Spring Lecture Series

Saturday's Schedule of Talks

Silvia Ghinassi, University of Washington

Title: Self-similar sets and Lipschitz graphs

Abstract: We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always intersect Lipschitz objects on a set of 1-measure zero, we ask whether these sets overlap with Lipschitz images or graphs at a dimension that is close to one. In an answer to this question, we show that one-dimensional attractors of iterated function systems satisfying the open set condition have subsets of dimension arbitrarily close to one that can be covered by Lipschitz graphs. Moreover, the Lipschitz constant of such graphs depends explicitly on the difference of the dimension of the original set and the subset that overlaps with the graph. This is joint work with Blair Davey and Bobby Wilson.

Steve Hofmann, University of Missouri

Title: A problem of free boundary type for caloric measure

Abstract: For an open set Ω⊂R^d with an Ahlfors regular boundary, solvability of the Dirichlet problem for Laplace’s equation, with boundary data in L^p for some p<∞, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak-A_∞ property) of harmonic measure with respect to surface measure on ∂Ω. A similar statement is true in the caloric setting. Thus, it is of interest to find geometric criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in the harmonic case. In this talk, we shall discuss recent progress in the caloric setting, in which we show that quantitative absolute continuity of caloric measure, with respect to “surface measure” on the parabolic Ahlfors regular (lateral) boundary Σ, implies parabolic uniform rectifiability of Σ. We observe that this result may be viewed as the solution of a certain 1-phase free boundary problem.
This is joint work with S. Bortz, J. M. Martell and K. Nyström.

José Maria Martell, Instituto de Ciencias Matemáticas (ICMAT)

Title: The Dirichlet problem with data in Hölder spaces in rough domains

Abstract: We consider the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. We work in open sets satisfying the capacity density condition (a quantitative version of the Wiener), without any further topological assumptions such as connectivity, and show that the Dirichlet boundary value problem is well-posed for boundary data in Hölder spaces with small enough exponent if Ω is either bounded, or unbounded with unbounded boundary. However, when Ω is unbounded with bounded boundary (e.g., the complement of a compact set), we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition.

José Maria Martell, Instituto de Ciencias Matemáticas (ICMAT)

Title: The Dirichlet problem with data in Hölder spaces in rough domains

Abstract: Unique continuation property is a fundamental property for harmonic functions, as well as a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular or critical set. In this talk, I will talk about some recent results together with C. Kenig on boundary unique continuation.

Sunday's Schedule of Talks

Joseph Feneuil, Australian National University

Title: Green functions, smooth distances, and uniform rectifiability

Abstract: The past 10 years have seen considerable achievements at the intersection of harmonic analysis, PDE, and geometric measure theory. One now better understands the relationship between the geometry of the boundary of a domain and the regularity of harmonic/elliptic solutions inside the domain. For instance, it was proved that the uniform rectifiability (UR) of a codimension 1 set is characterized by the A_∞-absolute continuity of its harmonic measure with respect to the surface measure - or equivalently the solvability of a L^p Dirichlet problem in the complement.
In this talk, I will show that another characterization of UR sets of codimension 1 can be obtained by comparing the Green function G with some regularized version of the distance to the boundary. Moreover, I will obtain a characterization of any UR set of any codimension by an estimate on ∇|∇G|.
   Those are joint works with Guy David, Linhan Li, and Svitlana Mayboroda.