## Brief Biography

Miguel Urbano received his PhD in Mathematical Analysis from the University of Lisbon in 1999. After a postdoc at Northwestern University in Chicago, he became an Assistant Professor at the University of Coimbra. He was promoted to Associate Professor with tenure in 2004 and awarded the Habilitation in Mathematics in 2005. Before joining KAUST in 2022, he had been a Full Professor at the University of Coimbra since 2009.

He is the author of *The Method of Intrinsic Scaling*, published in the *Lecture Notes in Mathematics* series, and of over 60 scientific papers in nonlinear PDEs. He served on panels to evaluate 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. He taught short courses at IMPA (Brazil), the University of Florence (Italy), Aalto University (Finland), and Seoul National University (South Korea). He was a Special Visiting Researcher in Brazil's *Science without Borders* program. He is a Corresponding Member of the *Lisbon Academy of Sciences* and Editor-in-Chief of *Portugaliae Mathematica*.

## Areas of Expertise and Research Interests

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.

## 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 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.

## 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.

## Career Recognition

- Corresponding Academician, Sciences Class, Mathematics Section, Lisbon Academy of Sciences, elected January 2021.
- Member of Portugal's National Council for Science and Technology, 2012–2015.
- José Anastácio da Cunha Prize (
*ex-aequo*), Portuguese Mathematical Society, 2002.

## Editorial Activities

- Editor-in-Chief,
*Portugaliae Mathematica*, EMS-Press, since January 2022. - Associate Editor,
*Nonlinear Analysis*, Elsevier, 2013 — 2021.

## Selected publications

- L.A. Caffarelli, R. Teymurazyan, and J.M. Urbano, Fully nonlinear integro-differential equations with deforming kernels,
*Comm. Partial Differential Equations*45 (2020), 847-871. - D.J. Araújo, E. V. Teixeira, and J.M. Urbano, A proof of the
*C*-regularity conjecture in the plane,^{p'}*Adv. Math.*316 (2017), 541-553. - P. Baroni, T. Kuusi, and J.M. Urbano, A quantitative modulus of continuity for the two-phase Stefan problem,
*Arch. Ration. Mech. Anal.*214 (2014), 545-573. - M. Sanchón and J.M. Urbano, Entropy solutions for the
*p(x)*-Laplace equation,*Trans. Amer. Math. Soc.*361 (2009), 6387-6405. - J.M. Urbano, The Method of Intrinsic Scaling,
*Lecture Notes in Mathematics*, vol. 1930, Springer, 2008.

## Recent preprints and ongoing work

- D.J. Araújo, R. Teymurazyan and J.M. Urbano,
*Free boundary regularity for the degenerate quenching problem*, in preparation. - D.J. Araújo, G.S. Sá, E.V. Teixeira and J.M. Urbano,
*The quenching problem in rough heterogeneous media*, in preparation. - D.J. Araújo, A. Sobral, E.V. Teixeira and J.M. Urbano,
*On free boundary problems with varying singularities*, in preparation. - V. Bianca, E.A. Pimentel and J.M. Urbano,
*Transmission problems: regularity theory, interfaces and beyond*, arXiv:2306.15570. - D.J. Araújo and J.M. Urbano,
*The ∞−Laplacian: from AMLEs to Machine Learning*, IMPA, 2023. - V. Bianca, E.A. Pimentel and J.M. Urbano,
*Improved regularity for a Hessian-dependent functional*, arXiv:2306.10995. - D.J. Araújo, G.S. Sá and J.M. Urbano,
*Sharp regularity for a singular fully nonlinear parabolic free boundary problem*, arXiv:2303.14520. - V. Bianca, E.A. Pimentel and J.M. Urbano,
*BMO-regularity for a degenerate transmission problem*, arXiv:2301.01171. - 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.