arXiv subjects

Victor Nistor

Publications and source records attributed to Victor Nistor.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness of a generalized Stokes operator on domains with cylindrical ends via layer-potentials

We study the \emph{generalized Stokes operator} \begin{equation*} \bsXi \ede \bsXi _{V,V_0} \ede \left(\begin{array}{ccc} \bsL + V & \nabla \\ \nabla^* & -V_0 \end{array}\right) \end{equation*} on a \emph{domain with straight cylindrical ends} $\Omega$ using \emph{the method of layer potentials} on $M \supset \Omega$. The operator $\bsXi_{0, 0}$ is the classical Stokes operator. Under suitable positivity assumptions on $V$ and $V_{0}$, we prove that $\bsXi$ is Fredholm. This allows us then to define the single- and double-layer potentials $\bsS$ and $\frac12 + \bsK$. Under further positivity assumptions, we prove that $\bsS$ and $\frac12 + \bsK$ are also Fredholm. Under slightly stronger assumptions on $V$ and $V_{0}$, we prove \emph{the invertibility} of the operators $\bsXi$, $\bsS$, and $\frac12 + \bsK$. The invertibility of these operators leads to \emph{well-posedness results} for the associated (linear) Stokes boundary value problem with Dirichlet boundary conditions on $\Omega$. The proofs of these results required us to develop many related tools. In particular, we develop an ``algebra tool kit'' to deal with \emph{limit and jump relations of layer potentials.} We also develop Green formulas and energy estimates for our generalized Stokes operator $\bsXi$ on manifolds with straight cylindrical ends, which requires a careful geometric study of the related differential operators, such as the deformation operator $\Def$. For completeness, we review suitable classes of pseudodifferential operators on manifolds with straight cylindrical ends that were studied in some previous papers of ours (including ``The Stokes operator on manifolds with cylindrical ends,'' J. Diff. Equations, 2024). As an application, we prove the well-posedness result for the Dirichlet problem for the generalized Navier-Stokes system with small data on a domain with cylindrical ends.

math.AP↗

arXiv2025

Well-posedness of a generalized Stokes operator on smooth bounded domains via layer-potentials

We prove the invertibility of the relevant single and double layer potentials associated to some generalizations of the Stokes operator on bounded domains. In order to do that, we first develop an ``algebra tool kit'' to deal with limit and jump relations of layer operators. We do that first on $\mathbb{R}^{n}$ for operators acting on a distribution supported on $\{x_{n} = 0\}$ and then in general on (possibly non-compact manifolds). We use these results to study the limit and jump relations of the layer potential operators associated to our generalized Stokes operators. In turn, we then use these results to prove the Fredholm property of single and double layer potentials of the generalized Stokes operator and even their invertibility when the auxiliary potentials satisfy suitable non-vanishing conditions. As an application, we obtain well-posedness results.

math.AP↗

arXiv2024

Schr\"odinger operators with non-integer power-law potentials and Lie-Rinehart algebras

We study Schr\"odinger operators $H:= -\Delta + V$ with potentials $V$ that have power-law growth (not necessarily polynomial) at 0 and at $\infty$ using methods of Lie theory (Lie-Rinehart algebras) and microlocal analysis. More precisely, we show that $H$ is ''generated'' in a certain sense by an explicit Lie-Rinehart algebra. This allows then to construct a suitable (microlocal) calculus of pseudodifferential operators that provides further properties of $H$. Classically, this microlocal analysis method was used to study $H$ when the power-laws describing the potential $V$ have integer exponents. Thus, the main point of this paper is that this integrality condition on the exponents is not really necessary for the microlocal analysis method to work. While we consider potentials following (possibly non-integer) power-laws both at the origin and at infinity, our results extend right away to potentials having power-law singularities at several points. The extension of the classical microlocal analysis results to potentials with non-integer power-laws is achieved by considering the setting of Lie-Rinehart algebras and of the continuous family groupoids integrating them. (The classical case relies instead on Lie algebroids and Lie groupoids.)

math.DG↗

arXiv2024

Uniform estimates for a family of Poisson problems: `rounding off the corners'

We prove \emph{uniform solvability estimates} for certain families of elliptic problems posed in a bounded family of domains (for example, a sequence that converges to another domain). We provide uniform estimates both in weighted and in usual Sobolev spaces. When the limit domain is a \emph{polygon} and the other domains are smooth, our results amount to rounding off'' the corners of the limit domain. The technique of proof is based on a suitable conformal modification of the metric, which makes the union of the domains a manifold with boundary and relative bounded geometry.

math.AP↗

arXiv2023

Polynomial bounds for the solutions of parametric transmission problems on smooth, bounded domains

We consider a \emph{family} $(P_\omega)_{\omega \in \Omega}$ of elliptic second order differential operators on a domain $U_0 \subset \mathbb{R}^m$ whose coefficients depend on the space variable $x \in U_0$ and on $\omega \in \Omega,$ a probability space. We allow the coefficients $a_{ij}$ of $P_\omega$ to have jumps over a fixed interface $\Gamma \subset U_0$ (independent of $\omega \in \Omega$). We obtain polynomial in the norms of the coefficients estimates on the norm of the solution $u_\omega$ to the equation $P_\omega u_\omega = f$ with transmission and mixed boundary conditions (we consider ``sign-changing'' problems as well). In particular, we show that, if $f$ and the coefficients $a_{ij}$ are smooth enough and follow a log-normal-type distribution, then the map $\Omega \ni \omega \to \|u_\omega\|_{H^{k+1}(U_0)}$ is in $L^p(\Omega)$, for all $1 \le p < \infty$. The same is true for the norms of the inverses of the resulting operators. We expect our estimates to be useful in Uncertainty Quantification.

math.AP↗

arXiv2023

Analysis on noncompact manifolds and Index Theory: Fredholm conditions and Pseudodifferential operators

We provide Fredholm conditions for compatible differential operators on certain Lie manifolds (that is, on certain possibly non-compact manifolds with nice ends). We discuss in more detail the case of manifolds with cylindrical, hyperbolic, and Euclidean ends, which are all covered by particular instances of our results. We also discuss applications to Schr\"odinger operators with singularities of the form r^{-2\gamma}$, $\gamma \in \RR_+$.

math.AP↗

arXiv2023

Layer potentials and essentially translation invariant pseudodifferential operators on manifolds with cylindrical ends

Motivated by the study of layer potentials on manifolds with straight conical or cylindrical ends, we introduce and study two classes (or calculi) of pseudodifferential operators defined on manifolds with cylindrical ends: the class of pseudodifferential operators that are ``translation invariant at infinity'' and the class of ``essentially translation invariant operators.'' These are ``minimal'' classes of pseudodifferential operators containing the layer potential operators of interest. Both classes are close to the $b$-calculus considered by Melrose and Schulze and to the $c$-calculus considered by Melrose and Mazzeo-Melrose. Our calculi, however, are different and, while some of their properties follow from those of the $b$- or $c$-calculi, many of their properties do not. In particular, we prove that the ``essentially translation invariant calculus'' is spectrally invariant, a property not enjoyed by the ``translation invariant at infinity'' calculus or the $b$-calculus. For our calculi, we provide easy, intuitive proofs of the usual properties: stability for products and adjoints, mapping and boundedness properties for operators acting between Sobolev spaces, regularity properties, existence of a quantization map, topological properties of our algebras, and the Fredholm property. Since our applications will be to the Stokes operator, we systematically work in the setting of \ADN-elliptic operators. We also show that our calculi behave well with respect to restrictions to (suitable) submanifolds, which is crucial for our applications to layer potential operators.

math.AP↗

arXiv2023

The $L^2$-unique continuation property on manifolds with bounded geometry and the deformation operator

A differential operator $T$ satisfies the $L^2$-unique continuation property if every $L^2$-solution of $T$ that vanishes on an open subset vanishes identically. We study the $L^2$-unique continuation property of an operator $T$ acting on a manifold with bounded geometry. In particular, we establish some connections between this property and the regularity properties of $T$. As an application, we prove that the deformation operator on a manifold with bounded geometry satisfies regularity and $L^2$-unique continuation properties. As another application, we prove that suitable elliptic operators are invertible (Hadamard well-posedness). Our results apply to compact manifolds, which have bounded geometry.

math.AP↗

arXiv2021

Approximate Solutions to Second-Order Parabolic Equations: evolution systems and discretization

We study the discretization of a linear evolution partial differential equation when its Green function is known. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method (DTCM) that we developed in J. Math. Phys. (2010). This approximation for short time, when combined with a bootstrap argument, gives an approximate solution on any fixed time interval within any prescribed tolerance.

math.NA↗

arXiv2020

A regularity result for the bound states of $N$-body Schr\"odinger operators: Blow-ups and Lie manifolds

We prove regularity estimates in weighted Sobolev spaces for the $L^2$-eigenfunctions of Schr\"odinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual $N$-body Hamiltonians with Coulomb-type singular potentials are covered by our result: in that case, the weight is $\delta_{\mathcal{F}}(x) := \min \{ d(x, \bigcup \mathcal{F}), 1\}$, where $d(x, \bigcup \mathcal{F})$ is the usual euclidean distance to the union $\bigcup\mathcal{F}$ of the set of collision planes $\bigcup\mathcal{F}$. The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification $\overline{X}$ of the underlying space $X$ and we first blow-up the spheres $\mathbb{S}_Y \subset \mathbb{S}_X$ at infinity of the collision planes $Y \in \bigcup\mathcal{F}$ to obtain the Georgescu-Vasy compactification. Then we blow-up the collision planes $\bigcup\mathcal{F}$. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher order differential operators, to certain classes of pseudodifferential operators, and to matrices of scalar operators.

math.AP↗

arXiv2020

Fredholm conditions for operators invariant with respect to compact Lie group actions

Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P \in ψ^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i \to M$, $i = 0,1$, and let $α$ be an irreducible representation of the group $G$. Then $P$ induces a map $π_α(P) : H^s(M; E_0)_α\to H^{s-m}(M; E_1)_α$ between the $α$-isotypical components. We prove that the map $π_α(P)$ is Fredholm if, and only if, $P$ is {\em transversally $α$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_i$.

math.FA↗

arXiv2020

Sobolev spaces and $\nabla$-differential operators on manifolds I: basic properties and weighted spaces

We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically denoted $\nabla$). These concepts arise naturally from Partial Differential Equations, including some that are formulated on plain Euclidean domains, such as the weighted Sobolev spaces used to study PDEs on singular domains. We prove several basic properties of the $\nabla$-Sobolev spaces and of the $\nabla$-differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and independence of the $\nabla$-Sobolev spaces on the choices of the connection $\nabla$ with respect to totally bounded perturbations. We introduce a {\em Fréchet finiteness condition} (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our $\nabla$-Sobolev spaces and of our $\nabla$-differential operators. We examine in more detail the particular case of domains in the Euclidean space, including the case of weighted Sobolev spaces. We also introduce and study the notion of a {\em $\nabla$-bidifferential} operator (a bilinear version of differential operators), obtaining results similar to those obtained for $\nabla$-differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

math.AP↗

arXiv2019

Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $Γ$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P \in ψ^m(M; E_0, E_1)$ be a $Γ$-invariant, classical, pseudodifferential operator acting between sections of two $Γ$-equivariant vector bundles $E_0$ and $E_1$. Let $α$ be an irreducible representation of the group $Γ$. Then $P$ induces by restriction a map $π_α(P) : H^s(M; E_0)_α\to H^{s-m}(M; E_1)_α$ between the $α$-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map $π_α(P)$ to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when $Γ$ is finite abelian. We prove then that the restriction $π_α(P)$ is Fredholm if, and only if, $P$ is "$α$-elliptic", a condition defined in terms of the principal symbol of $P$. If $P$ is elliptic, then $P$ is also $α$-elliptic, but the converse is not true in general. However, if $Γ$ acts freely on a dense open subset of $M$, then $P$ is $α$-elliptic for the given fixed $α$ if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra $ψ^{m}(M; E)^Γ$ of classical, $Γ$-invariant pseudodifferential operators acting on sections of the vector bundle $E \to M$ and of the structure of its restrictions to the isotypical components of $Γ$. These structures are described in terms of the isotropy groups of the action of the group $Γ$ on $E \to M$.

math.OA↗

arXiv2019

A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications

We provide new insight into the analysis of N-body problems by studying a compactification $M_N$ of $\mathbb{R}^{3N}$ that is compatible with the analytic properties of the $N$-body Hamiltonian $H_N$. We show that our compactification coincides with the compactification introduced by Vasy using blow-ups in order to study the scattering theory of N-body Hamiltonians and with a compactification introduced by Georgescu using $C^*$-algebras. In particular, the compactifications introduced by Georgescu and by Vasy coincide (up to a homeomorphism that is the identity on $\mathbb{R}^{3N}$). Our result has applications to the spectral theory of $N$-body problems and to some related approximation properties. For instance, results about the essential spectrum, the resolvents, and the scattering matrices of $H_N$ (when they exist) may be related to the behavior near $M_N\setminus \mathbb{R}^{3N}$ (i.e. "at infinity") of their distribution kernels, which can be efficiently studied using our methods. The compactification $M_N$ is compatible with the action of the permutation group $S_N$, which allows to implement bosonic and fermionic (anti-)symmetry relations. We also indicate how our results lead to a regularity result for the eigenfunctions of $H_N$.

math-ph↗

arXiv2019

The strong Legendre condition and the well-posedness of mixed Robin problems on manifolds with bounded geometry

Let $M$ be a smooth manifold with boundary $\partial M$ and bounded geometry, $\partial_D M \subset \partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $\partial M \smallsetminus \partial_D M$. We prove the regularity and well-posedness of the mixed Robin boundary value problem $$Pu = f \mbox{ in } M,\ u = 0 \mbox{ on } \partial_D M,\ \partial^P_νu + bu = 0 \mbox{ on } \partial M \setminus \partial_D M$$ under some natural assumptions. Our operators act on sections of a vector bundle $E \to M$ with bounded geometry. Our well-posedness result is in the Sobolev spaces $H^s(M; E)$, $s \geq 0$. The main novelty of our results is that they are formulated on a non-compact manifold. We include also some extensions of our main result in different directions. First, the finite width assumption is required for the Poincaré inequality on manifolds with bounded geometry, a result for which we give a new, more general proof. Second, we consider also the case when we have a decomposition of the vector bundle $E$ (instead of a decomposition of the boundary). Third, we also consider operators with non-smooth coefficients, but, in this case, we need to limit the range of $s$. Finally, we also consider the case of uniformly strongly elliptic operators. In this case, we introduce a \emph{uniform Agmon condition} and show that it is equivalent to the Gårding inequality. This extends an important result of Agmon (1958).

math.AP↗

arXiv2019

Analysis and boundary value problems on singular domains: an approach via bounded geometry

We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry.

math.AP↗

arXiv2019

Uniform Shapiro-Lopatinski conditions and boundary value problems on manifolds with bounded geometry

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincare inequality of (Ammann-Grosse-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, well-known theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition.

math.AP↗

arXiv2018

A volatility-of-volatility expansion of the option prices in the SABR stochastic volatility model

We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for $\beta = 1$, namely $\partial_{\tau}u = \sigma^2 \big [ \frac{1}{2} (\partial^2_xu - \partial_xu) + \nu \rho \partial_x\partial_\sigma u + \frac{1}{2} \nu^2 \partial^2_\sigma u \, \big ] + \kappa (\theta - \sigma) \partial_\sigma$, by choosing $\nu$ as small parameter. This yields $u = u_0 + \nu u_1 + \nu^2 u_2 + \ldots$, with $u_j$ independent of $\nu$. The terms $u_j$ are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of $u$ that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" $\tau$ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of $\nu$, similar to Hagan's formula, but including also the {\em mean reverting term.} We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility.

q-fin.CP↗