Marius Mitrea
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View article: On the <i>L</i> <sup> <i>p</i> </sup> -boundedness of Calderón-Zygmund operators
On the <i>L</i> <sup> <i>p</i> </sup> -boundedness of Calderón-Zygmund operators Open
The main result in this paper is that, for singular integral operators associated with standard kernels, local L 1 -estimates imply global L p -estimates for every p ∈ (1, ∞ ). When combined with the result of Melnikov-Verdera, this yields…
View article: Sharp Boundary Trace Theory and Schrödinger Operators on Bounded Lipschitz Domains
Sharp Boundary Trace Theory and Schrödinger Operators on Bounded Lipschitz Domains Open
We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is do…
View article: Navier-Stokes equations on Lipschitz domains in Riemannian manifolds
Navier-Stokes equations on Lipschitz domains in Riemannian manifolds Open
The Navier-Stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous, incompressible fluid. One ingredient in the analysis of this system is the stationary, linear system known as the Stokes system, a bo…
View article: Extension and Representation of Divergence-free Vector Fields on Bounded Domains
Extension and Representation of Divergence-free Vector Fields on Bounded Domains Open
Let Ω ⊂ Rn be a bounded, connected domain, with b + 1 boundary components, ∂Ω = Γ0∪...∪Γb. Say O0,...,Ob are the connected components of Rn\nΩ, O0 being the unbounded component, and Γj = ∂Oj . If b > 0, pick yj ∊ Oj ; 1 ≤ j ≤ b, and set (1…
View article: Boundary Layer Methods for Lipschitz Domains in Riemannian Manifolds
Boundary Layer Methods for Lipschitz Domains in Riemannian Manifolds Open
We extend to the variable coefficient case boundary layer techniques that have been successful in the treatment of the Laplace equation and certain other constant coefficient elliptic partial differential equations on Lipschitz domains in …
View article: Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem
Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem Open
We continue a program to develop layer potential techniques for PDE on Lipschitz domains in Riemannian manifolds. Building on Lp and Hardy space estimates established in previous papers, here we establish Sobolev and Besov space estimates …
View article: Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains
Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains Open
In the first part of this paper we give intrinsic characterizations of the classes of Lipschitz and C1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domai…
View article: Lipschitz Domains, Domains with Corners, and the Hodge Laplacian
Lipschitz Domains, Domains with Corners, and the Hodge Laplacian Open
We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators’ domains a…
View article: A Sharp Divergence Theorem with Nontangential Traces
A Sharp Divergence Theorem with Nontangential Traces Open
PerspectiveThe Fundamental Theorem of Calculus, one of the most spectacular scientific achievements, stands as beautiful, powerful, and relevant today as it did more than three centuries ago when it first emerged onto the mathematical scen…
View article: Erratum to “Non Self Adjoint Operators, Infinite Determinants, and Some Applications,” Russ. J. Math. Phys. 12, 443–471 (2005)
Erratum to “Non Self Adjoint Operators, Infinite Determinants, and Some Applications,” Russ. J. Math. Phys. 12, 443–471 (2005) Open
The additional hypothesis: each connected component of C\ σ e (H 0 ) contains a point of ρ(H), and each connected component of C\ σ e (H) contains a point of ρ(H 0 ), should be made in Theorem 4.5, Remark 4.6, Theorem 5.4, and Remark 5.5.F…
View article: The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO
The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO Open
We prove that for any homogeneous, second-order, constant complex coefficient elliptic system L in ℝ, the Dirichlet problem in ℝ with boundary data in BMO.ℝ/ is well-posed in the class of functions u for which the Littlewood-Paley measure …
View article: The $L^p$ Dirichlet boundary problem for second order Elliptic Systems with rough coefficients
The $L^p$ Dirichlet boundary problem for second order Elliptic Systems with rough coefficients Open
Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, i…
View article: Coupling of symmetric operators and the third Green identity
Coupling of symmetric operators and the third Green identity Open
The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension T of a symmetric operator S in a Hilbert space $$\mathfrak {H}$$ H , employing the technique of quasi boundary tri…
View article: The Dirichlet problem for elliptic systems with data in Köthe function spaces
The Dirichlet problem for elliptic systems with data in Köthe function spaces Open
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 …
View article: 𝐿^{𝑝}-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets
𝐿^{𝑝}-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets Open
We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local T (b) theorem and applying it to show that such estimates are stable under the so-called big pieces functor.More generally, we c…
View article: Coupling of symmetric operators and the third Green identity
Coupling of symmetric operators and the third Green identity Open
The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension $T$ of a symmetric operator $S$ in a Hilbert space $\mathfrak H$, employing the technique of quasi boundary triple…
View article: A bound for the eigenvalue counting function for Krein--von Neumann and\n Friedrichs extensions
A bound for the eigenvalue counting function for Krein--von Neumann and\n Friedrichs extensions Open
For an arbitrary open, nonempty, bounded set $\\Omega \\subset \\mathbb{R}^n$,\n$n \\in \\mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider\nthe closed, strictly positive, higher-order differential operator $A_{\\Omega,…
View article: Lp−BOUNDS FOR THE RIESZ TRANSFORMS ASSOCIATED WITH THE HODGE LAPLACIAN IN LIPSCHITZ SUBDOMAINS OF RIEMANNIAN MANIFOLDS
Lp−BOUNDS FOR THE RIESZ TRANSFORMS ASSOCIATED WITH THE HODGE LAPLACIAN IN LIPSCHITZ SUBDOMAINS OF RIEMANNIAN MANIFOLDS Open
We prove Lp-bounds for the Riesz transforms d/ √−∆, δ/√− ∆ associated with the Hodge-Laplacian ∆ = −δd − dδ equipped with absolute and relative boundary conditions in a Lipschitz subdomain Ω of (smooth) Riemannian manifoldM, for p in a cer…
View article: The method of layer potentials in<i>L<sup>p</sup></i>and endpoint spaces for elliptic operators with<i>L<sup>∞</sup></i>coefficients
The method of layer potentials in<i>L<sup>p</sup></i>and endpoint spaces for elliptic operators with<i>L<sup>∞</sup></i>coefficients Open
We consider layer potentials associated to elliptic operators $Lu=-{\\rm\ndiv}(A \\nabla u)$ acting in the upper half-space $\\mathbb{R}^{n+1}_+$ for\n$n\\geq 2$, or more generally, in a Lipschitz graph domain, where the\ncoefficient matri…
View article: Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus
Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus Open
We show that operators of layer potential type on surfaces that are locally graphs of Lipschitz functions with gradients in vmo are equal, modulo compacts, to pseudodifferential operators (with rough symbols), for which a symbol calculus i…
View article: The Krein-von Neumann Realization of Perturbed Laplacians on Bounded Lipschitz Domains
The Krein-von Neumann Realization of Perturbed Laplacians on Bounded Lipschitz Domains Open
In this paper we study the self-adjoint Krein-von Neumann realization $A_K$ of the perturbed Laplacian $-Δ+V$ in a bounded Lipschitz domain $Ω\subset\mathbb{R}^n$. We provide an explicit and self-contained description of the domain of $A_K…
View article: The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains
The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains Open
We investigate the Navier-Stokes equations in a suitable functional setting, in a three-dimensional bounded Lipschitz domain Ω, equipped with “free boundary ” conditions. In this context, we employ the Fujita-Kato method and prove the exis…