Miguel Urbano

Professor Urbano’s research interests are in regularity theory for nonlinear partial differential equations, in particular of singular or degenerate type, arising from different applications, like phase transitions, flows in porous media or semi-supervised learning. He is also interested in free boundary problems, focusing on understanding the local behaviour of weak solutions and the geometric properties of interfaces.

Biography

Professor Miguel Urbano, who joined KAUST in 2022, received his Ph.D. in Mathematical Analysis in 1999 from the University of Lisbon, Portugal. Following a postdoctoral position at Northwestern University in the United States, he became an assistant professor at the University of Coimbra (UC), Portugal. He was promoted to associate professor with tenure in 2004 at UC and awarded a habilitation in mathematics in 2005 before becoming a full professor in 2009.

Professor Urbano is the author of The Method of Intrinsic Scaling, published in the Lecture Notes in Mathematics series, and over 70 scientific papers on nonlinear partial differential equations (PDEs). He has served on panels evaluating grants and research projects for the European Union, the European Research Council, the Academy of Finland, the Latvian Council of Science, the Serrapilheira Institute of Brazil and the Portuguese Science Foundation.

Urbano served on Portugal's National Council for Science and Technology from 2012 to 2015, won the José Anastácio da Cunha Prize from the Portuguese Mathematical Society in 2002, and was an associate editor for Nonlinear Analysis from 2013 to 2021. He is a corresponding academician of the Lisbon Academy of Sciences (elected in January 2021) and has been the editor-in-chief of Portugaliae Mathematica since January 2022.

Research Interests

Professor Miguel Urbano is an expert on free boundary problems and regularity theory for nonlinear PDEs, particularly on the method of intrinsic scaling for singular or degenerate-type equations.

He has made several contributions leading to a better understanding of the local behaviour of weak solutions, e.g., the derivation of a quantitative modulus of continuity for weak solutions of the two-phase Stefan problem, which models a phase transition at a constant temperature or the proof of a long-standing conjecture on the optimal regularity for solutions of the p-Poisson equation in the plane.

Awards and Distinctions

  • Corresponding Academician, Sciences Class, Mathematics Section, Lisbon Academy of Sciences, 2021
  • José Anastácio da Cunha Prize, Portuguese Mathematical Society, 2002

Education

Habilitation
Mathematics, University of Coimbra, Portugal, 2005
Doctor of Philosophy (Ph.D.)
Mathematical Analysis, University of Lisbon, Portugal, 1999
Bachelor of Science (B.S.)
Pure Mathematics, University of Coimbra, Portugal, 1992

Recent Works

Recent preprints and ongoing work

  • C. Alcantara, E.A. Pimentel and J.M. Urbano, Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation, submitted.
  • D.J. Araújo, G.S. Sá, E.V. Teixeira and J.M. Urbano, Oscillatory free boundary problems in stochastic materials, arXiv:2404.03060.
  • E.A. Pimentel and J.M. Urbano (Eds.), Modern methods in the analysis of free boundary problems, Coimbra Mathematical Texts, Springer Cham, to appear.
  • N. Igbida and J.M. Urbano, A granular model for crowd motion and pedestrian flow, arXiv:2402.17361.
  • D.J. Araújo, R. Teymurazyan and J.M. Urbano, Hausdorff measure estimates for the degenerate quenching problem, arXiv:2402.11536.
  • D.J. Araújo, A. Sobral, E.V. Teixeira and J.M. Urbano, On free boundary problems shaped by oscillatory singularities, arXiv:2401.08071.
  • V. Bianca, E.A. Pimentel and J.M. Urbano, Transmission problems: regularity theory, interfaces and beyond,  Coimbra Mathematical Texts, Springer Cham, to appear.
  • D.J. Araújo and J.M. Urbano, The ∞−Laplacian: from AMLEs to Machine Learning, IMPA, 2023, ISBN 978-85-244-0529-7.
  • V. Bianca, E.A. Pimentel and J.M. Urbano, Improved regularity for a Hessian-dependent functional, Proc. Amer. Math. Soc., to appear.
  • D.J. Araújo, G.S. Sá and J.M. Urbano, Sharp regularity for a singular fully nonlinear parabolic free boundary problem, J. Differential Equations 389 (2024), 90-113.
  • V. Bianca, E.A. Pimentel and J.M. Urbano, BMO-regularity for a degenerate transmission problem, Anal. Math. Phys. 14 (2024), article 9 (26 pages).
  • D. dos Prazeres, A. Sobral and J.M. Urbano, Cordes-Nirenberg type results for nonlocal equations with deforming kernels, arXiv:2212.07228.
  • D. Jesus, E.A. Pimentel and J.M. Urbano, Fully nonlinear Hamilton-Jacobi equations of degenerate type, Nonlinear Anal. 227 (2023), Paper No. 113181, 15 pp.

Questions and Answers

Why KAUST

The research opportunities granted by KAUST are unique. The scientific environment around campus is vibrant and defiant, and the possibility to bridge different research areas through collaborative work in projects driven by concrete applications is challenging and inspiring.

Why your area of research?

Free boundary problems involve a priori unknown boundaries whose location is part of the solution. They are ubiquitous in applications as they model various phenomena, from ocean-atmosphere interactions and flows in porous media to the propagation of fire fronts and option pricing of financial assets. As applied mathematicians, we aim to understand the behaviour of the weak solutions to the problems and provide information on the regularity and geometric properties of the pertaining interfaces. 

Singular and degenerate nonlinear PDEs appear in different contexts, such as the melting of crushed ice, the stochastic approach to tug-of-war games, the collapse of sand piles, or semi-supervised learning. The study of their solutions amounts to understanding to what extent the collapse in ellipticity (an algebraic property) results in a collapse of regularity (an analytical property). This is paramount regarding the justification of the theoretical models and the implementation of efficient numerical schemes to approximate the solutions.

Selected Publications

  • Urbano, J. M. ., Araújo, D. ., & Sá, G. . (2024). Sharp regularity for a singular fully nonlinear parabolic free boundary problem. J. Differential Equations, 389.
  • Urbano, J. M. ., & Araújo, D. . (2023). The ∞−Laplacian: from AMLEs to Machine Learning. In Colóquio Brasileiro de Matemática. IMPA.
  • Urbano, J. M. ., Caffarelli, L. ., & Teymurazyan, R. . (2020). Fully nonlinear integro-differential equations with deforming kernels. Comm. Partial Differential Equations, 45.
  • Urbano, J. M. ., Teixeira, E. ., & Araújo, D. . (2017). A proof of the C^p’-regularity conjecture in the plane. Adv. Math., 316.
  • Urbano, J. M. ., Kuusi, T. ., & Baroni, P. . (2014). A quantitative modulus of continuity for the two-phase Stefan problem. Arch. Ration. Mech. Anal., 214.
  • Urbano, J. M. ., & Sanchón, M. . (2009). Entropy solutions for the p(x)-Laplace equation. Trans. Amer. Math. Soc., 361.
  • Urbano, J. M. . (2008). The Method of Intrinsic Scaling. In Lecture Notes in Mathematics (Vol. 1930). Springer.

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