI 2023), May 23-26, 2023 Timisoara, Romania. Presented: Kirchhoff-type energy functionals and spectral gaps on Heisenberg groups
Mezei I:
9th International Conference on Mathematics and Informatics, Sapientia Hungarian University of Transylvania, Târgu-Mureş/Marosvásárhely, Romania, 7-8 September, 2023. Presented: Nonsmooth elliptic problems on Riemannian manifolds
Kajántó S:
Hungarian Day of Science in Transylvania, MTNE, 2023, Cluj-Napoca, 16-17 November 2023, Title of the talk: Riccati pairs and functional equations
20th EUROpt Workshop: continuous optimization working group is coming home, Corvinus University of Budapest, Hungary, 23-25 August 2023. Presented: Saturation of a nonlocal eigenvalue problem on Riemannian manifolds
Colloquium on Finsler Geometry and its Applications, University of Debrecen, Hungary, 12-16 June 2023. Presented: The flag and S-curvatures on the interpolated Poincaré metric: unexpected behaviours
17th International Symposium on Applied Computational Intelligence and Informatics (SACI 2023), May 23-26, 2023 Timisoara, Romania. Presented: Saturation phenomena of a Riemannian non-local eigenvalue problem
Summary 2022
Scientific papers
Balogh Z, Kristály A, Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature, MATH ANNALEN, 385 (2023), no. 3-4, 1747–1773. [pdf]
Kristály A, Lord Rayleigh’s conjecture for positively curved vibrating clamped plates, GEOM FUNCT ANAL (GAFA), 32 (2022), 881–937. [pdf]
Farkas C, Kajántó S, Varga C, Lower semicontinuity of Kirchhoff-type energy functionals and spectral gaps on non-euclidean structures, TOPOL. METHODS NONLINEAR ANAL., 61(2023), no. 2 , 743 - 760. [pdf]
Kristály A, Mester Á, Mezei I. I, Sharp Morrey–Sobolev inequalities and eigenvalue problems on Riemannian–Finsler manifolds with nonnegative Ricci curvature, COMMUN CONTEMP MATH, 25 (2023), no. 10, Paper No. 2250063 [pdf]
Kristály A, Mezei I. I, Szilák K, Elliptic differential inclusions on non-compact Riemannian manifolds, NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, 69 (2023), 103740 [pdf]
Disseminations, talks, visits
Kristály A:
Special Session at the 2022 Spring Sectional Meeting of the AMS, Purdue University: “Differential Operators: Spectral Estimation and Optimization”, March 26-27, 2022. Presented: Clamped plates in curved spaces
AMS special session “Geometric and Functional Inequalities and Applications to PDEs”, May 14-15 2022. Presented: Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature
6th CIAS International Workshop, Corvinus Institute for Advanced Studies, Corvinus University of Budapest, Budapest, Hungary. Presented: Sharp geometric and functional inequalities on curved structures.
Corvinus Centre for Operations Research (CCOR) Seminar, Corvinus Institute for Advanced Studies, Corvinus University of Budapest, Budapest, Hungary, September 8, 2022. Presented: Sharp geometric inequalities on non-euclidean settings: an optimal mass transport approach
14th Joint Conference on Mathematics and Computer Science, MACS, Cluj-Napoca, Romania, 24-27 November 2022. Presented: On Csaba Varga’s mathematical contributions
Hungarian Day of Science in Transylvania, MTNE, 2022, Cluj-Napoca, 24-27 November 2022. Presented: Sharp geometric inequalities on non-euclidean settings
Visit/Scientific collaboration: University of Bern, 12-30 June, 2022 (Collaboration with Z. Balogh, S. Don, F. Tripaldi).
Visit/Scientific collaboration: Fields Institute of Mathematics, University of Toronto, 14-20 November, 2022 (Collaboration with Z. Balogh, R. McCann, S. Ohta, L. Mazzieri)
Farkas C:
Corvinus Centre for Operations Research (CCOR) Seminar, Corvinus Institute for Advanced Studies, Corvinus University of Budapest, Budapest, Hungary, September 15, 2022. Presented: Compact Sobolev embeddings on non-compact manifolds with applications
Hungarian Day of Science in Transylvania, MTNE, 2022, Cluj-Napoca, 24-27 November 2022. Presented: Partial differential equations with double-phase operator
Mezei I:
14th Joint Conference on Mathematics and Computer Science, MACS, Cluj-Napoca, Romania, 24-27 November 2022. Presented: Differential inclusions on non-compact Riemannian manifolds
Kajántó S:
Hungarian Day of Science in Transylvania, MTNE, 2022, Cluj-Napoca, 24-27 November 2022. Presented: Saturation problems on space-forms
Summary 2021
Scientific papers
Kristály A, Zhao W, On the geometry of irreversible metric-measure spaces: Convergence, Stability and Analytic aspects, J MATH PURES APPL (Liouville Journal), Volume 158, pp. 216-292, 2022. [pdf]
Disseminations, talks, visits
Kristály A:
Analysis and PDE Seminar, Durham University, 10 December 2021, Presented: Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature
Visit/Scientific collaboration: University of Bern, 15-30 October, 2021 (Collaboration with Z. Balogh, S. Don, F. Tripaldi).
Farkas C:
Analysis and PDE Seminar, Durham University, 5 November 2021, Presented: Compact Sobolev embeddings on non-compact manifolds with applications
Submitted papers
Balogh Z, Don S, Kristály A, Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications [pdf]
Balogh Z, Don S, Kristály A, Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications [pdf]
Kajántó S, Kristály A, Peter I R, Zhao W, A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities [pdf]
Farkas C, Kajántó S, Kristály A, Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds [pdf]
Faraci F, Farkas C, Multiplicity of global minima for critical Choquard-Kirchhoff type energy functionals [pdf]