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Marius Mitrea

Publications and source records attributed to Marius Mitrea.

At least 37 records · Page 2Linked to original sources
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.

math.AP↗
arXiv2014

Square Function Estimates in Spaces of Homogeneous Type and on Uniformly Rectifiable Euclidean Sets

We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation results for $L^2$ square function estimates related to the analysis of integral operators that act on Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The inductive scheme is a natural application of the local $T(b)$ theorem and it implies the stability of $L^2$ square function estimates under the so-called big pieces functor. In particular, this analysis implies $L^p$ and Hardy space square function estimates for integral operators on uniformly rectifiable subsets of the Euclidean space.

math.AP↗
arXiv2014

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,Ω}$, the Krein--von Neumann extension of the perturbed Laplacian $-Δ+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(Ω)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Ω\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.

math.SP↗
arXiv2013

The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^\infty$ coefficients

We consider layer potentials associated to elliptic operators $Lu=-{\rm div}(A \nabla u)$ acting in the upper half-space $\mathbb{R}^{n+1}_+$ for $n\geq 2$, or more generally, in a Lipschitz graph domain, where the coefficient matrix $A$ is $L^\infty$ and $t$-independent, and solutions of $Lu=0$ satisfy interior estimates of De Giorgi/Nash/Moser type. A "Calderón-Zygmund" theory is developed for the boundedness of layer potentials, whereby sharp $L^p$ and endpoint space bounds are deduced from $L^2$ bounds. Appropriate versions of the classical "jump-relation" formulae are also derived. The method of layer potentials is then used to establish well-posedness of boundary value problems for $L$ with data in $L^p$ and endpoint spaces.

math.AP↗
arXiv2013

The Integrability of Negative Powers of the Solution of the Saint Venant Problem

We initiate the study of the finiteness condition $\int_Ωu(x)^{-β}\,dx\leq C(Ω,β)<+\infty$ where $Ω\subseteq{\mathbb{R}}^n$ is an open set and $u$ is the solution of the Saint Venant problem $Δu=-1$ in $Ω$, $u=0$ on $\partialΩ$. The central issue which we address is that of determining the range of values of the parameter $β>0$ for which the aforementioned condition holds under various hypotheses on the smoothness of $Ω$ and demands on the nature of the constant $C(Ω,β)$. Classes of domains for which our analysis applies include bounded piecewise $C^1$ domains in ${\mathbb{R}}^n$, $n\geq 2$, with conical singularities (in particular polygonal domains in the plane), polyhedra in ${\mathbb{R}}^3$, and bounded domains which are locally of class $C^2$ and which have (finitely many) outwardly pointing cusps. For example, we show that if $u_N$ is the solution of the Saint Venant problem in the regular polygon $Ω_N$ with $N$ sides circumscribed by the unit disc in the plane, then for each $β\in(0,1)$ the following asymptotic formula holds: % {eqnarray*} \int_{Ω_N}u_N(x)^{-β}\,dx=\frac{4^βπ}{1-β} +{\mathcal{O}}(N^{β-1})\quad{as}\,\,N\to\infty. {eqnarray*} % One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions $v$ satisfying $v(0)=0$, $\nabla v(0)=0$ and $Δv\geq c>0$.

math.AP↗
arXiv2013

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions II

The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $L^2(\Om; d^n x)$-realizations, $n\in\bbN$, $n\geq 2$, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$, where $$ Lu = - \sum_{j,k=1}^n\partial_j a_{j,k}\partial_k u. $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ ν\cdot A\nabla u + Θ\big[u\big|_{\partial\Om}\big]=0 \, \text{on} \, \partialΩ, $$ where $Θ$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, and $ν$ denotes the outward pointing normal unit vector on $\partial\Om$.

math.AP↗
arXiv2013

Heat Kernel Bounds for Elliptic Partial Differential Operators in Divergence Form with Robin-Type Boundary Conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{Θ, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$. In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions $L$ which act as $$ Lu = - \biggl(\sum_{j,k=1}^n\partial_j\bigg(\sum_{β= 1}^m a^{α,β}_{j,k}\partial_k u_β\bigg) \bigg)_{1\leqα\leq m}, \quad u=(u_1,...,u_m). $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ ν\cdot A D u + Θ\big[u\big|_{\partial \Om}\big] = 0 \, \text{on $\partial \Om$}, $$ where $Θ$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, $ν$ denotes the outward pointing normal unit vector on $\partial\Om$, and $Du:=\bigl(\partial_j u_α\bigr)_{\substack{1\leqα\leq m 1\leq j\leq n}}$. Assuming $Θ\geq 0$ in the scalar case $m=1$, we prove Gaussian heat kernel bounds for $L_{Θ, \Om}$ by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on $\partial \Om$. We also discuss additional zero-order potential coefficients $V$ and hence operators corresponding to the form sum $L_{Θ, \Om} + V$.

math.AP↗
arXiv2013

$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets

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 consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local $T(b)$ theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on the entire set. Extrapolation results for $L^p$ and Hardy space versions of these estimates are also established. Moreover, we prove square function estimates for integral operators associated with variable coefficient kernels, including the Schwartz kernels of pseudodifferential operators acting between vector bundles on subdomains with uniformly rectifiable boundaries on manifolds.

math.AP↗
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.

math.AP↗
arXiv2010

Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $Θ$ which give rise to self-adjoint Laplacians $-Δ_{Θ, Ω}$ in $L^2(Ω; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $Ω\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $Ω$, following an approach introduced by Filonov for this type of problems.

math.SP↗
arXiv2010

On Dirichlet-to-Neumann Maps and Some Applications to Modified Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators in $L^2(Ω; d^n x)$, $n=2,3$, where $Ω$ is an open set with a compact, nonempty boundary satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in $L^2(Ω; d^n x)$ to modified Fredholm perturbation determinants associated with operators in $L^2(\partialΩ; d^{n-1}σ)$, $n=2,3$. This leads to a two- and three-dimensional extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schrodinger operator on the half-line $(0,\infty)$ to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation.

math.SP↗
arXiv2010

On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators describing nonlocal interactions in $L^2(Ω; d^n x)$, $n\geq 2$, where $Ω$ is an open set with a compact, nonempty boundary satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of Fredholm perturbation determinants associated with operators in $L^2(Ω; d^n x)$ to Fredholm perturbation determinants associated with operators in $L^2(\partialΩ; d^{n-1}σ)$. This leads to an extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schrödinger operator on the half-line $(0,\infty)$, in the case of local interactions, to a Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation.

math.SP↗
arXiv2010

Boundary Data Maps for Schrodinger Operators on a Compact Interval

We provide a systematic study of boundary data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrodinger operators on a compact interval [0,R] with separated boundary conditions at 0 and R. Most of our results are formulated in the non-self-adjoint context. Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schrodinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrodinger operators corresponding to different (separated) boundary conditions, and a derivation of the Herglotz property of boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the special self-adjoint case.

math.SP↗
arXiv2010

The Krein-von Neumann Extension and its Connection to an Abstract Buckling Problem

We prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq εI_{\mathcal{H}}$ for some $ε>0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. In the concrete case where $S=\bar{-Δ|_{C_0^\infty(Ω)}}$ in $L^2(Ω; d^n x)$ for $Ω\subset\mathbb{R}^n$ an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian $S_K$ (i.e., the Krein--von Neumann extension of $S$), \[ S_K v = λv, \quad λ\neq 0, \] is in one-to-one correspondence with the problem of {\em the buckling of a clamped plate}, \[ (-Δ)^2u=λ(-Δ) u \text{in} Ω, \quad λ\neq 0, \quad u\in H_0^2(Ω), \] where $u$ and $v$ are related via the pair of formulas \[ u = S_F^{-1} (-Δ) v, \quad v = λ^{-1}(-Δ) u, \] with $S_F$ the Friedrichs extension of $S$. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.).

math.SP↗
arXiv2010

Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains

We study spectral properties for $H_{K,Ω}$, the Krein--von Neumann extension of the perturbed Laplacian $-Δ+V$ defined on $C^\infty_0(Ω)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Ω\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula \[ #\{j\in\mathbb{N} | λ_{K,Ω,j}\leqλ\} = (2π)^{-n} v_n |Ω| λ^{n/2}+O\big(λ^{(n-(1/2))/2}\big) {as} λ\to\infty, \] where $v_n=π^{n/2}/ Γ((n/2)+1)$ denotes the volume of the unit ball in $\mathbb{R}^n$, and $λ_{K,Ω,j}$, $j\in\mathbb{N}$, are the non-zero eigenvalues of $H_{K,Ω}$, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein--von Neumann extension of $-Δ+V$ defined on $C^\infty_0(Ω)$) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980's. Our work builds on that of Grubb in the early 1980's, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.

math.SP↗