Svitlana Mayboroda

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

A. We consider divergence form elliptic operators L = − div A(x)∇, defined in the half space Rn+1 + , n ≥ 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 “!-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∞ with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in Lp, for some p < ∞). Previously, these results had been known only in the case n = 1.

Universal mechanism for Anderson and weak localization

Localization of stationary waves occurs in a large variety of vibrating systems, whether mechanical, acoustical, optical, or quantum. It is induced by the presence of an inhomogeneous medium, a complex geometry, or a quenched disorder. One of its most striking and famous manifestations is Anderson localization, responsible for instance for the metal-insulator transition in disordered alloys. Yet, despite an enormous body of related literature, a clear and unified picture of localization is still to be found, as well as the exact relationship between its many manifestations. In this paper, we demonstrate that both Anderson and weak localizations originate from the same universal mechanism, acting on any type of vibration, in any dimension, and for any domain shape. This mechanism partitions the system into weakly coupled subregions. The boundaries of these subregions correspond to the valleys of a hidden landscape that emerges from the interplay between the wave operator and the system geometry. The height of the landscape along its valleys determines the strength of the coupling between the subregions. The landscape and its impact on localization can be determined rigorously by solving one special boundary problem. This theory allows one to predict the localization properties, the confining regions, and to estimate the energy of the vibrational eigenmodes through the properties of one geometrical object. In particular, Anderson localization can be understood as a special case of weak localization in a very rough landscape.

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.

Rectifiability of harmonic measure

In the present paper we prove that for any open connected set

Boundedness of the square function and rectifiability

{\Omega\subset\mathbb{R}^{n+1}}

The Poisson Problem for the Lamé System on Low-dimensional Lipschitz Domains

Ω⊂Rn+1,

Fundamental matrices and Green matrices for non-homogeneous elliptic systems

{n\geq 1}