Publications

Books, articles, edited volumes, special issues, and lecture notes.

Books

  1. The Infinity-Laplacian: from AMLEs to Machine Learning
    (with D.J. Araújo)
    Brazilian Mathematics Colloquium, vol. 34-09, IMPA, 2023.

  2. The Method of Intrinsic Scaling
    Lecture Notes in Mathematics, vol. 1930, Springer, 2008.

Articles

  1. Lipschitz regularity for the parabolic (s,p)-obstacle problem
    (with A. Sobral and D. Jesus)
    submitted.

  2. A coupled prediction-correction Hughes' model for congested crowd motion
    (with H. Ennaji, N. Igbida and G. Jradi)
    submitted.

  3. An L1-theory for p-Schrödinger equations with confinement in measure
    (with N. J. Alves)
    submitted.

  4. The two-phase Alt-Phillips problem for quasilinear operators
    (with Y. Alamri)
    submitted.

  5. Fractional p-caloric functions are Lipschitz
    (with A. Sobral and D. Jesus)
    submitted.

  6. Regularity estimates for doubly degenerate fully nonlinear evolution equations
    (with J. Bessa and J.V. da Silva)
    submitted.

  7. On the regularity of minimizers arising from singular minimal surface-like energies
    (with A. Sobral and E.V. Teixeira)
    submitted.

  8. Regularity estimates for PDEs with nonlocal degeneracies
    (with D.J. Araújo, A. Sobral and E.V. Teixeira)
    submitted.

  9. Improved moduli of continuity for degenerate phase transitions
    (with U. Gianazza and N. Liao)
    submitted.

  10. Oscillatory free boundary problems in heterogeneous media
    (with D.J. Araújo, G.S. Sá and E.V. Teixeira)
    submitted.

  11. Borderline regularity in singular free boundary problems
    (with D.J. Araújo, A. Sobral and E.V. Teixeira)
    Comm. Partial Differential Equations, to appear.

  12. On the fully nonlinear quenching problem
    (with S. Kim)
    Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), to appear.

  13. Improved regularity for a nonlocal dead-core problem
    (with D. dos Prazeres and R. Teymurazyan)
    Indiana Univ. Math. J., to appear.

  14. Geometric properties of free boundaries in degenerate quenching problems
    (with D.J. Araújo and R. Teymurazyan)
    SIAM J. Math. Anal. 58 (2026), 3599-3621.

  15. Higher Hölder regularity for degenerate elliptic PDEs with data in Morrey spaces
    (with G. Di Fazio and R. Teymurazyan)
    Ann. Mat. Pura Appl. 205 (2026), 929-940.

  16. Gradient regularity for a class of singular or degenerate elliptic equations
    (with C. Alcantara and M. Santos)
    J. Differential Equations 458 (2026), 114040, 32 pp.

  17. On free boundary problems shaped by varying singularities
    (with D.J. Araújo, A. Sobral and E.V. Teixeira)
    Calc. Var. Partial Differential Equations 65 (2026), Article No. 24, 53 pp.

  18. A granular model for crowd motion and pedestrian flow
    (with N. Igbida)
    J. Lond. Math. Soc. (2) 111 (2025), Paper No. e70184, 26 pp.

  19. Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation
    (with C. Alcantara and E.A. Pimentel)
    C. R. Math. Acad. Sci. Paris 363 (2025), 533-539.

  20. Degenerate free boundary problems with oscillatory singularities
    (with A. Sobral and E.V. Teixeira)
    Partial Differ. Equ. Appl. 6 (2025), Paper No. 14, 16 pp.

  21. Transmission problems: regularity theory, interfaces and beyond
    (with V. Bianca and E.A. Pimentel)
    in: Proceedings of the Brazil-Portugal joint meeting in Mathematics, Coimbra Mathematical Texts, vol. 3, Springer Cham, 2025.

  22. Cordes-Nirenberg type results for nonlocal equations with deforming kernels
    (with D. dos Prazeres and A. Sobral)
    J. Funct. Anal. 287 (2024), Paper No. 110593, 33 pp.

  23. Improved regularity for a Hessian-dependent functional
    (with V. Bianca and E.A. Pimentel)
    Proc. Amer. Math. Soc. 152 (2024), 4393-4403.

  24. Sharp regularity for a singular fully nonlinear parabolic free boundary problem
    (with D.J. Araújo and G.S. Sá)
    J. Differential Equations 389 (2024), 90-113.

  25. BMO-regularity for a degenerate transmission problem
    (with V. Bianca and E.A. Pimentel)
    Anal. Math. Phys. 14 (2024), Article No. 9, 26 pp.

  26. Fully nonlinear Hamilton-Jacobi equations of degenerate type
    (with D. Jesus and E. A. Pimentel)
    Nonlinear Anal. 227 (2023), Paper No. 113181, 15 pp.

  27. On a two-phase free boundary problem ruled by the infinity Laplacian
    (with D.J. Araújo and E. V. Teixeira)
    Israel J. Math. 245 (2021), 773-785.

  28. Existence and improved regularity for a nonlinear system with collapsing ellipticity
    (with E. A. Pimentel)
    Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), 1385-1400.

  29. Geometric tangential analysis and sharp regularity for degenerate pdes
    (with E. V. Teixeira)
    in: Harnack Inequalities and Nonlinear Operators - Proceedings of the INdAM conference to celebrate the 70th birthday of Emmanuele DiBenedetto
    Springer INdAM Series 46 (2021), 175-192.

  30. Sharp Hölder regularity for the inhomogeneous Trudinger's equation
    (with N. M. L. Diehl)
    Nonlinearity 33 (2020), 7054-7066.

  31. Fully nonlinear integro-differential equations with deforming kernels
    (with L.A. Caffarelli and R. Teymurazyan)
    Comm. Partial Differential Equations 45 (2020), 847-871.

  32. Sharp regularity for the inhomogeneous porous medium equation
    (with D.J. Araújo and A. F. Maia)
    J. Anal. Math. 140 (2020), 395-407.

  33. Singularly perturbed fully nonlinear parabolic problems and their asymptotic free boundaries
    (with G. C. Ricarte and R. Teymurazyan)
    Rev. Mat. Iberoam. 35 (2019), 1535-1558.

  34. Nash equilibria in N-person games with super-quadratic Hamiltonians
    (with C. Ebmeyer and J. Vogelgesang)
    J. Lond. Math. Soc. (2) 99 (2019), 609-628.

  35. Towards the Cp'-regularity conjecture in higher dimensions
    (with D.J. Araújo and E. V. Teixeira)
    Int. Math. Res. Not. IMRN 2018 (2018), no. 20, 6481-6495.

  36. Existence and boundary regularity for degenerate phase transitions
    (with P. Baroni, C. Lindfors and T. Kuusi)
    SIAM J. Math. Anal. 50 (2018), 456-490.

  37. On the interior regularity of weak solutions to the 2-D incompressible Euler equations
    (with J. Siljander)
    Calc. Var. Partial Differential Equations 56 (2017), 56:126.

  38. A proof of the Cp'-regularity conjecture in the plane
    (with D.J. Araújo and E. V. Teixeira)
    Adv. Math. 316 (2017), 541-553.

  39. A free boundary optimization problem for the ∞-Laplacian
    (with R. Teymurazyan)
    J. Differential Equations 263 (2017), 1140-1159.

  40. On the bulk velocity of Brownian ratchets
    (with S. Kondratyev and D. Vorotnikov)
    SIAM J. Math. Anal. 48 (2016), 950-980.

  41. Analysis of adaptive forward-backward diffusion flows with applications in image processing
    (with V.B. Surya Prasath and D. Vorotnikov)
    Inverse Problems 31 (2015), 105008 (30pp).

  42. Optimal regularity at the free boundary for the infinity obstacle problem
    (with J.D. Rossi and E. V. Teixeira)
    Interfaces Free Bound. 17 (2015), 381-398.

  43. Regularity for anisotropic fully nonlinear integro-differential equations
    (with L.A. Caffarelli and R. Leitão)
    Math. Ann. 360 (2014), 681-714.

  44. A quantitative modulus of continuity for the two-phase Stefan problem
    (with P. Baroni and T. Kuusi)
    Arch. Ration. Mech. Anal. 214 (2014), 545-573.

  45. A geometric tangential approach to sharp regularity for degenerate evolution equations
    (with E. V. Teixeira)
    Anal. PDE 7 (2014), 733-744.

  46. An intrinsic Liouville theorem for degenerate parabolic equations
    (with E. V. Teixeira)
    Arch. Math. (Basel) 102 (2014), 483-487.

  47. Local Hölder continuity for doubly nonlinear parabolic equations
    (with T. Kuusi and J. Siljander)
    Indiana Univ. Math. J. 61 (2012), 399-430.

  48. Hölder continuity for Trudinger's equation in measure spaces
    (with T. Kuusi, R. Laleoglu and J. Siljander)
    Calc. Var. Partial Differential Equations 45 (2012), 193-229.

  49. Limits of anisotropic and degenerate elliptic problems
    (with A. Di Castro and M. Pérez-Llanos)
    Commun. Pure Appl. Anal. 11 (2012), 1217-1229.

  50. On the well-posedness of a two-phase minimization problem
    (with D. Vorotnikov)
    J. Math. Anal. Appl. 378 (2011), 159-168.

  51. Limits as p(x) → ∞ of p(x)-harmonic functions
    (with J.J. Manfredi and J.D. Rossi)
    Nonlinear Anal. 72 (2010), 309-315.

  52. p(x)-Harmonic functions with unbounded exponent in a subdomain
    (with J.J. Manfredi and J.D. Rossi)
    Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 2581-2595.

  53. Entropy solutions for the p(x)-Laplace equation
    (with M. Sanchón)
    Trans. Amer. Math. Soc. 361 (2009), 6387-6405.

  54. On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding
    (with M. Bendahmane, R. Burger and R. Ruiz-Baier)
    Math. Methods Appl. Sci. 32 (2009), 1704-1737.

  55. The nonlinear N-membranes evolution problem
    (with J.F. Rodrigues and L. Santos)
    J. Math. Sci. (N. Y.) 159 (2009), 559-572.
    [published in Zap. Nauchn. Sem. POMI 362 (2008), 303-324.]

  56. The obstacle problem for nonlinear elliptic equations with variable growth and L1-data
    (with J.F. Rodrigues and M. Sanchón)
    Monatsh. Math. 154 (2008), 303-322.

  57. Quasi-steady Stokes flow of multiphase fluids with shear-dependent viscosity
    (with C. Ebmeyer)
    European J. Appl. Math. 18 (2007), 417-434.

  58. On a two-sidedly degenerate chemotaxis model with volume-filling effect
    (with M. Bendahmane and K.H. Karlsen)
    Math. Models Methods Appl. Sci. 17 (2007), 783-804.

  59. Intrinsic scaling for PDEs with an exponential nonlinearity
    (with E. Henriques)
    Indiana Univ. Math. J. 55 (2006), 1701-1722.

  60. New global a priori estimates for the third-grade fluid equations
    (with M. Steinhauer and J.H. Videman)
    Math. Methods Appl. Sci. 29 (2006), 1339-1348.

  61. Forest fire spread models: the local quasi-equilibrium approach
    (with J. André and D. Xavier Viegas)
    Combust. Sci. and Tech. 178 (2006), 2115-2143.

  62. Boundary regularity at {t=0} for a singular free boundary problem
    (with E. Henriques)
    in: Free Boundary Problems: theory and applications, pp. 231-240, Int. Ser. Num. Math. 154, Birkhäuser, 2006.

  63. On the doubly singular equation γ(u)t = Δpu
    (with E. Henriques)
    Comm. Partial Differential Equations 30 (2005), 919-955.

  64. The smoothing property for a class of doubly nonlinear parabolic equations
    (with C. Ebmeyer)
    Trans. Amer. Math. Soc. 357 (2005), 3239-3253.

  65. Current issues on singular and degenerate evolution equations
    (with E. DiBenedetto and V. Vespri)
    in: Handbook of Differential Equations, Evolutionary Equations, vol. 1, pp. 169-286, Elsevier, 2004.

  66. The geometry of some nonlinear PDEs
    in: The J.A. Sampaio Martins anniversary volume, pp. 121-133, Textos Mat. Sér. B 34, DMUC, 2004.

  67. Regularity in Sobolev spaces for doubly nonlinear parabolic equations
    (with C. Ebmeyer)
    J. Differential Equations 187 (2003), 375-390.

  68. Uniqueness for nonlinear degenerate problems
    (with N. Igbida)
    NoDEA Nonlinear Differential Equations Appl. 10 (2003), 287-307.

  69. Regularity for partial differential equations: from De Giorgi-Nash-Moser theory to intrinsic scaling
    in: CIM Bulletin #12, pp. 8-14, June 2002.

  70. Unidirectional steady flow of a viscoelastic fluid with a free surface
    (with J.H. Videman)
    Interfaces Free Bound. 3 (2001), 447-463.

  71. A free boundary problem: contributions from modern analysis
    in: European Congress of Mathematics, pp. 319-329, Progress in Mathematics 202, Birkhäuser, 2001.

  72. Hölder continuity of local weak solutions for parabolic equations exhibiting two degeneracies
    Adv. Differential Equations 6 (2001), 327-358.

  73. Continuous solutions for a degenerate free boundary problem
    Ann. Mat. Pura Appl. 178 (2000), 195-224.

  74. A singular-degenerate parabolic problem: regularity up to the Dirichlet boundary
    in: FBPs: Theory and Applications I, pp. 399-410, GAKUTO Int. Series Math. Sci. Appl. 13, 2000.

  75. On a three-dimensional convective Stefan problem for a non-newtonian fluid
    (with J.F. Rodrigues)
    in: Applied Nonlinear Analysis, pp. 457-468, Kluwer Academic/Plenum Publishers, 1999.

  76. Degenerate elliptic problems in a class of free domains
    (with J.F. Rodrigues)
    J. Math. Pures Appl. 78 (1999), 819-840.

  77. On the Stefan problem with convection and nonlinear diffusion in a porous medium
    in: Nonlinear Evolution Equations and Their Applications, pp. 225-236, World Scientific, 1999.

  78. On a Darcy-Stefan problem arising in freezing and thawing of saturated porous media
    (with J.F. Rodrigues)
    Contin. Mech. Thermodyn. 11 (1999), 181-191.

  79. On the mathematical analysis of a valley glacier model
    (with J.F. Rodrigues)
    in: FBPs: theory and applications, pp. 237-245, Res. Notes Math. 409, Chapman & Hall/CRC, 1999.

  80. On the stationary Boussinesq-Stefan problem with constitutive power-laws
    (with J.F. Rodrigues)
    Internat. J. Non-Linear Mech. 33 (1998), no. 4, 555-566.

  81. A free boundary problem with convection for the p-Laplacian
    Rend. Mat. Appl. 17 (1997), 1-19.

  82. A stationary Stefan problem with convection and nonlinear diffusion
    Port. Math. 52 (1995), 481-497.

  83. Some convergence properties of matrix sets
    (with D. P. Stanford)
    SIAM J. Matrix Anal. Appl. 15 (1994), 1132-1140.

Books
(as editor)

  1. Modern methods in the analysis of free boundary problems
    (E.A. Pimentel and J.M. Urbano eds.)
    Coimbra Mathematical Texts, vol. 6, Springer, 2026.

  2. Trends in Partial Differential Equations of Mathematical Physics
    (J.F. Rodrigues, G. Seregin and J.M. Urbano eds.)
    Progress in Nonlinear Differential Equations and Their Applications, vol. 61, Birkhäuser, 2005.

  3. School and Workshop on Oceanography, Lakes and Rivers
    (J.H. Videman, J.M. Urbano and D. Bresch eds.)
    Monographs and Proceedings, vol. 25, CIM, 2004.

Special Issues
(as editor)

  1. Advances in nonlinear elliptic and parabolic PDE
    (Guest editors: D. Gomes and J.M. Urbano)
    Partial Differential Equations and Applications, vol. 6, Springer, 2025.

  2. Harnack Estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations
    (Guest editors: E. Dibenedetto, U. Gianazza, M. Safonov, J.M. Urbano and V. Vespri)
    Bound. Value Probl., vol. 2007.

Lecture Notes

  1. A very short introduction to regularity theory for linear elliptic equations
    Short course, KAUST, 2026.

  2. Análise Infinitesimal II
    Notas de curso, Universidade de Coimbra, 2022.

  3. An introduction to intrinsic scaling
    Short course, Seoul National University, 2018.

  4. An introduction to the infinity-Laplacian
    Short course, Fortaleza/Aalto/KAUST, 2017.

  5. Análise Matemática II
    Notas de curso, Universidade de Coimbra, 2017.