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Svitlana Mayboroda

Publications and source records attributed to Svitlana Mayboroda.

89 records · Page 5Linked to original sources
arXiv2014

Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $Ω:= \mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure estimates, and are "$\varepsilon$-approximable". Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure, and surface measure.

math.AP↗
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

The Regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $t$-independent complex bounded measurable coefficients ($t$ being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in $L^{p'}$, subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in $L^p$. Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with $t$-independent real (possibly non-symmetric) coefficients there exists a $p>1$ such that the Regularity problem is well-posed in $L^p$.

math.AP↗
arXiv2013

Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm condition

We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In addition, we suppose that $A$ satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy $A(x,t) -A(x,0)$ satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in $L^p$, Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for $L$, with data in $L^p(\mathbb{R}^n)$, $BMO(\mathbb{R}^n)$, and $C^α(\mathbb{R}^n)$, and solvability of the Neumann and Regularity problems, with data in the spaces $L^p(\mathbb{R}^n)/H^p(\mathbb{R}^n)$ and $L^p_1(\mathbb{R}^n)/H^{1,p}(\mathbb{R}^n)$ respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials in for the $t$-independent operator $L_0:= -\nabla\cdot(A(\cdot,0)\nabla)$.

math.AP↗
arXiv2013

Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted Sobolev classes. We establish mapping properties for the double and single layer potentials, as well as the Newton potential, on Besov spaces. We prove extrapolation-type solvability results: that is, we show that solvability of the Dirichlet or Neumann boundary value problem at any given L^p space automatically assures their solvability in an extended range of Besov spaces. We also establish well-posedness for non-homogeneous boundary value problems. In particular, we prove well-posedness of the non-homogeneous Dirichlet problem with data in Besov spaces for operators with real, not necessarily symmetric coefficients.

math.AP↗
arXiv2013

The Dirichlet problem for higher order equations in composition form

The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=\Div B^*\nabla(a\Div A\nabla u)=0, where A and B are elliptic matrices with complex-valued bounded measurable coefficients and a is an accretive function. Elliptic operators of this type naturally arise, for instance, via a pull-back of the bilaplacian Δ^2 from a Lipschitz domain to the upper half-space. More generally, this form is preserved under a Lipschitz change of variables, contrary to the case of divergence-form fourth order differential equations. We establish well-posedness of the Dirichlet problem for the equation Lu=0, with boundary data in L^2, and with optimal estimates in terms of nontangential maximal functions and square functions.

math.AP↗
arXiv2012

Uniform Rectifiability and Harmonic Measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a closed, Ahlfors-David regular set of dimension $n$ satisfying the "Riesz Transform bound" $$\sup_{\varepsilon>0}\int_E\left|\int_{\{y\in E:|x-y|>\varepsilon\}}\frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)\right|^2 dH^n(x) \leq C \int_E|f|^2 dH^n .$$ Assume further that $E$ is the boundary of a domain $Ω\subset \mathbb{R}^{n+1}$ satisfying the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition. Then $E$ is uniformly rectifiable.

math.CA↗
arXiv2012

Regularity of solutions to the polyharmonic equation in general domains

The present paper establishes boundedness of $[m-\frac n2+\frac 12]$ derivatives for the solutions to the polyharmonic equation of order $2m$ in arbitrary bounded open sets of $\RR^n$, $2\leq n\leq 2m+1$, without any restrictions on the geometry of the underlying domain. It is shown that this result is sharp and cannot be improved in general domains. Moreover, it is accompanied by sharp estimates on the polyharmonic Green function.

math.AP↗
arXiv2012

Square function/non-tangential maximal function estimates and the dirichlet problem for non-symmetric elliptic operators

We consider divergence form elliptic operators L = - div A(x)\nabla, defined in the half space R^{n+1}_+, n \geq 2, where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation Lu = 0, and we then combine these estimates with the method of "ε-approximability" to show that L-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class A_\infty with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in L^p, for some p < \infty). Previously, these results had been known only in the case n = 1.

math.AP↗
arXiv2011

The Hidden Landscape of Localization

Wave localization occurs in all types of vibrating systems, in acoustics, mechanics, optics, or quantum physics. It arises either in systems of irregular geometry (weak localization) or in disordered systems (Anderson localization). We present here a general theory that explains how the system geometry and the wave operator interplay to give rise to a "landscape" that splits the system into weakly coupled subregions, and how these regions shape the spatial distribution of the vibrational eigenmodes. This theory holds in any dimension, for any domain shape, and for all operators deriving from an energy form. It encompasses both weak and Anderson localizations in the same mathematical frame and shows, in particular, that Anderson localization can be understood as a special case of weak localization in a very rough landscape.

cond-mat.dis-nn↗
arXiv2010

Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show that the known ranges of boundedness in $L^p$ for the heat semigroup and Riesz transform of $L$, are sharp. In particular, the heat semigroup $e^{-tL}$ need not be bounded in $L^p$ if $p\not\in [2n/(n+2),2n/(n-2)]$. Then we provide a complete description of {\it all} Sobolev spaces in which $L$ admits a bounded functional calculus, in particular, where $e^{-tL}$ is bounded. Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces associated to $L$, that serves the range of $p$ beyond $[2n/(n+2),2n/(n-2)]$. It includes, in particular, characterizations by the sharp maximal function and the Riesz transform (for certain ranges of $p$), as well as the molecular decomposition and duality and interpolation theorems.

math.FA↗
arXiv2009

Boundedness of the square function and rectifiability

Following a recent paper by X. Tolsa [JFA, 2008] we show that the finiteness of square function associated with the Riesz transforms with respect to Hausdorff measure $H^n$ ($n$ is interger) on a set $E$ implies that $E$ is rectifiable.

math.CA↗
arXiv2009

Square function and Riesz transform in non-integer dimensions

Following a recent paper by X. Tolsa and A. Ruiz de Villa [Non existence of principal values of signed Riesz transforms of non integer dimension, preprint], we show that the finiteness of square function associated with the Riesz transforms with respect to Hausdorff measure $H^s$ implies that $s$ is integer.

math.CA↗
arXiv2007

Hardy and BMO spaces associated to divergence form elliptic operators

Consider the second order divergence form elliptic operator $L$ with complex bounded coefficients. In general, the operators related to it (such as Riesz transform or square function) lie beyond the scope of the Calderón-Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some $L^p$ spaces. In this work we generalize the classical approach and develop a theory of Hardy and BMO spaces associated to $L$, which includes, in particular, molecular decomposition, maximal function characterization, duality of Hardy and BMO spaces, John-Nirenberg inequality, and allows to handle aforementioned operators.

math.AP↗