Harmonic Measure and free Boundary Problems

1. Boundary regularity of harmonic functions

-Basic properties of harmonic functions
-Existence via Perron’s method
-Dirichlet and Neumann boundary conditions
-Boundary regularity in C^k,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.

[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.