Affiliations

• Applied Mathematics and Computational Science

Education Profile

• Habilitation in Mathematics, University of Coimbra, Portugal, 2005
• Postdoctoral Fellow, Northwestern University, Chicago, USA, 1999
• PhD in Mathematical Analysis, University of Lisbon, Portugal, 1999
• BSc in Pure Mathematics, University of Coimbra, Portugal, 1992

Research Interests

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, with a focus on the understanding of the local behaviour of weak solutions and the geometric properties of interfaces.

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 Cp'-regularity conjecture in the plane, 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.
• E. V. Teixeira and J.M. Urbano, A geometric tangential approach to sharp regularity for degenerate evolution equations, Anal. PDE 7 (2014), 733-744.
• 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.