Project NoDES

Brief Description

This project (which started in March 1, 2021) addresses key contemporary questions in Partial Differential Equations (PDEs), focusing on several nonlinear problems which appear at the intersection of the dispersive and elliptic worlds and which require a multidisciplinary approach. It is structured around 4 main topics:

- Singularly Perturbed Nonlinear Systems;

- Solitons in Nonlinear Schrödinger (NLS) Equations; 

- Nonlinear Smoothing for Dispersive PDEs;

- Self-Similar Solutions for Dispersive PDEs.

The goal of this project is to build consolidated theories to tackle these problems, going from existence and regularity of stationary solutions exhibiting special symmetry, to the existence of a well-defined dynamical flow around them and concluding with spectral and stability analysis. We work both in Euclidean spaces as well as on metric graphs, being a multidisciplinary team of experts drawn from these different fields.

The Research Team

Publications

• Francisco Agostinho, Simão Correia, Hugo Tavares, Classification and stability of positive solutions to the NLS equation on the T-metric graph, Nonlinearity 37 (2024) 025005 (47pp).

• V. Barros, S. Correia, F. Oliveira, On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity. To appear in Differential and Integral Equations (2022)

• Gregory Berkolaiko, James B. Kennedy, Pavel Kurasov, Delio Mugnolo, Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap, Proc. Amer. Math. Soc. 151 (2023), 3439-3455

• D. Buoso and J. B. Kennedy, The Bilaplacian with Robin boundary conditions, SIAM J. Math. Anal. 54 (2022), 36-78.

• Marco Cirant, Gianmaria Verzini, Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms, Advances in Mathematics 409, Part B, 19 November 2022, 108700

• M. Clapp, A. Pistoia and H. Tavares, Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation, arXiv:2106.00579 (2021), 49 pp, to appear in the Journal of the European Math. Soc.

• F. Colassuono, B. Noris and G. Verzini, Multiplicity of solutions on a Nehari set in an invariant cone. arXiv:2109.14443 (2021), 20 pp. To appear in Minimax Theorems.

• Simão Correia, Mário Figueira, A note on bifurcations from eigenvalues of the Dirichlet-Laplacian with arbitrary multiplicity, NoDEA - Nonlinear Differ. Equ. Appl. 30 (2023), Paper no. 37.

• Manuel Dias and Hugo Tavares, Optimal uniform bounds for competing variational elliptic systems with variable coefficients, Nonlinear Analysis 235 (2023), Paper No. 113348, 59 pp.

• Rodrigo Duarte, Jorge Drumond Silva, Weighted Gagliardo-Nirenberg Interpolation Inequalities, Journal of Functional Analysis 285 (2023), 110009

• Lorenzo Ferreri, Gianmaria Verzini. Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions, Nonlinear Analysis 224, November 2022, 113103

• M. Hofmann and J. B. Kennedy, Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs, Lett. Math. Phys. 111 (2021), article no. 96.

• M. Hofmann, J. B. Kennedy, D. Mugnolo and M. Plümer, On Pleijel's nodal domain theorem for quantum graphs, Ann. Henri Poincaré 22 (2021), 3841-3870.

• Riccardo Molle, Giuseppe Riey, Gianmaria Verzini, Existence of normalized solutions to mass supercritical Schrodinger equations with negative potential, Journal of Differential Equations 333 (2022), 302-331

• Dario Mazzoleni, Benedetta Pellacci and Gianmaria Verzini, Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problems, SIAM Journal on Mathematical Analysis 55 (2023), 4162-4192

• E. Moreira dos Santos, G. Nornberg, D. Schiera and H. Tavares, Principal spectral curves of Lane-Emden fully nonlinear type systems and applications, Calculus of Variations and PDE 62 (2023), no. 2, Paper No. 49, 38 pp.

• Benedetta Pellacci, Giovanni Pisante, Delia Schiera, Spectral optimization for weighted anisotropic problems with Robin conditions, Journal of Differential Equations 378, 5 January 2024, Pages 303-338

• Angela Pistoia, Alberto Saldaña and Hugo Tavares, Existence of solutions to a slightly supercritical pure Neumann problem,accepted in SIAM Journal on Mathematical Analysis (SIMA) 55 (2023), no. 4, 3844-3887.

• Angela Pistoia, Delia Schiera and Hugo Tavares, Existence of solutions on the critial hyperbola for a pure Lane-Emden system with Neumann boundary conditions, IMRN Vo2024, Issue 1, January 2024, Pages 745–803.

• A. Saldaña, H. Tavares, On the least-energy solutions of the pure Neumann Lane-Emden equation, NoDEA - Nonlinear Differential Equations and Applications 29, Article number: 30 (2022).

• D. Schiera, A family of nonlocal degenerate operators: maximum principles and related properties, NoDEA - Nonlinear Differential Equations and Applications 31, article number 1, (2024)

• N. Soave, H. Tavares and A. Zilio, Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius. Mathematische Annalen 386 (2023), no. 1-2, 551-585.

• Hugo Tavares, Topics in elliptic problems: from semilinear equations to shape optimization. Accepted in Communications in Mathematics (2024)

• H. Tavares, S. You and W. Zou, Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: the higher dimensional case. Journal of Functional Analysis 283 (2022), issue 2,109497 .

Preprints

• S. Correia, R. Côte, Sharp blow-up stability for self-similar solutions of the modified Korteweg-de Vries equation (2024), arXiv:2402.16423 

• Nicola Abatangelo, Alberto Saldaña, Hugo Tavares, An asymptotic relationship between Lane-Emden systems and the 1-bilaplacian equation, arXiv:2312.16696 

• Mohammed Ahrami, Zakaria El Allali, Evans M Harrell II, James B. Kennedy, Optimizing the Fundamental Eigenvalue Gap of Quantum Graphs, arXiv:2401.04344 

• P. Andrade, E. Moreira dos Santos, M. Santos and H. Tavares, Spectral partition problems with volume and inclusion constraints. arXiv:2305.02870

• P. Andrade, J. Vitor da Silva, G. Rampasso and M. Santos, Sharp regularity estimates for a singular inhomogeneous for (m,p)-Laplacian equation. arXiv:2305.02841

• P. Andrade, D. dos Prazeres and M. Santos. Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy. arXiv:2302.00423

• Luís N. Baptista, James B. Kennedy, Delio Mugnolo, Mean distance on metric graphs, arXiv:2312.04952. 

• Luccas Campos, Simão Correia and Luiz Gustavo Farah, Some well-posedness and ill-posedness results for the INLS equation, arXiv:2210.07060 (2022), 21 pp.

• M. Cirant, A. Cosenza, G. Verzini, Ergodic Mean Field Games: existence of local minimizers up to the Sobolev critical case, arXiv:2301.11692

• Simão Correia, Improved global well-posedness for the quartic Korteweg-de Vries equation, arXiv:2310.13497 

• Simão Correia, Raphaël Cöte, Perturbation at blow-up time of self-similar solutions for the modified Korteweh-de Vries equation. arXiv:2201.03240 (2022), 28 pp.

• Simão Correia, Filipe Oliveira, Jorge Drumond Silva, Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estimates (2023), arXiv: 2302.03575

• J. V. da Silva and M. Santos, Schauder and Calderón-Zygmund type estimates for fully nonlinear parabolic equations under "small ellipticity aperture" and applications, arXiv:2311.02524.

• Marco Düfel, James B. Kennedy, Delio Mugnolo, Marvin Plümer, Matthias Täufer, Boundary Conditions Matter: On The Spectrum Of Infinite Quantum Graphs (2022), arXiv:2207.04024  

• L. Ferreri and G. Verzini, Asymptotic properties of an optimal principal Dirichlet eigenvalue arising in population dynamics (2023) arXiv:2304.08421

• Pedro Freitas, James B. Kennedy, On domain monotonicity of Neumann eigenvalues of convex domains, arXiv:2307.06593  

• Matthias Hofmann, James B. Kennedy, Andrea Serio. Spectral minimal partitions of unbounded metric graphs, arXiv:2209.03658 

• James B. Kennedy, João P. Ribeiro, Cheeger cuts and Robin spectral minimal partitions of metric graphs, arXiv:2310.02701

• Benedetta Pellacci, Angela Pistoia, Giusi Vaira, Gianmaria Verzini, Partially concentrating standing waves for weakly coupled Schrödinger systems, arXiv:2311.11648 

• Ariel Salort, Susanna Terracini, Gianmaria Verzini, Alessandro Zilio, Rotating Spirals in segregated reaction-diffusion systems (2022), arXiv:2202.10369  

Other

• J. Drumond Silva, "A Extraordinária Influência de Fourier no Desenvolvimento da Análise Matemática", Revista Ponto Fixo, n.3, 2021.

• H. Tavares, Semilinear elliptic problems: old and new, CIM Bulletin 43 (2021), 11–18.