Questions tagged [differential-operators]

Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.

Filter by
Sorted by
Tagged with
7
votes
2answers
221 views

Differentiation of functions over graphs

In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary. Definitions. Let $G=(V,E)$ be a directed ...
3
votes
1answer
217 views

Differential inequalities under which a flat function must be identically zero

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
3
votes
0answers
430 views

Another question from Villani's monograph “Hypocoercivity”

I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 62 (the first snapshot), he defined a new inner product $((\cdot,\cdot))$ as in (...
1
vote
0answers
108 views

A question on Villani's monograph “Hypocoercivity”

I can not figure out the appearance of the term $\int h_0\,d\mu$ in the statement of Theorem 35 above. Here are some background information: $L$ is an unbunded operator on a Hilbert space $\mathcal{H}^...
2
votes
1answer
283 views

A possible error in Villani's monograph “Hypocoercivity”

I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 48, he wrote "For the second term in (6.9), we use the identity $$\nabla\...
5
votes
1answer
67 views

Spectrum of an elliptic operator in divergence form with a reflecting boundary condition

Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$ \begin{align} L ...
3
votes
1answer
169 views

Ricci flow for manifold learning

I know that mean curvature and diffusion-type flows are common in manifold learning because of their smoothing effects. I haven't seen Ricci flow used as much. Given that Ricci and diffusion-type ...
3
votes
0answers
53 views

Range of divergence operator on the space of traceless symmetric $(0,2)$ tensors; conformal vector fields on an arbitrary metric on $S^2$

Let $\gamma$ be a metric on $S^2$. I am trying to solve the following PDE on a $(0,2)$ symmetric traceless tensor $A$: $$div_{\gamma} A = \omega$$ where $\omega$ is a 1-form. It is known that there ...
3
votes
1answer
154 views

Is it possible that $\int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0$ for all compactly supported $\phi$?

When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral $$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$ where $P$ is a linear ...
4
votes
1answer
126 views

Degenerate second-order Lagrangians

Let $M$ be a smooth $m$ dimensional manifold, let $\pi:E\rightarrow M$ be a smooth fibred manifold over $M$. Let us write generic fibred coordinates as $(x^i,y^\sigma)$ with $x^i$ being the base ...
3
votes
0answers
142 views

Dirichlet to Neumann operator and the Riesz transform

Consider the manifold $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $\gamma$ be the induced metric on $\partial M$....
4
votes
0answers
61 views

Analyticity of the regularized $\eta$-invariant

The APS $\eta$- invariant of an operator $B$ with eigenvalues $\lambda$ is defined as $$\eta = \sum_\lambda sgn (\lambda)$$ which is a divergent sum and it can be regularized as follows: $$\eta(s) = \...
4
votes
1answer
187 views

About the index theorems

I am looking for some introductory book/paper/notes about the several index theorems and their applications. By several I mean the "classical" Atiyah-Singer theorem, the local index theorem (...
0
votes
0answers
61 views

Two definitions of elliptic symbols

There are two definitions of elliptic symbol. A smooth matrix function $p(x,\xi)$ is an elliptic symbol of order $m\in\mathbb{R}$ if there exists a constant $c>0$ such that for all $|\xi|>c$ we ...
2
votes
0answers
50 views

Singularity of reproducing kernel for elliptic operator

Let $(M,g)$ be a smooth compact Riemannian manifold and dimension $2$, $\Gamma$ a smooth vector bundle over $M$, and suppose $L: W^{k,2}(\Gamma)\to W^{k-2,2}(\Gamma)$ is a second order strongly ...
7
votes
0answers
242 views

$D(\mathcal{O}(n))$ via generators and relations

Let $V$ be a complex vector space. Consider the algebra $D(\mathbb{P}(V),\mathcal{O}(n)))$ of global differential operators from line bundle $\mathcal{O}(n)$ to itself, here $n \in \mathbb{Z}_{\...
0
votes
0answers
75 views

Characterization of convolution operators via the Fourier transform

Let $\mathcal{L}$ be a linear and continuous operator from the space of tempered distributions $\mathcal{S}'(\mathbb{R})$ to itself. The Fourier transform of a tempered distribution $f$ is denoted by $...
4
votes
1answer
511 views

Analytic functions where all derivatives vanish at infinity and which are bounded

Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$. I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...
1
vote
0answers
87 views

Is this definition of a Fuchsian operator correct?

In Bjork, Analytic D-modules and applications, the following definition of a Fuchsian operator is given: Here, I believe, $D(0)=\mathcal{O}$, the zeroth filtered piece of the ring of germs of ...
5
votes
0answers
75 views

An application of Leray-Schauder degree theory for Nirenberg problem on the 2-sphere

I'm studying the article "The scalar curvature equation on 2- and 3-spheres" by Chang, Gursky and Yang and I'm particulary interested in the 2-sphere case. They prove that if $K:S^2\...
2
votes
1answer
187 views

Elliptic operators and Leibniz rule

Let $M$ be a manifold. Does it necessarily admit an elliptic operator on $C^{\infty}(M)$ which satisfy Leibniz rule? Let $M$ be a symplectic manifold with the standard Poisson structure on $C^{\...
1
vote
0answers
90 views

Sturm-Liouville Problem: When does $w y^2$ vanish at a singular boundary point?

It is well known (e.g. Courant, Hilbert - Methods of Mathematical Physics) that solutions of the Sturm-Liouville problem on an interval $J=(a,b)$ \begin{equation} \tag{1} \left(p y' \right)' - qy \; = ...
2
votes
0answers
89 views

Spectrum of the Witten Laplacian on compact Riemannian manifolds

Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$ How generally is it true that this ${\rm ...
2
votes
0answers
40 views

Analysis of coefficients for quickly vanishing analytic vector field

Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
5
votes
2answers
165 views

Analytic approximations of smooth vector fields

Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with $$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$ on $\mathbb{R}^3$ for any $\alpha,K$. Further, we ...
5
votes
1answer
197 views

A question on moduli space of Hitchin's equations

I am reading Hitchin's Self-Duality paper. In section 5 (page 85), he is trying to prove that $Dim H^1=12(g-1)$. In doing so, he defines an operator $d^*_2+d_1$, where $d^*_2$ and $d_1$ are given by $...
1
vote
0answers
66 views

What types of semigroups have a Laplacean type operator as infinitesimal generator?

Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded connected set having Lipschitz uniform boundary. Moreover let $d\in L^{\infty}(\Omega,\mathbb{R}^M),\ d_1(x),d_2(x),\dots, d_M(x)>d>0,\ \...
2
votes
0answers
89 views

Laplacian coupled with another equation over a two-dimensional rectangular region

I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation which is: $$\frac{\partial t}{\partial x}+\alpha(t-T)=0 \tag 1$$ where it is known that $t(x=0)=t_i$. The ...
5
votes
0answers
239 views

On the definition of Gysin homomorphism

According to Lawson-Michelsohn's book (p239), the Gysin homomorphism for a continuous map between (compact) manifolds $f:Y\to X$ is defined by setting $$ f_!=PD_X^{-1}\circ f_*\circ PD_Y $$ where $PD$ ...
6
votes
0answers
127 views

Differential birational equivalence

Suppose the base field algebraically closed and of zero characteristic. There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion ...
6
votes
1answer
248 views

Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. ...
3
votes
1answer
275 views

When is the exterior derivation $d$ a Lie algebra morphism?

In this question we search for some conditions under which the exterior derivation $d:\Omega^i(M)\to \Omega^{i+1}(M)$ on a differentiable manifold $M$ is a Lie algebra morphism in a certain sense. We ...
1
vote
0answers
79 views

Fredholmness of elliptic operator on Hölder spaces

Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
1
vote
0answers
89 views

Derivations of differential operators

For a smooth affine variety $\operatorname{Spec} A$ over a ring $R$ we have an algebra of differential operators $\mathcal{D}_A$ (here I mean not the Grothendieck differential operators but PD-ones). ...
1
vote
0answers
68 views

Pseudo-differential operators and differetial operator

Hello I am totally new to Pseudo-differential operators and I’m wondering if a differential operator is a pseudo-differential operator. So, I want to show , using the definition of the symbol given ...
2
votes
0answers
103 views

Is this $1$-form harmonic?

Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \...
5
votes
0answers
281 views

On Grothendieck's abstract definition of differential operators

I have heard that there is the following abstract definition due to Grothendieck of differential operators on a module $M$ over a commutative associative unital algebra $A$ over a field of ...
5
votes
1answer
187 views

Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials

There are some known criteria for the Sturm-Liouville Problem \begin{equation} \tag{1} \frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y \...
7
votes
0answers
209 views

Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$

The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by $$ \eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
1
vote
0answers
22 views

Reference request: Transverse parabolic Schauder estimates

Is there a version of the parabolic Schauder estimates for transversely parabolic linear PDE's on a manifold with a Riemannian foliation for functions that are constant on the leaves of the foliation? ...
4
votes
0answers
152 views

About generalized binomial theorem and Grünwald-Letnikov fractional derivative

I have run into a problem while computing the fractional derivatives of order $\alpha$ for the Riemann zeta function. My Theorem states Let $s\in\mathbb{C}$, $\mathfrak{Re}(s)>1$, then the ...
2
votes
0answers
134 views

Differential operators and vector fields [closed]

Let $M$ be a smooth manifold. It is well known that there is a bijective correspondance between vector fields on $M$ and differential operators of order 1. My question is: if we take a differential ...
3
votes
0answers
73 views

Understanding the geometric fibre twisted differential operators

Let $\mathfrak{g}$ be a Complex semisimple Lie algebra with decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}$. For $\lambda \in \mathfrak{h}^*$, we let $\mathcal{D}_{\lambda}$ be ...
5
votes
0answers
169 views

Index of the Fredholm operator

I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...
2
votes
1answer
89 views

Multidifferential operators with vanishing integrals

(Moved from math.stackexchange.) Is the following proposition true? Given a multidifferential operator $D$ on $\mathbb{R}^n$ with constant coefficients, i.e. for all functions $f_1,\dots,f_k \in C^...
6
votes
0answers
309 views

Is there an analog of the Levi–Civita connection for schemes?

Is there an analog of the Levi–Civita connection for schemes? There exists algebraic de Rham theory, $n$-forms on vector bundles (algebraically), and familiar constructions from differential geometry....
4
votes
0answers
187 views

Equivalent definitions of differential operator

This puzzles me from some time and is in parts connected to the questions Symmetrized derivatives version and Symmetrized derivatives version II. For me the linear DO between vector bundles $E$ and $...
6
votes
1answer
176 views

Elliptic operator with finite spectrum?

Is it possible for a (non-symmetric) elliptic differential operator to have finite spectrum. If so, is there an explicit example?
1
vote
2answers
144 views

Link btw. exponential and derivatives from an algebraic perspective [closed]

I have been attempting to understand my math education (as a bachelor in electrical engineering) from a more algebraic perspective recently. I would like to understand more about the link between ...
2
votes
0answers
44 views

Reference request on the basics of differential operators and symbols

Perhaps it is too much to ask, but I'm searching for some published article or book, to cite some results from it, that treats infinite degree differential opertors (with constant coefficients) and ...

1
2 3 4 5
9