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Harmonic Measure and free Boundary Problems

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Advanced course / School
From November 07, 2023
to December 21, 2023

Location: Universitat de Barcelona

Rooms: T1 on Tuesdays and B2 on Thursdays

Registration deadline 29 / 10 / 2023

SChedule

Period: Tuesdays and Thursdays from November 7 to December 21, 2023
Time Schedule: 11h a 13h
November: days 7,9,21,23,28 and 30
December: days 12,14,19 and 21

Introduction

Harmonic measure and free boundary problems are classical topics in Analysis and PDE which have seen enormous development in recent years. While the motivation and origin of these two topics of research are quite different, there have been important connections between them, and the development we have seen in the last years would not have been possible without such deep connections. Both topics have been strongly influenced by Geometric Measure Theory, and are still objects of intensive research.

Syllabus

1. Boundary regularity of harmonic functions

-Basic properties of harmonic functions
-Existence via Perron’s method
-Dirichlet and Neumann boundary conditions
-Boundary regularity in Ck,α domains
-Schauder estimates

2. Harmonic measure and Green function

-Harmonic measure
-Behaviour of harmonic measure in the plane under conformal maps
-The Riesz brothers’ theorem
-The Green function

3. Harmonic measure in NTA domains

-The doubling property and other related properties
-The change of pole formula
-Dahlberg’s theorem for Lipschitz domains
-Lp solvability of the Dirichlet problem in Lipschitz domains

4. Relationship between harmonic measure and Hausdorff measures

-Hausdorff measures and the dimension of a measure
-Survey of advanced results: the theorems of Jones-Wolff, Wolff, and Bourgain about the dimension of harmonic measure, the connection between harmonic measure and rectifiability
-Related free boundary problems

5. The Bernoulli (or Alt-Caffarelli) free boundary problem

-The energy functional: minimizers and critical points
-Existence and regularity of solutions
-Blow-up limits
-Regularity of free boundaries
-Survey of advanced results: dimension of the singular set, open problems.

References

[1] L. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68. American Mathematical Society, Providence, RI, 2005.
[2] X. Fernández-Real, X. Ros-Oton, Regularity Theory for Elliptic PDE, Zurich Lectures in Advanced Mathematics. EMS books, 2022.
[3] J. Garnett, D. Marshall, Harmonic Measure, Cambridge University Press, 2005.
[4] D. S. Jerison, C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), no. 1, 80{147.
[5] X. Ros-Oton, Mirando hacia el futuro: Problemas de frontera libre, La Gaceta de la RSME 24 (2021), 399-416.
[6] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione Matematica Italiana, 2023.

Organisers

Xavier Ros Oton | ICREA-UB-CRM
Xavier Tolsa | ICREA-UAB-CRM

lecturers

Xavier Ros Oton

Xavier Ros Oton

ICREA-UB-CRM

Xavier is ICREA Research Professor and Catedràtic d’Universitat at UB since 2020. Previously, he has been Assistant Professor at Universität Zürich, as well as R. H. Bing Instructor at the University of Texas at Austin. He is a mathematician who works on Partial Differential Equations (PDE). Specifically, he studies the regularity of solutions to elliptic and parabolic PDE, and he is mostly known for his results on free boundary problems and integro-differential equations. He is the PI of an ERC Starting Grant (2019-2024), has received several awards for young mathematicians in Spain, as well as the Scientific Research Award from the Fundación Princesa de Girona in 2019. Furthermore, in 2021 he was awarded the Stampacchia Gold Medal, an international prize awarded every three years in recognition of outstanding contributions to the Calculus of Variations.

Xavier Tolsa

Xavier Tolsa

ICREA-UAB-CRM

I was born in Barcelona in 1966. First I studied engineering, but later I turned to mathematics. After obtaining my PhD in mathematics in 1998 (UAB), I spent about one year in Goteborg (University of Goteborg – Chalmers) and another year in Paris (Université de Paris-Sud), until I came back to Barcelona (UAB) by means of a “Ramón y Cajal” position. Since 2003 I am an ICREA Research Professor. My current research in mathematics focuses in Fourier analysis, geometric measure theory, and geometric function theory.

My main scientific achievements are the proof of the semiadditivity of analytic capacity and contribution to the Painlevé problem (2003), and the solution of the David Semmes problem in codimension 1, with F. Nazarov and A. Volberg (2014).

LIST OF PARTICIPANTS

Name Institution
Nurbek Kakharman Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Philip Pita Universitat de Barcelona
Daniel Pascuas Universitat de Barcelona
Maria Carmen Cascante Canut Universitat de Barcelona
Marvin Weidner Universitat de Barcelona
Clara Torres Latorre Universitat de Barcelona
Núria Fagella Rabionet Universitat de Barcelona
Tomás Sanz Perela Universitat de Barcelona
Joaquín James Universitat de Barcelona
Òscar Burés Universitat de Barcelona
Pablo García Arias Universitat de Barcelona
Anna Jové Campabadal Universitat de Barcelona
Lucas Esquenet Universitat de Barcelona
Robert Florido Llinàs Universitat de Barcelona
Gerard Codina Universitat Autònoma de Barcelona
Matteo Levi Universitat Autònoma de Barcelona
Martí Prats Soler Universitat Autònoma de Barcelona
Bruno Poggi Cevallos Universitat Autònoma de Barcelona
Adem Limani Universitat Autònoma de Barcelona
Antoni Fierro Universitat Autònoma de Barcelona
Ignasi Guillén-Mola Universitat Autònoma de Barcelona
Josep M Gallegos Universitat Autònoma de Barcelona
Teodoro Mayayo Universitat Autònoma de Barcelona
Joan Hernández Universitat Autònoma de Barcelona
Juan Carlos Cantero Universitat Autònoma de Barcelona
Marc Magaña Centelles Universitat Autònoma de Barcelona
Kristina Oganesyan Universitat Autònoma de Barcelona
Elena Isasi Theus Universitat Politècnica de Catalunya
Renzo Bruera Universitat Politècnica de Catalunya
Bartomeu Garau Verger Universitat Politècnica de Catalunya
Antoni Burón i Palau Universitat Politècnica de Catalunya
Bernat Espigule Universitat de Girona
Mittu Walia Indian Institute of Technology Roorkee
GABRIELE ALFONSO University of Naples Federico II
Niyaz Tokmagambetov Centre de Recerca Matemàtica
Alberto Maione Centre de Recerca Matemàtica
Miquel Saucedo Centre de Recerca Matemàtica
Gustavo Rodrigues Ferreira Centre de Recerca Matemàtica

 

For inquiries about this event please contact the Scientific Events Coordinator Ms. Núria Hernández at nhernandez@crm.cat​​