# Questions tagged [differential-operators]

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

422
questions

**7**

votes

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

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**0**answers

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

**1**answer

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

**1**answer

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 ...

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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**

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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

**1**answer

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

**0**answers

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

**0**answers

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**

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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**

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**0**answers

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

**1**answer

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**

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**0**answers

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

**0**answers

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

**1**answer

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**

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**0**answers

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

**0**answers

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 ...

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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

**2**answers

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

**1**answer

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**

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**0**answers

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**

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**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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**

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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 ...

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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). ...

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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 ...

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**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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**

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**0**answers

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 ...

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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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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 ...