Mihalis Mourgoglou

activity

20102026

most citedRectifiability of harmonic measure

61 citations · 237 across the 33 of their papers we have counts for

collaborators

49 papers

math.AP2026

The Dirichlet problem as the boundary of the Poisson problem: A sharp approximation result

Mihalis Mourgoglou, Bruno Poggi

On a bounded domain $Ω \subset R^{n + 1}$ , $n \geq 2$ , satisfying the corkscrew condition and with Ahlfors regular boundary, we characterize the dual space to the space ${\bf N}_…

math.CA2025

Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries

Jonas Azzam, Mihalis Mourgoglou, Michele Villa

Given a corkscrew domain with uniformly rectifiable boundary, we construct a surjective trace map onto the $L^{p}$ Hajlasz-Sobolev space on the boundary from the space of functions o…

math.AP2025

Layer potentials for elliptic operators with DMO-type coefficients: big pieces $T b$ theorem, quantitative rectifiability, and free boundary problems

Andrea Merlo, Mihalis Mourgoglou, Carmelo Puliatti

For $n \geq 2$ , we consider the operator $L_{A} = - div (A (\cdot) \nabla)$ , where $A$ is a uniformly elliptic $(n + 1) \times (n + 1)$ matrix with variable coefficients, a Radon mea…

math.AP2024★ 1 cited

Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains

Mihalis Mourgoglou, Xavier Tolsa

Let $Ω \subset R^{n + 1}$ be a bounded chord-arc domain, let $L = - div A \nabla$ be an elliptic operator in $Ω$ associated with a matrix $A$ having Dini mean osci…

math.CA2023

Quantitative differentiability on uniformly rectifiable sets

Jonas Azzam, Mihalis Mourgoglou, Michele Villa

We prove $L^{p}$ quantitative differentiability estimates for functions defined on uniformly rectifiable subsets of the Euclidean space. More precisely, we show that a Dorronsoro-typ…

math.AP2023

Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

Josep M. Gallegos, Mihalis Mourgoglou, Xavier Tolsa

Let $Ω \subset R^{n + 1}$ , $n \geq 2$ , be an open set satisfying the corkscrew condition with $n$ -Ahlfors regular boundary $\partial Ω$ , but without any connectivity assumption.…

math.AP2023

Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with $L^{\infty}$ coefficients

Mihalis Mourgoglou, Thanasis Zacharopoulos

Let $Ω \subset R^{n + 1}$ , $n \geq 1$ , be an open set with $s$ -Ahlfors regular boundary $\partial Ω$ , for some $s \in (0, n]$ , such that either $s = n$ and $Ω$ is a corkscrew doma…

math.MG2022

On the density problem in the parabolic space

Andrea Merlo, Mihalis Mourgoglou, Carmelo Puliatti

In this work we extend many classical results concerning the relationship between densities, tangents and rectifiability to the parabolic spaces, namely $R^{n + 1}$ equipped…

math.AP2022

Solvability of the Poisson-Dirichlet problem with interior data in $L^{p^{'}}$ -Carleson spaces and its applications to the $L^{p}$ -regularity problem

Mihalis Mourgoglou, Bruno Poggi, Xavier Tolsa

We prove that the $L^{p^{'}}$ -solvability of the homogeneous Dirichlet problem for an elliptic operator $L = - div A \nabla$ with real and merely bounded coefficients is equ…

math.AP2021

$L^{2}$ -boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

Alejandro Molero, Mihalis Mourgoglou, Carmelo Puliatti +1

We consider a uniformly elliptic operator $L_{A}$ in divergence form associated with an $(n + 1) \times (n + 1)$ -matrix $A$ with real, merely bounded, and possibly non-symmetric coefficien…

math.AP2021★ 1 cited

The regularity problem for the Laplace equation in rough domains

Mihalis Mourgoglou, Xavier Tolsa

Let $Ω \subset R^{n + 1}$ , $n \geq 2$ , be a bounded open and connected set satisfying the corkscrew condition with uniformly $n$ -rectifiable boundary. In this paper we study t…

math.AP2020

Blow-ups of caloric measure in time varying domains and applications to two-phase problems

Mihalis Mourgoglou, Carmelo Puliatti

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in $R^{n + 1}$ , $n \geq 2$ , a…

math.CA2019★ 58 cited

Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$ -solvability of the Dirichlet problem

Jonas Azzam, Steve Hofmann, José María Martell +2

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- $A_{\infty}$ property) of harmonic measure with respect to surface measure, on the b…

math.AP2019

Regularity theory and Green's function for elliptic equations with lower order terms in unbounded domains

Mihalis Mourgoglou

We consider elliptic operators in divergence form with lower order terms of the form $Lu = -$ div $\nabla u + b u) - c \nabla u - d u$ , in an open set $Ω \subset R^{n}$ , $n \geq 3$ , with p…

math.AP2018

Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$ solvability of the Dirichlet problem. Part II

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

Let $Ω \subset R^{n + 1}$ be an open set with $n$ -AD-regular boundary. In this paper we prove that if the harmonic measure for $Ω$ satisfies the so-called weak- $A_{\infty}$ condi…

math.CA2018

Failure of $L^{2}$ boundedness of gradients of single layer potentials for measures with zero low density

José M. Conde-Alonso, Mihalis Mourgoglou, Xavier Tolsa

Consider a totally irregular measure $μ$ in $R^{n + 1}$ , that is, the upper density $lim sup_{r \to 0} \frac{μ ( B ( x , r ))}{( 2 r ) ^{n}}$ is positive $μ$ -a.e.\ in $R^{n + 1}$ , an…

math.AP2017

A two-phase free boundary problem for harmonic measure and uniform rectifiability

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

We assume that $Ω_{1}, Ω_{2} \subset R^{n + 1}$ , $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their bound…

math.AP2017★ 8 cited

Tangent measures of elliptic harmonic measure and applications

Jonas Azzam, Mihalis Mourgoglou

Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its h…

math.CA2016★ 29 cited

Uniform rectifiability from Carleson measure estimates and $ε$ -approximability of bounded harmonic functions

John Garnett, Mihalis Mourgoglou, Xavier Tolsa

Let $Ω \subset R^{n + 1}$ , $n \geq 1$ , be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial Ω$ is uniformly $n$ -rectifiable if every bo…

math.CA2016★ 18 cited

Uniform rectifiability, elliptic measure, square functions, and $ε$ -approximability via an ACF monotonicity formula

Jonas Azzam, John Garnett, Mihalis Mourgoglou +1

Let $Ω \subset R^{n + 1}$ , $n \geq 2$ , be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ i…

math.CA2016

On a two-phase problem for harmonic measure in general domains

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa +1

We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rec…

math.CA2016★ 2 cited

Non-homogeneous square functions on general sets: suppression and big pieces methods

Henri Martikainen, Mihalis Mourgoglou, Emil Vuorinen

We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local $T b$ theorems. The setting is new: we consider conical square f…

math.CA2016★ 3 cited

The one-phase problem for harmonic measure in two-sided NTA domains

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

We show that if $Ω \subset R^{3}$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $VMO (σ)$ , where $σ$ is th…

math.CA2016

Absolute continuity of harmonic measure for domains with lower regular boundaries

Murat Akman, Jonas Azzam, Mihalis Mourgoglou

We study absolute continuity of harmonic measure with respect to surface measure on domains $Ω$ that have large complements. We show that if $Γ \subset R^{d + 1}$ is $d$ -Ahlf…

The Dirichlet problem as the boundary of the Poisson problem: A sharp approximation result

Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries

Layer potentials for elliptic operators with DMO-type coefficients: big pieces Tb theorem, quantitative rectifiability, and free boundary problems

Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains

Quantitative differentiability on uniformly rectifiable sets

Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with L∞ coefficients

On the density problem in the parabolic space

Solvability of the Poisson-Dirichlet problem with interior data in Lp′-Carleson spaces and its applications to the Lp-regularity problem

L2-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

The regularity problem for the Laplace equation in rough domains

Blow-ups of caloric measure in time varying domains and applications to two-phase problems

Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem

Regularity theory and Green's function for elliptic equations with lower order terms in unbounded domains

Harmonic measure and quantitative connectivity: geometric characterization of the Lp solvability of the Dirichlet problem. Part II

Failure of L2 boundedness of gradients of single layer potentials for measures with zero low density

A two-phase free boundary problem for harmonic measure and uniform rectifiability

Tangent measures of elliptic harmonic measure and applications

Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions

Uniform rectifiability, elliptic measure, square functions, and ε-approximability via an ACF monotonicity formula

On a two-phase problem for harmonic measure in general domains

Non-homogeneous square functions on general sets: suppression and big pieces methods

The one-phase problem for harmonic measure in two-sided NTA domains

Absolute continuity of harmonic measure for domains with lower regular boundaries

Layer potentials for elliptic operators with DMO-type coefficients: big pieces $T b$ theorem, quantitative rectifiability, and free boundary problems

Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with $L^{\infty}$ coefficients

Solvability of the Poisson-Dirichlet problem with interior data in $L^{p^{'}}$ -Carleson spaces and its applications to the $L^{p}$ -regularity problem

$L^{2}$ -boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$ -solvability of the Dirichlet problem

Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$ solvability of the Dirichlet problem. Part II

Failure of $L^{2}$ boundedness of gradients of single layer potentials for measures with zero low density

Uniform rectifiability from Carleson measure estimates and $ε$ -approximability of bounded harmonic functions

Uniform rectifiability, elliptic measure, square functions, and $ε$ -approximability via an ACF monotonicity formula