arXiv subjects
Publications and source records attributed to Tatiana Toro.
We characterize Radon measures $μ$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $μ$-measure zero by countably many $d$-dimensional Lipschitz graphs and $μ\ll \mathcal{H}^{d}$. The characterization is in terms of a Jones function involving the so-called $α$-numbers. This answers a question left open in a former work by Azzam, David, and Toro.
We consider second order divergence form elliptic operators with $W^{1,1}$ coefficients, in a uniform domain $Ω$ with Ahlfors regular boundary. We show that the $A_\infty$ property of the elliptic measure associated to any such operator implies that $Ω$ is a set of locally finite perimeter whose boundary, $\partialΩ$, is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional information about $Ω$.
We consider a certain class of second order, variable coefficient divergence form elliptic operators, in a uniform domain $Ω$ with Ahlfors regular boundary, and we show that the $A_\infty$ property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact NTA (and hence chord-arc). The converse was already known, and follows from work of Kenig and Pipher.
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.
We study the structure of the support of a doubling measure by analyzing its self-similarity properties, which we estimate using a variant of the $L^1$ Wasserstein distance. We show that measure satisfying certain self-similarity conditions admits a unique (up to multiplication by a constant) flat tangent measure at almost every point. This allows us to decompose the support into rectifiable pieces of various dimensions.
Let $μ$ be a doubling measure in $\mathbb{R}^n$. We investigate quantitative relations between the rectifiability of $μ$ and its distance to flat measures. More precisely, for $x$ in the support $Σ$ of $μ$ and $r > 0$, we introduce a number $α(x,r)\in (0,1]$ that measures, in terms of a variant of the $L^1$-Wasserstein distance, the minimal distance between the restriction of $μ$ to $B(x,r)$ and a multiple of the Lebesgue measure on an affine subspace that meets $B(x,r/2)$. We show that the set of points of $Σ$ where $\int_0^1 α(x,r) \frac{dr}{r} < \infty$ can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of $μ$ when we assume that some Carleson measure estimates hold. Soit $μ$ une mesure doublante dans $\mathbb{R}^n$. On étudie des relations quantifiées entre la rectifiabilité de $μ$ et la distance entre $μ$ et les mesures plates. Plus précisément, on utilise une variante de la $L^1$-distance de Wasserstein pour définir, pour $x$ dans le support $Σ$ de $μ$ et $r>0$, un nombre $α(x,r)$ qui mesure la distance minimale entre la restriction de $μ$ à $B(x,r)$ et une mesure de Lebesgue sur un sous-espace affine passant par $B(x,r/2)$. On décompose l'ensemble des points $x\in Σ$ tels que $\int_0^1 α(x,r) \frac{dr}{r} < \infty$ en parties rectifiables de dimensions diverses, et on obtient un meilleur contrôle de ces parties et de la taille de $μ$ quand les $α(x,r)$ vérifient certaines conditions de Carleson.
We show that if $Ω\subset \mathbb{R}^{n+1}$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $Ω$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $Ω$ is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew conditions, and an interior Harnack Chain condition. We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain free boundary problems.
Let $E$ be a set in $\mathbb R^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that $\liminf_{r\to0}r^{-n} H^n(B(x,r)\cap E)>0$ for $H^n$-a.e. $x\in E$. In this paper it is shown that $E$ is $n$-rectifiable if and only if $$\int_0^1 \left|\frac{H^n(B(x,r)\cap E)}{r^n} - \frac{H^n(B(x,2r)\cap E)}{(2r)^n}\right|^2\,\frac{dr}r < \infty$$ for $H^n$-a.e. $x\in E$; and also if and only if $$ \lim_{r\to0}\left(\frac{H^n(B(x,r)\cap E)}{r^n} - \frac{H^n(B(x,2r)\cap E)}{(2r)^n}\right) = 0$$ for $H^n$-a.e. $x\in E$. Other more general results involving Radon measures are also proved.
A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of ${\Bbb R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:{\Bbb R}^N\to{\Bbb R}^N$ . We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz $n$-manifold and for a quasiplane to have big pieces of bi-Lipschitz images of ${\Bbb R}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension $N-n$. To establish the big pieces criterion, we prove new extension theorems for "almost affine" maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.
In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_Ω|\nabla u(x)|^2 +q^2_+(x)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(Ω)$. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see \cite{AC}, \cite{ACF}, \cite{CJK}, \cite{W}). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers.
We study the boundary regularity of solutions to divergence form operators which are small perturbations of operators for which the boundary regularity of solutions is known. An operator is a small perturbation of another operator if the deviation function of the coefficients satisfies a Carleson measure condition with small norm. We extend Escauriaza's result on Lipschitz domains to chord arc domains with small constant. In particular we prove that if $L_1$ is a small perturbation of $L_0$ and $\log k_0$ has small BMO norm so does $\log k_1$. Here $k_i$ denotes the density of the elliptic measure of $L_i$ with respect to the surface measure of the boundary of the domain.
We obtain Dini conditions with "exponent 2" that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any e>0 integrability of (esssup_{1-t < |x| < 1+t} K_f(x)-1)^{2-e} dt/t implies that the image of the unit sphere under a global quasiconformal homeomorphism f is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global K-quasiconformal map in neighborhoods with maximal dilatation close to 1.
We prove local well-posedness of the Schrödinger flow from $R^n$ into a compact K\{"a}hler manifold $N$ with initial data in $H^{s+1}(R^n, N)$ for $s\geq n/2+4$.
We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of $E$ by a $d$-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers $β_1(x,r)$. In particular, it applies in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz images of $\R^d$.
In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the inside and outside harmonic measures of domains in n-dimensional Euclidean space, with n bigger than or equal to 3.
This paper has been withdrawn by the authors.
In this paper we consider the question of whether the doubling character of a measure supported on a subset of $\RR^m$ determines the regularity of its support (in a classical sense). This problem was studied by David, Kenig and Toro for codimension 1 sets under the assumption that the support be flat. Here we study the higher codimension case and remove the flatness hypothesis.
One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said to be doubling with constant C if C times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center. We prove that if the doubling constant of a measure on \R^{n+1} is close to the doubling constant of the n-dimensional Lebesgue measure then its support is well approximated by n-dimensional affine spaces, provided that the support is relatively flat to start with. Primarily we consider sets which are boundaries of domains in \R^{n+1}. The n-dimensional Hausdorff measure may not be defined on the boundary of a domain in R^{n+1}. Thus we turn our attention to the harmonic measure which is well behaved under minor assumptions. We obtain a new characterization of locally flat domains in terms of the doubling properties of their harmonic measure. Along these lines we investigate how the ``weak'' regularity of the Poisson kernel of a domain determines the geometry of its boundary.