Supported by Fundação para a Ciência e Tecnologia, project PTDC/MAT-PUR/1788/2020 with DOI identifier 10.54499/PTDC/MAT-PUR/1788/2020
Brief Description
This project was developed between March 2021-February 2025. It addressed 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 was 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.
This project built 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 worked both in Euclidean spaces as well as on metric graphs, being a multidisciplinary team of experts drawn from these different fields.
The Research Team
Some Indicators:
46 publications
1 book
13 preprints, submitted and waiting for a decision
65 talks at international conferences
34 talks at national events
14 organization of conferences, workshops and other events
3 PhD thesis
10 Master thesis
Final list of published papers in international journals
1. N. Abatangelo, A. Saldaña and H. Tavares, An asymptotic relationship between Lane-Emden systems and the 1-bilaplacian equation. Annali della Scuola Normale Superiore di Pisa, Classe di Science. DOI: 10.2422/2036-2145.202401_009 http://dx.doi.org/10.2422/2036-2145.202401_009
2. F. Agostinho, S. Correia and H. Tavares, Classification and stability of positive solutions to the NLS equation on the T-metric graph, Nonlinearity 37 (2024), no.2. https://iopscience.iop.org/article/10.1088/1361-6544/ad1535
3. M. Ahrami, Z. El Alalli, E. M. Harrell II and J. B. Kennedy, Optimizing the fundamental eigenvalue gap of quantum graphs, J. Phys. A: Math. Theor. 57 (2024) 385205 (26pp)
https://iopscience.iop.org/article/10.1088/1751-8121/ad6410
4. P. Andrade, D. dos Prazeres and M. Santos, Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy, Nonlinearity 37 (2024), no. 4, Paper No. 045009, 29 pp.
https://iopscience.iop.org/article/10.1088/1361-6544/ad2c22
5. P. Andrade, J.V. Da Silva, G. Rampasso, M. Santos, Sharp Regularity Estimates for a Singular Inhomogeneous (m, p)-Laplacian Equation. Potential Anal (2024). https://doi.org/10.1007/s11118-024-10164-2
6. P. Andrade, E. Moreira dos Santos, M. Santos and H. Tavares, Spectral partition problems with volume and inclusion constraints, SIAM Journal on Mathematical Analysis 56 (2024), no. 6, 7136–7169.
https://epubs.siam.org/doi/abs/10.1137/23M161553X?journalCode=sjmaah
7. L. Baptista, J. B. Kennedy and D. Mugnolo, Mean distance on metric graphs, J Geom Anal 34, 137 (2024). https://doi.org/10.1007/s12220-024-01574-0
8. V. Barros, S. Correia, F. Oliveira, On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity. Differential Integral Equations 35 (2022): 371-392
https://doi.org/10.57262/die035-0708-371
9. G. Berkolaiko, J. B. Kennedy, P. Kurasov and D. Mugnolo, Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap, Proc. Amer. Math. Soc., Volume 151, Number 8, August 2023, Pages 3439–3455
https://doi.org/10.1090/proc/16322
10. D. Buoso and J. B. Kennedy, The Bilaplacian with Robin boundary conditions, SIAM J. Math. Anal. 54 (2022), 36-78.
https://doi.org/10.1137/20M1363984
11. L. Campos, S. Correia, L. G. Farah, Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation, Nonlinear Analysis: Real World Applications (2025) 85,104336
https://www.sciencedirect.com/science/article/pii/S1468121825000227
12. M. Cirant, M. Cosenza and G. Verzini, Ergodic mean field games: existence of local minimizers up to the Sobolev critical case. Calc. Var. Partial Differential Equations 63 (2024), no. 5, Paper No. 134, 23 pp.
https://link.springer.com/article/10.1007/s00526-024-02744-2
13. M. Cirant and G. Verzini, Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms. Adv. Math. 409 (2022), Paper No. 108700, 16 pp. https://www.sciencedirect.com/science/article/abs/pii/S0001870822005175
14. M. Clapp, A. Pistoia and H. Tavares, Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation, accepted for publication in the Journal of the European Mathematical Society. J. Eur. Math. Soc. (2024), published online first
https://ems.press/journals/jems/articles/14297804
15. F. Colasuonno, B. Noris, G. Verzini, Multiplicity of solutions on a Nehari set in an invariant cone. Minimax Theory and its Applications 7 (2), (2022).
https://journalmta.com/index.php/jmta/article/view/164
16. S. Correia, Improved global well-posedness for the quartic Korteweg-de Vries equation, Proc. Amer. Math. Soc. 152 (2024), 5117-5136.
https://doi.org/10.1090/proc/16911
17. S. Correia, R. Côte, Perturbation at blow-up time of self-similar solutions for the modified Korteweg-de Vries equation, Arch Rational Mech Anal 248, 25 (2024). https://doi.org/10.1007/s00205-024-01969-x
18. S. Correia, M. Figueira, A note on bifurcations from eigenvalues of the Dirichlet- Laplacian with arbitrary multiplicity, Nonlinear Differ. Equ. Appl. (2023) 30, 37. https://link.springer.com/article/10.1007/s00030-023-00846-y
19. S. Correia, F. Oliveira, J. Drumond Silva, Mass-transfer instability of ground-states for Hamiltonian Schrödinger systems, J. Anal. Math. (2022), 148, 681–710. https://doi.org/10.1090/proc/16322
20. S. Correia, F. Oliveira and J. Drumond Silva. Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estimates. SIAM J. Math. Anal. 56 (2024), 5604-5633.
https://epubs.siam.org/doi/10.1137/23M156923X
21. J.V. Da Silva and M. Santos, Schauder and Calderòn-Zygmund type estimates for fully nonlinear parabolic equations under “small ellipticity aperture” and applications, Nonlinear Anal. 246 (2024), Paper No. 113578, 17 pp.
https://www.sciencedirect.com/science/article/pii/S0362546X2400097X
22. M. Dias and H. Tavares, Optimal uniform bounds for competing variational elliptic systems with variable coefficients, Nonlinear Analysis 235 (2023), Paper No. 113348, 59 pp.
https://www.sciencedirect.com/science/article/abs/pii/S0362546X23001402
23. R. Duarte, J. Drumond Silva, Weighted Gagliardo-Nirenberg Interpolation Inequalities, J. Functional Analysis 285 (5), 110009 https://doi.org/10.1016/j.jfa.2023.110009
24. L. Ferreri and G. Verzini, Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions. Nonlinear Anal. 224 (2022), Paper No. 113103, 25 pp. https://www.sciencedirect.com/science/article/abs/pii/S0362546X22001912
25. L. Ferreri and G. Verzini, Asymptotic properties of an optimal principal Dirichlet eigenvalue arising in population dynamics. J. Funct. Anal. 287 (2024), no. 7, Paper No. 110543, 51 pp.
https://www.sciencedirect.com/science/article/pii/S0022123624002313
26. M. Hofmann and J. B. Kennedy, Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs, Lett. Math. Phys. 111 (2021), 96. https://doi.org/10.1007/s11005-021-01438-6
27. 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. https://doi.org/10.1007/s00023-021-01077-6
28. M. Hofmann, J.B. Kennedy and A. Serio, Spectral minimal partitions of unbounded metric graphs, J. Spectr. Theory 13 (2023), 593-622
https://ems.press/content/serial-article-files/33937
29. J. B. Kennedy, Geometric spectral theory of quantum graphs, Communications in Mathematics 32 (2024), no. 3, 393–439.
https://doi.org/10.46298/cm.12380
30. J. B. Kennedy and J. P. Ribeiro, Cheeger cuts and Robin spectral minimal partitions of metric graphs. J. Anal. Math., in press.
http://www.math.huji.ac.il/~w-jam/articles.html
31. L. Machado, J. Natário and J. Drumond Silva. Free-falling motion of an elastic rigid rod towards a Schwarzschild black hole. Classical Quantum Gravity 41 (2024), 215002.
https://iopscience.iop.org/article/10.1088/1361-6382/ad7a4a
32. D. Mazzoleni, B. Pellacci and G. Verzini, Singular Analysis of the Optimizers of the Principal Eigenvalue in Indefinite Weighted Neumann Problems. SIAM J. Math.Anal. 55 (2023), 4162–4192.
https://epubs.siam.org/doi/10.1137/22M1490600
33. R. Molle, G. Riey and G. Verzini, Normalized solutions to mass supercritical Schrodinger equations with negative potential. J. Differential Equations 333 (2022), 302-331. https://www.sciencedirect.com/science/article/abs/pii/S0022039622003801
34. E. Moreira dos Santos, G.S. Nornberg, D. Schiera and H. Tavares, Principal spectral curves for Lane-Emden fully nonlinear type systems and applications, Calc. Var. Partial Differ. Equ. 62 (2023), no 49. https://link.springer.com/article/10.1007/s00526-022-02386-2
35. B. Pellacci, G. Pisante and D. Schiera, Spectral optimization for weighted anisotropic problems with Robin conditions, J. Differential Equations 378 (2024), 303–338. https://www.sciencedirect.com/science/article/abs/pii/S0022039623006289
36. B. Pellacci, A. Pistoia, G. Vaira and G. Verzini, Partially concentrating standing waves for weakly coupled Schrödinger systems. Math. Ann. 390 (2024), no. 3, 3691-3722.
https://link.springer.com/article/10.1007/s00208-024-02842-8
37. D. Pierotti, G. Verzini and J. Yu, Normalized solutions for Sobolev critical Schrödinger equations on bounded domains. SIAM J. Math. Anal. 57 (2025), no. 1, 262-285.
https://epubs.siam.org/doi/abs/10.1137/24M1656281?journalCode=sjmaah
38. A. Pistoia, A. Saldaña and H. Tavares, Existence of solutions to a slightly supercritical pure Neumann problem, SIAM Journal on Mathematical Analysis 55 (2023), no. 4, 3844-3887.
https://epubs.siam.org/doi/10.1137/22M1520360
39. A. Pistoia and D. Schiera, On a critical Hamiltonian system with Neumann boundary conditions, to appear in Topological Methods and Nonlinear Analysis.
40. A. Pistoia, D. Schiera and H. Tavares, Existence of solutions on the critical hyperbola for a pure Lane- Emden system with Neumann boundary conditions, Int. Math. Res. Not. - IMRN, Volume 2024, Issue 1, 745–803.
https://academic.oup.com/imrn/article-abstract/2024/1/745/7221390?redirectedFrom=fulltext
41. H. Tavares and Alberto Saldaña, On the least-energy solutions of the pure Neumann Lane-Emden equation, Nonlinear Differ. Equ. Appl. 29, 30 (2022). https://doi.org/10.1007/s00030-022-00762-7
42. A. Salort, S. Terracini, G. Verzini and A. Zilio, Rotating Spirals in segregated reaction-diffusion systems, Analysis & PDE 18 No. 3 (2025), 549-590
https://msp.org/apde/2025/18-3/p01.xhtml
43. D. Schiera, A family of nonlocal degenerate operators: maximum principles and related properties, Nonlinear Differ. Equ. Appl. 31.1 (2024). https://link.springer.com/article/10.1007/s00030-023-00892-6
44. N. Soave, H. Tavares and A. Zilio, Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius. Math. Ann. 386, 551–585 (2023).https://doi.org/10.1007/s00208-022-02406-8
45. H. Tavares, Topics in elliptic problems: from semilinear equations to shape optimization, Communications in Mathematics, March 14, 2024, Volume 32 (2024), Issue 3 (Special issue: Portuguese Mathematics).
https://cm.episciences.org/13231
46. 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. J. Functional Analysis 283 (2022) 109497.
https://www.sciencedirect.com/science/article/pii/S0022123622001173
Books:
1. S. Salsa and G. Verzini, Partial Differential Equations in Action. From Modelling to Theory. 2022, Springer (4th ed.) https://link.springer.com/book/10.1007/978-3-031-21853-8
Preprints submitted to international journals and waiting for a decision.
Submitted during the period: 1-03-2024 to 29-02-2025
1. F. Agostinho, S. Correia and H. Tavares, A comprehensive study of bound-states for the nonlinear Schrödinger equation on single-knot metric graphs. arXiv:2502.14097
2. C. Alcantara and M. Santos. Global fractional Sobolev regularity for fully nonlinear elliptic equations, preprint. arXiv:2411.15311
3. S. Correia and P. Leite, Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities. arXiv:2412.11808
4. S. Correia, F. Linares and J. Drumond Silva, Sharp Local Well-Posedness for the Schrödinger-Korteweg-DeVries System. arXiv:2408.10028
5. L. Ferreri, D. Mazzoleni, B. Pellacci and G. Verzini, Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem. arXiv:2407.17931
6. James B. Kennedy, Delio Mugnolo and Matthias Täufer, Towards a theory of eigenvalue asymptotics on infinite metric graphs: the case of diagonal combs. arXiv:2403.10708
7. D. Mazzoleni, M. Santos and H. Tavares, Free boundary regularity for a spectral optimal partition problem with volume and inclusion constraints. arXiv:2409.14916
8. D. Prazeres and M. Santos, Optimal Regularity for Fully Nonlinear Nonlocal Equations with Unbounded Source Terms, preprint. arXiv:2409.03216
9. A. Saldaña, D. Schiera and H. Tavares, On least energy solutions to a pure Neumann Lane-Emden system: convergence, symmetry breaking, and multiplicity. arXiv:2412.09512
Submitted during the period: 1-03-2023 to 29-02-2024
10. S. Correia and R. Côte, Sharp blow-up stability for self-similar solutions of the modified Korteweg-de Vries equation. arXiv:2402.16423
11. . P. Freitas and J.B. Kennedy, On domain monotonicity of Neumann eigenvalues of convex domains, arXiv:2307.06593
Submitted during the period: 1-03-2022 to 28-02-2023
12. M. Cirant, A. Cosenza and G. Verzini, Ergodic Mean Field Games: existence of local minimizers up to the Sobolev critical case, arXiv:2301.1169
13. M. Düfel, J. Kennedy, D. Mugnolo, M. Plümer and M. Täufer, Boundary Conditions Matter: On the Spectrum of Infinite Quantum Graphs. arXiv:2207.04024