A Two-Phase Free Boundary Problem for Harmonic Measure

We study a 2-phase free boundary problem for harmonic measure first considered by Kenig and Toro and prove a sharp H\"older regularity result. The central difficulty is that there is no a priori non-degeneracy in the free boundary condition. Thus we must establish non-degeneracy by means of monotonicity formulae.Comment: 45 pages. This version has minor revisions as suggested by the refere

Non-Uniqueness of Bubbling for Wave Maps

We consider wave maps from $\mathbb R^{2+1}$ to a $C^\infty$-smooth Riemannian manifold, $\mathcal N$. Such maps can exhibit energy concentration, and at points of concentration, it is known that the map (suitably rescaled and translated) converges weakly to a harmonic map, known as a bubble. We give an example of a wave map which exhibits a type of non-uniqueness of bubbling. In particular, we exhibit a continuum of different bubbles at the origin, each of which arise as the weak limit along a different sequence of times approaching the blow-up time. This is the first known example of non-uniqueness of bubbling for dispersive equations. Our construction is inspired by the work of Peter Topping [Topping 2004], who demonstrated a similar phenomena can occur in the setting of harmonic map heat flow, and our mechanism of non-uniqueness is the same 'winding' behavior exhibited in that work.Comment: 26 pages, two figures. Comments welcom

Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional

In this paper we prove uniqueness of blow-ups and $C^{1,\log}$-regularity for the free-boundary of minimizers of the Alt-Caffarelli functional at points where one blow-up has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free-boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a $C^{1,\alpha}$ regularity result. We also show that these additional assumptions are satisfied by the De Silva-Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the \L ojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem.Comment: 37 pages. To appear in Duke Math Journa

Structure of sets which are well approximated by zero sets of harmonic polynomials

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how "degree $k$ points" sit inside zero sets of harmonic polynomials in $\mathbb R^n$ of degree $d$ (for all $n\geq 2$ and $1\leq k\leq d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree $k$ points" ($k\geq 2$) without proving uniqueness of blowups or aid of PDE methods such as monotonicity formulas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of $k$. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro.Comment: 40 pages, 2 figures (v2: streamlined several proofs, added statement of Lojasiewicz inequality for harmonic polynomials [Theorem 3.1]

Review of Erica Fudge, Ruth Gilbert and Susan Wiseman, eds., At the Borders of the Human: Beasts, Bodies, and Natural Philosophy in the Early Modern Period.

Erica Fudge, Ruth Gilbert and Susan Wiseman, eds.,At the Borders of the Human: Beasts, Bodies, and Natural Philosophy in the Early Modern Period. New York: St. Martin’s Press, 1999. 269 pp. ISBN 0312220383

BRCA Previvors: Medical and Social Factors That Differentiate Them From Previvors With Other Hereditary Cancers

Commentaire / CommentaryDans cet article, je décris quelques-unes des raisons pour lesquelles les « previvors » de BRCA (c-a-d. « survivants d ’ u n e prédisposition au cancer ») sont différents des previvors avec d’autres cancers héréditaires. J’examine comment l’absence d’une norme de soins pour le risque de cancer du sein chez les femmes ayant une mutation BRCA, associée à un large éventail de pénétration génétique et une mortalité plus faible, rend le BRCA différent des autres cancers héréditaires qui ont des directives claires et établies. En plus de ces différences médicales, des facteurs sociaux tels que la prééminence culturelle du cancer du sein et la signification sociale des seins ont engendré une identité prédictive individuelle plus complexe et une réponse culturelle aux femmes ayant une mutation BRCA.In this paper, I outline some of the reasons why BRCA “previvors” (i.e., “survivors of a predisposition to cancer”) are different from previvors with other hereditary cancers. I examine how the absence of a standard of care for breast cancer risk for women with a BRCA mutation, coupled with a broad range of genetic penetrance and lower mortality, makes BRCA different than other hereditary cancers that have clear and established guidelines. In addition to these medical differences, social factors like the cultural prominence of breast cancer and the social significance of breasts have engendered a more complicated individual previvor identity for and cultural response to women with a BRCA mutation