arXiv subjects

Irina Mitrea

Publications and source records attributed to Irina Mitrea.

9 recordsLinked to original sources

arXiv2026

The Solution to Agmon's Conjecture for the polylaplacian

In his 1957 paper on the Dirichlet problem for higher-order elliptic equations in the plane, Agmon conjectured that there should exist higher-order equations in dimensions greater than two for which a potential-theoretic approach based on multi-layer integral operators can be successfully implemented. We show that this is indeed the case for the polyharmonic operator $Δ^m$ of arbitrary order $m\in\mathbb{N}$ in $\mathbb{R}^n$, $n\geq 2$, even in geometrically rough domains permitting singularities beyond the scope of the existing theory of higher-order elliptic boundary value problems. Central to our approach is a new genre of multi-layer singular integral operators, which play for $Δ^m$ the role assumed by the classical harmonic double layer potential in the treatment of the Dirichlet problem for harmonic functions.

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arXiv2026

The Green Function for Elliptic Systems in the Upper-Half Space

Let $L$ be a second-order, homogeneous, constant (complex) coefficient elliptic system in ${\mathbb{R}}^n$. The goal of this article is provide a qualitative and quantitative study of the nature of the Green function associated with the system $L$ in the upper-half space. Starting with a definition of the Green function which brings forth the minimal features which identify this object uniquely, we establish optimal nontangential maximal function estimates and regularity results up to the boundary for the said Green function. The main tools employed in the proof include the Agmon-Douglis-Nirenberg construction of a Poisson kernel for the system $L$, the Agmon-Douglis-Nirenberg a priori regularity estimates near the boundary, and the brand of Divergence Theorem from the book Geometric Harmonic Analysis Vol. I by the last three authors of this paper in which the boundary trace of the corresponding vector field is taken in nontangential pointwise sense.

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arXiv2025

Riesz Transform Characterizations of $H^1$ and {\rm BMO} on Ahlfors Regular Sets with Small Oscillations

We employ the Riesz transform as a means for describing geometric properties of sets in ${\mathbb{R}}^n$, and study the extent to which they can be used to characterize function spaces defined on said sets. In particular, characterizations of the end-point spaces on the Lebesgue scale $L^p$ with $1<p<\infty$, namely the Hardy space $H^1$ and the John-Nirenberg space {\rm BMO}, are produced in terms of the Riesz transforms on Ahlfors regular sets in ${\mathbb{R}}^n$ with small oscillations (quantified in terms of the {\rm BMO} nature of the outward unit normal). These generalize the celebrated results of C.~Fefferman and E.~Stein in the flat Euclidean setting.

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arXiv2019

Fatou-Type Theorems and Boundary Value Problems for Elliptic Systems in the Upper Half-Space

We survey recent progress in a program aimed at proving general Fatou-type results and establishing the well-posedness of a variety of boundary value problems in the upper half-space ${\mathbb{R}}^n_{+}$ for second-order, homogeneous, constant complex coefficient, elliptic systems $L$, formulated in a manner that emphasizes pointwise nontangential boundary traces of the null-solutions of $L$ in ${\mathbb{R}}^n_{+}$.

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arXiv2018

The BMO-Dirichlet problem for elliptic systems in the upper-half space and quantitative characterizations of VMO

We prove that for any homogeneous, second order, constant complex coefficient elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with boundary data in BMO is well-posed in the class of functions $u$ with $dμ_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We establish a Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions $u$ of such systems satisfying the said Carleson measure condition. These imply that BMO can be characterized as the collection of nontangential pointwise traces of smooth null-solutions $u$ to the elliptic system $L$ with the property that $μ_u$ is a Carleson measure. We establish a regularity result for the BMO-Dirichlet problem in the upper-half space: the nontangential pointwise trace of any given smooth null-solutions of $L$ satisfying the above Carleson measure condition belongs to Sarason's space VMO if and only if $μ_u$ satsifies a vanishing Carleson measure condition. Moreover, we obtain the well-posedness of the Dirichlet problems when the boundary data are prescribed in Morrey-Campanato and solutions are required to satisfy a vanishing Carleson measure condition of fractional order. As a consequence, we characterize the space VMO as the closure in BMO of classes of smooth functions contained in BMO within which uniform continuity may be suitably quantified (such as the class of smooth functions satisfying a Hölder or Lipschitz condition), improving Sarason's classical result describing VMO as the closure in BMO of the space of uniformly continuous functions with bounded mean oscillations. Finally, we show that any Calderón-Zygmund operator $T$ satisfying $T(1)=0$ extends as a bounded linear operator in $\mathrm{VMO}$, and characterize the membership to $\mathrm{VMO}$ via the action of various classes of singular integral operators.

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arXiv2016

The Dirichlet problem for elliptic systems with data in Köthe function spaces

We show that the boundedness of the Hardy-Littlewood maximal operator on a Köthe function space ${\mathbb{X}}$ and on its Köthe dual ${\mathbb{X}}'$ is equivalent to the well-posedness of the $\mathbb{X}$-Dirichlet and $\mathbb{X}'$-Dirichlet problems in $\mathbb{R}^{n}_{+}$ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space $H^1$, and the Beurling-Hardy space ${\rm HA}^p$ for $p\in(1,\infty)$. Based on the well-posedness of the $L^p$-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.

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arXiv2014

On the $L^p$-Poisson semigroup associated with elliptic systems

We study the infinitesimal generator of the Poisson semigroup in $L^p$ associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the $L^p$-based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may adapted to treat the case of higher order systems in graph Lipschitz domains.

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arXiv2014

The higher order regularity Dirichlet problem for elliptic systems in the upper-half space

We identify a large class of constant (complex) coefficient, second order elliptic systems for which the Dirichlet problem in the upper-half space with data in $L^p$-based Sobolev spaces, $1<p<\infty$, of arbitrary smoothness $\ell$, is well-posed in the class of functions whose nontangential maximal operator of their derivatives up to, and including, order $\ell$ is $L^p$-integrable. This class includes all scalar, complex coefficient elliptic operators of second order, as well as the Lamé system of elasticity, among others.

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arXiv2012

Extending Sobolev Functions with Partially Vanishing Traces from Locally (epsilon,delta)-Domains and Applications to Mixed Boundary Problems

We prove that given any positive integer $k$, for each open set $Ω$ and any closed subset $D$ of its closure such that $Ω$ is locally an (epsilon,delta)-domain near points in the boundary of $Ω$ not contained in $D$ there exists a linear and bounded extension operator $E$ mapping, for each $p\in[1,\infty]$, the space $W^{k,p}_D(Ω)$ into $W^{k,p}_D({\mathbb{R}}^n)$. Here, with $O$ denoting either $Ω$ or the entire ambient, the space $W^{k,p}_D(O)$ is defined as the completion in the classical Sobolev space $W^{k,p}(O)$ of compactly supported smooth functions whose supports are disjoint from $D$. In turn, this result is used to develop a functional analytic theory for the class $W^{k,p}_D(Ω)$ (including intrinsic characterizations, boundary traces and extensions results, interpolation theorems, among other things) which is then employed in the treatment of mixed boundary value problems formulated in locally (epsilon,delta)-domains.

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