Borderline regularity in singular free boundary problems
Miércoles 3/12, 11.30h
Seminario | José Miguel Urbano
El Departamento de Matemáticas y Estadística de la Universidad Torcuato Di Tella invita al seminario “Borderline regularity in singular free boundary problems”, a cargo de José Miguel Urbano (KAUST).
Abstract
We address the borderline regularity of local minimizers of singular energy functionals. For bounded and measurable potentials, we show that sign-changing minimizers are Log-Lipschitz continuous, which is optimal in this generality. In the one-phase case, however, we derive gradient bounds along the free boundary, uncovering a structural gain in regularity. Our first main result establishes sharp Lipschitz regularity for a merely bounded potential. Most notably, we prove that if the potential is further assumed to be a modulus of continuity, then minimizers become continuously differentiable. We thus identify a sharp threshold for differentiability in terms of the regularity of the potential.
This is joint work with D. Araújo (UFPB), A. Sobral (KAUST), and E. Teixeira (OSU).
El seminario se dictará en inglés, sin traducción.
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.
Abstract
We address the borderline regularity of local minimizers of singular energy functionals. For bounded and measurable potentials, we show that sign-changing minimizers are Log-Lipschitz continuous, which is optimal in this generality. In the one-phase case, however, we derive gradient bounds along the free boundary, uncovering a structural gain in regularity. Our first main result establishes sharp Lipschitz regularity for a merely bounded potential. Most notably, we prove that if the potential is further assumed to be a modulus of continuity, then minimizers become continuously differentiable. We thus identify a sharp threshold for differentiability in terms of the regularity of the potential.
This is joint work with D. Araújo (UFPB), A. Sobral (KAUST), and E. Teixeira (OSU).
El seminario se dictará en inglés, sin traducción.
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.