arXiv subjects

Guy David

Publications and source records attributed to Guy David.

At least 37 records · Page 2Linked to original sources

arXiv2020

Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one

In this paper, we characterize the rectifiability (both uniform and not) of an Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that, in a sharp contrast with the usual setting of co-dimension one, a converse to the Dahlberg's theorem (see [Da] and [DFM2]) must be false on lower dimensional boundaries without additional assumptions.

math.AP↗

arXiv2020

Good elliptic operators on Cantor sets

It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely unrectifiable) Cantor sets in ${\mathbb{R}}^2$ whose elliptic measure is absolutely continuous, and in fact, essentially proportional to the Hausdorff measure.

math.AP↗

arXiv2019

Free Boundary Regularity for Almost-Minimizers

In this paper we study the free boundary regularity for almost-minimizers of the functional \begin{equation*} J(u)=\int_{\mathcal O} |\nabla u(x)|^2 +q^2_+(x)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{u<0\}}(x)\ dx \end{equation*} where $q_\pm \in L^\infty(\mathcal O)$. Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [AC], [ACF], [CJK], [W]). Nevertheless we succeed in proving that, under a non-degeneracy assumption on $q_\pm$, the free boundary is uniformly rectifiable. Furthermore, when $q_-\equiv 0$, and $q_+$ is Hölder continuous we show that the free boundary is almost-everywhere given as the graph of a $C^{1,α}$ function (thus extending the results of [AC] to almost-minimizers).

math.AP↗

arXiv2019

A local description of 2-dimensional almost minimal sets bounded by a curve

We study the local regularity of sliding almost minimal sets of dimension 2 in $R^n$ , bounded by a smooth curve $L$. These are a good way to model soap films bounded by a curve, and their definition is similar to Almgren's. We aim for a local description, in particular near L and modulo $C^{1+$ε$}$ diffeomorphisms, of such sets $E$, but in the present paper we only obtain a full description when $E$ is close enough to a half plane, a plane or a union of two half planes bounded by the same line, or a transverse minimal cone of type $Y$ or $T$. The main tools are adapted near monotonicity formulae for the density, including for balls that are not centered on L, and the same sort of construction of competitors as for the generalization of J. Taylor's regularity result far from the boundary.

math.CA↗

arXiv2018

Sliding almost minimal sets and the Plateau problem

We present some old and recent regularity results concerning minimal and almost minimal sets in domains of the Euclidean space. We concentrate on a sliding variant of Almgren's notion of minimality, which is well suited in the context of Plateau problems relative to soap films. We are especially interested in regularity properties near a boundary curve, where we would like to get a local C1 description of 2-dimensional almost minimal sets in the spirit of J. Taylor's theorem, but we first study weaker and more general results (local Ahlfors regularity, rectifiability, limits, monotonicity of density), which we describe far from the boundary for simplicity. There we insist on some simpler techniques, in particular the use of Federer-Fleming projections.

math.CA↗

arXiv2018

Localization of eigenfunctions via an effective potential

We consider the localization of eigenfunctions for the operator $L=-\mbox{div} A \nabla + V$ on a Lipschitz domain $Ω$ and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to $Lu=1$, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework that reveals a richly detailed portrait of the eigenfunctions and eigenvalues. We show that the reciprocal of the landscape function, $1/u$, acts as an effective potential. Hence from the single measurement of $u$, we obtain, via $1/u$, explicit bounds on the exponential decay of the eigenfunctions of the system and estimates on the distribution of eigenvalues near the bottom of the spectrum. (This version strengthens and simplifies the results of the first one by replacing a global bi-Lipschitz hypothesis on the domain with a local bi-Lipschitz hypothesis. It improves on the second version by adding pictures and numerical examples. This version is identical to the third version; all that is changed is to correct some tex mistakes in symbols in this abstract. There are no changes to the paper itself.)

math.AP↗

arXiv2018

Computing spectra without solving eigenvalue problems

The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation of the spectrum of a Schrödinger operator with a disordered potential. Unlike plane waves or Bloch waves that arise as Schrödinger eigenfunctions for periodic and other ordered potentials, for many forms of disordered potentials the eigenfunctions remain essentially localized in a very small subset of the initial domain. A celebrated example is Anderson localization, for which, in a continuous version, the potential is a piecewise constant function on a uniform grid whose values are sampled independently from a uniform random distribution. We present here a new method for approximating the eigenvalues and the subregions which support such localized eigenfunctions. This approach is based on the recent theoretical tools of the localization landscape and effective potential. The approach is deterministic, predicting quantities that depend sensitively on the particular realization, rather than furnishing statistical or probabilistic results about the spectrum associated to a family of potentials with a certain distribution. These methods, which have only been partially justified theoretically, enable the calculation of the locations and shapes of the approximate supports of the eigenfunctions, the approximate values of many of the eigenvalues, and of the eigenvalue counting function and density of states, all at the cost of solving a single source problem for the same elliptic operator. We study the effectiveness and limitations of the approach through extensive computations in one and two dimensions, using a variety of piecewise constant potentials with values sampled from various different correlated or uncorrelated random distributions.

math.NA↗

arXiv2018

A new elliptic measure on lower dimensional sets

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of $n-1$ dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.

math.AP↗

arXiv2017

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n-1$ in $\mathbb R^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond. In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph $Γ$ of dimension $d$ in $\mathbb R^n$, $d<n-1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator $L$ such that the corresponding harmonic measure $ω_L$ is absolutely continuous with respect to the Hausdorff measure on $Γ$. More generally, we provide sufficient conditions on the matrix of coefficients of $L$ which guarantee the mutual absolute continuity of $ω_L$ and the Hausdorff measure.

math.AP↗

arXiv2016

Harmonic measure on sets of codimension larger than one

We introduce a new notion of a harmonic measure for a $d$-dimensional set in $\R^n$ with $d<n-1$, that is, when the codimension is strictly bigger than 1. Our measure is associated to a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the $d$-dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.

math.AP↗

arXiv2016

Effective confining potential of quantum states in disordered media

The amplitude of localized quantum states in random or disordered media may exhibit long range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that the Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random.

cond-mat.dis-nn↗

arXiv2014

Wasserstein Distance and the Rectifiability of Doubling Measures: Part II

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.

math.MG↗

arXiv2014

A monotonicity formula for minimal sets with a sliding boundary condition

We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure $\cal{H}^d$, subject to a sliding boundary constraint where competitors for $E$ are obtained by deforming $E$ by a one-parameter family of functions $φ_t$ such that $φ_t(x) \in L$ when $x\in E$ lies on the boundary $L$. In the simple case when $L$ is an affine subspace of dimension $d-1$, the monotone or almost monotone functional is given by $F(r) = r^{-d} \cal{H}^d(E \cap B(x,r)) + r^{-d} \cal{H}^d(S \cap B(x,r))$, where $x$ is any point of $E$ (not necessarily on $L$) and $S$ is the shade of $L$ with a light at $x$. We then use this, the description of the case when $F$ is constant, and a limiting argument, to give a rough description of $E$ near $L$ in two simple cases. ----- On donne une formule de monotonie pour des ensembles minimaux ou presque minimaux pour la mesure de Hausdorff $\cal{H}^d$, avec une condition de bord où les compétiteurs de $E$ sont obtenus en déformant $E$ par une famille à un paramètre de fonctions $φ_t$ telles que $φ_t(x)\in L$ quand $x\in E$ se trouve sur la frontière $L$. Dans le cas simple où $L$ est un sous-espace affine de dimension $d-1$, la fonctionelle monotone ou presque monotone est donnée par $F(r) = r^{-d} \cal{H}^d(E \cap B(x,r)) + r^{-d} \cal{H}^d(S \cap B(x,r))$, où $x$ est un point de $E$, pas forcément dans $L$, et $S$ est l'ombre de $L$, éclairée depuis $x$. On utilise ceci, la description des cas où $F$ est constante, et un argument de limite, pour donner une description de $E$ près de $L$ dans deux cas simples.

math.CA↗

arXiv2014

Wasserstein Distance and the Rectifiability of Doubling Measures: Part I

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.

math.CA↗

arXiv2014

A free boundary problem for the localization of eigenfunctions

We study a variant of the Alt, Caffarelli, and Friedman free boundary problem with many phases and a slightly different volume term, which we originally designed to guess the localization of eigenfunctions of a Schrödinger operator in a domain. We prove Lipschitz bounds for the functions and some nondegeneracy and regularity properties for the domains.

math.CA↗

arXiv2014

Local regularity properties of almost- and quasiminimal sets with a sliding boundary condition

We study the boundary regularity of almost minimal and quasiminimal sets that satisfy sliding boundary conditions. The competitors of a set $E$ are defined as $F = φ_1(E)$, where $\{ φ_t \}$ is a one parameter family of continuous mappings defined on $E$, and that preserve a given collection of boundary pieces. We generalize known interior regularity results, and in particular we show that the quasiminimal sets are locally Ahlfors-regular, rectifiable, and some times uniformly rectifiable, that our classes are stable under limits, and that for almost minimal sets the density of Hausdorff measure in balls centered on the boundary is almost nondecreasing.

math.CA↗

arXiv2013

Regularity for almost minimizers with free boundary

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.

math.AP↗

arXiv2012

Approximation of a Reifenberg-flat set by a smooth surface

We show that if $E ı\R^n$ is a Reifenberg flat set $E$ of dimension $d$ at scale $r_0$, we can find a smooth surface $Σ_0$ of dimension $d$ which is close to $E$ at the scale $r_0$. When $E$ is a Reifenberg flat set, this allows us to apply a result of G. David and T. Toro [Memoirs of the AMS 215 (2012), 1012], and get a bi-Hölder homeomorphism of $\R^n$ that sends $Σ_0$ to $E$. If in addition $d=n-1$ and $E$ is compact and connected, then $Σ_0$ is orientable, and $\R^n \sm E$ has exactly two connected components, which we can approximate from the inside by smooth domains.

math.CA↗