Estimates on the generalized critical strata of Green's function

In this paper, we obtain quantitative estimates on the fine structure of Green's functions for pairs of complementary domains, $\Omega^+, \Omega^- \subset \mathbb{R}^n$ which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary, $\partial \Omega^{\pm},$ and on how critical set of the Green's functions approach the boundary. These estimates are not obtainable by naively combining boundary and interior estimates.Comment: 72 pages, 1 figur

Unique Continuation on Convex Domains

In this paper, we adapt powerful tools from geometric analysis to get quantitative estimates on the quantitative strata of the generalized critical set of harmonic functions which vanish continuously on an open subset of the boundary of a convex domain. These estimates represent a significant improvement upon existing results for boundary analytic continuation in the convex case.Comment: 69 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1904.0936

Non-existence of cusps for a Free-boundary Problem for Water Waves

In arXiv:0908.1031, Varvaruca and Weiss eliminate the existence of cusps for a free-boundary problem for two-dimensional water waves under assumptions that hold for solutions such that $\{u>0\}$ is a "strip-like" domain in the sense of arXiv:0708.4371. In this paper it is proven that cusps do not exists in the natural setting for these free-boundary problems. In particular, non strip-like domains are also allowed. This builds upon recent work on non-existence of cusps in arXiv:2202.00616

Critical Allard regularity: pointwise tilt-excess estimates

The main results of this paper provide VMO-type estimates for the quadratic tilt-excess on varifolds with critical generalized mean curvature. These estimates apply to varifolds with "almost-integral" density which are close to a multiplicity one $m$-disc in a ball in the usual senses. The class of almost-integral varifolds allows for varifolds with non-perpendicular mean curvature. Moreover, the estimates hold \emph{uniformly for every point} in a relatively open set in $\text{spt}||V||$ and naturally imply a Reifenberg-type parametrization. The proof relies upon generalizing the $Q$-valued Lipschitz approximation and Sobolev-Poincar\'e estimates of arXiv:0808.3660 to almost-integral rectifiable varifolds

Making Memorial Student-Ready: Reflections on the First Year Success Experience

In eleven short chapters faculty, academic advising staff and student union representatives discuss aspects of Memorial’s First Year Success Program (piloted as a Teaching Learning Framework initiative 2012-2017). Teaching approaches, curriculum content and policy rationales are covered in a broad view of how and why students identified as least likely to succeed at university can be academically supported. Contributors identify the singular importance of the community that First Year Success provided them and its student participants

Minkowski-type Estimates on the Quantitative Strata of the Generalized Critical set of Green's functions for Two-Sided NTA Domains arising from a Free-Boundary Problem for Harmonic Measure

Thesis (Ph.D.)--University of Washington, 2019In this work, we prove three things. The main results are two different results on Minkowski-type estimates on the quantitative strata of the generalized critical set of Green's functions of 2-Sided NTA domains arising from a free-boundary problem for harmonic measure. The first uses simpler techniques and obtains weaker results. The second employs much more complicated machinery and obtains a much stronger result which completely subsumes the results of the first approach. The third result contained in this work is the construction of two families of rectifiable sets which fail to be uniformly rectifiable as dramatically as possible which still retaining nice topological and measure theoretic properties. This third result is independent of the first two, and represents joint work with Max Goering

One-phase free-boundary problems with degeneracy

In this paper, we study local minimizers of a degenerate version of the Alt-Caffarelli functional. Specifically, we consider local minimizers of the functional $J_{Q}(u, \Omega):= \int_{\Omega} |\nabla u|^2 + Q(x)^2\chi_{\{u>0\}}dx$ where $Q(x) = dist(x, \Gamma)^{\gamma}$ for $\gamma>0$ and $\Gamma$ a $C^{1, \alpha}$ submanifold of dimension $0 \le k \le n-1$. We show that the free boundary may be decomposed into a rectifiable set, on which we prove effective estimates, and a degenerate cusp set, about which little can be said in general with the current techniques. Work in the theory of water waves and the Stokes wave serves both as our inspiration and as an application. However, the main thrust of this paper is to inaugurate a study of the geometry of the free boundary for degenerate one-phase Bernoulli free-boundary problems