**2**

votes

**0**answers

60 views

### Rellich Embedding Theorem for the $2$-Sphere

I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the $2$-sphere. To be precise, for $S$ the spinor bundle of $S^2$; $L^2(S^2)$ the space of square integrable ...

**4**

votes

**1**answer

55 views

### Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
...

**3**

votes

**2**answers

337 views

### Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ ...

**0**

votes

**1**answer

81 views

### Estimate for differential of inverse map

Let $f: M \to N$ be a diffeomorphism between two riemannian Manifolds. Suppose there exist constants $0 < c \leq C$ such that for all $p \in M$, we have $c \leq |df_p| \leq C$. Here, $df$ denotes ...

**4**

votes

**0**answers

215 views

### The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on ...

**0**

votes

**0**answers

25 views

### Deciding whether linear Equations are solvable over specific subrings of K(x_1,..,x_n)

The Definition of 'linear equations are solvable' which is meant here is
1)
Let $R$ be a commutative Ring (associative and with unity)
For a given $m \in \mathbb{N} $ and $b \in R$
It is decideable ...

**5**

votes

**0**answers

64 views

### Ring of SO(n)-invariant differential operators on M_n,m

I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).
There comes a point in the paper (Lemma 2.8) ...

**2**

votes

**0**answers

68 views

### $\eta$-invariants of Riemann Surface

I am curious about a concrete computation of $\eta$-invariants for Riemann surface, e.g. Torus.
Is there any nice review or notes talking about the computation? Or is it possible to express it as ...

**5**

votes

**0**answers

89 views

### $\eta$ invariants of Pin+ manifolds $\mathbb{RP}^{8k}$

In general, I am curious about the 'quantization' of $\eta$-invariants on Pin+ manifold, i.e., what is the 'minimal unit' of $\eta$-invariants on a manifold with certain choice of Pin+ structure.
...

**4**

votes

**1**answer

120 views

### Differential Operators On A Curve And On Osculating Circle

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...

**1**

vote

**0**answers

36 views

### Equivalence of fractional power of second-order positive differential operator as pseudodifferential operator and a fractional definition

Let $A$ be a second-order differential operator on a closed manifold $M$ satisfying
$$(Au,u) \geq 0$$
with $A=-\Delta$ the Laplace-Beltrami the model case. One can define for $s \in (0,1)$ the ...

**2**

votes

**0**answers

359 views

### Differential and pre-differential of Jacobi identity

Let M be a manifold.
To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied?
That is a Lie algebra structure for which $[X,fY]=f[X,Y]$.
(For ...

**3**

votes

**0**answers

72 views

### Chain rule for weakly differentiable functions

Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ...

**4**

votes

**1**answer

124 views

### [This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale

I have a stupid question:
Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space $H^{-s}(\partial \Omega)$? We usually encounter trace theorem as
...

**2**

votes

**1**answer

127 views

### derivative of the adiabatic limit of the eta invariant

To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ...

**3**

votes

**2**answers

71 views

### Density of Laplace image

Let $D$ define the differential operator $-\frac{\partial^2}{\partial x^2}$ on $\mathbb R$.
Let $\xi\notin\mathbb R$ be a complex number.
Is it true that
$$
(D-\xi)C_c^\infty({\mathbb R})
$$
is dense ...

**1**

vote

**1**answer

130 views

### Ask the validity of Tauberian lemma in Sogge's book

In C.D.Sogge's Fourier Integrals in Classical Analysis pp.128-129, he proved Lemma4.2.3(Tauberian Lemma):
Lemma. Let$g(\lambda)$ be a piece-wise continuous tempered function of $\mathbb{R}$. Assume ...

**0**

votes

**0**answers

44 views

### comparison principle for viscosity solution to linear nonlocal equation with drift

I met a problem about comparison principle for nonlocal equation when I study SDEs driven by Levy noise. Since, I have no background about PDEs, it may be a stupid question:
$$Iu=\lambda ...

**2**

votes

**0**answers

37 views

### what and where is Dubrovin's tri-differential operator?

IFF memory serves, Dubrovin made good use of a tridiffereential operator
long before H & R fluxes. What did he use it for and where?

**2**

votes

**1**answer

69 views

### Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over ...

**14**

votes

**1**answer

377 views

### Harmonic spinors on closed hyperbolic manifolds

Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial?
I'm mainly interested in the 3-dimensional case ...

**0**

votes

**1**answer

320 views

### Operator theory of the Hessian

How can I learn more about the operator theory of the Hessian? The Hessian of a function $u : \Omega \rightarrow \mathbb R$ over a domain $\Omega \subseteq \mathbb R^n$ is the matrix of second ...

**2**

votes

**0**answers

127 views

### Intuition behind Stokes operator?

I know that the definition of the Stokes operator (which appears in the functional form of the Navier-Stokes equations) is
$$A = -P_L Δ$$
where $Δ$ is the Laplacian, and $P_L$ is the Leray ...

**1**

vote

**0**answers

34 views

### Recursive formula for symbol of resolvent on noncompact manifold

On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: ...

**3**

votes

**1**answer

191 views

### A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$
Does for any ...

**15**

votes

**2**answers

691 views

### Are there any natural differential operators besides $d$?

Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ and $\mu = (\mu_1, \ldots, \mu_r)$ be partitions such that $\mu_j = \lambda_j +1$ for one index $j$ and $\mu_i = \lambda_i$ for all other $i$. Then there ...

**1**

vote

**2**answers

137 views

### Motivating the Bessel translation operator

In a paper I am reading on the Hankel transform (this paper to be exact), I've come across a somewhat peculiar definition for a generalized translation operator. The operator is designed with a ...

**0**

votes

**1**answer

153 views

### Diffusion on a semi-Riemannian manifold?

A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, ...

**2**

votes

**0**answers

135 views

### Estimates of eigenvalues of elliptic operators on compact manifolds

The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula
...

**2**

votes

**0**answers

120 views

### Discrete p-Laplacian

One of the definitions of the discrete (weighted) $p$-Laplacian is the following:
$$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$
Consider the one dimensional case. Then the free ...

**3**

votes

**1**answer

198 views

### adjoint of this closed (?) operator

I am currently dealing with an unbounded operator
$T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow ...

**2**

votes

**1**answer

115 views

### Proper domain for operators

in this paper on arxiv in equation 27, two operators
$$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$
and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + ...

**0**

votes

**1**answer

144 views

### Functional Calculus and Fredholm index

Let $-\Delta: W^{2,2} \subset L^2(\mathbb{S}^2) \rightarrow L^2(\mathbb{S}^2)$. Then it is "easy" to show that $-\Delta $ is self-adjoint. Now, I am looking for closed operators $T$ and $T^*$ of order ...

**0**

votes

**0**answers

89 views

### Uniqueness of a Integro-parabolic differential equation?

Let $r, q,\lambda,\sigma,\kappa,\mu$ are positive real numbers and let $c(t)$ is a differential function of $t$. $\Gamma(\eta)$ is a probability density function.
When I consider price of American ...

**14**

votes

**2**answers

883 views

### Exact Definition of Dirac Operator

Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ...

**0**

votes

**1**answer

185 views

### Cauchy problem for an overdetermined system of PDE

This question is strictly related to this one. Let us consider the differential system with constant coefficients
$$\left(\begin{array}{ccc}
B_{11} & B_{12} & 0\\
...

**1**

vote

**1**answer

112 views

### Heat Kernel estimate at the level of the form

Let $(M,g)$ be a compact Riemannian manifold. It is known there exist Gaussian estimates of the heat kernel and its derivatives acting on functions on $M$.
The kind of estimate I'm looking for could ...

**0**

votes

**1**answer

181 views

### How to solve this differential equation with an infinite sum?

I would like to find solutions of the following differential equation:
$ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$
For example in space of function from $\mathbb R^*$ to $\mathbb ...

**2**

votes

**1**answer

84 views

### Local fractional Sobolev inequality

If $\mathcal{X}$ is a smooth cutoff near 0 in $\mathbb{R}^n$, then $M_0 = \mathcal{X}(-\Delta+Id)\mathcal{X}$ is a self-adjoint operator in $L^2(\mathbb{R}^n)$. Because $M_0$ is semi-positive and the ...

**13**

votes

**4**answers

822 views

### Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...

**2**

votes

**0**answers

101 views

### “simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series
Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...

**7**

votes

**1**answer

386 views

### Killing vector fields on sphere

Let $u$ be a smooth function on $\mathbb S^2$, and assume that for every killing vector field $V$ on $\mathbb S^2$.
$$\int_{\mathbb S^2} V(u) x_j dS=0\text{,}\forall j=1,2,3$$
Is $u$ necessarily ...

**9**

votes

**1**answer

199 views

### When are the Dolbeault and de Rham dgas homotopy equivalent?

Let $M$ be a compact Kahler manifold. Then the Hodge decomposition says that the Dolbeault dga (of forms of all bidegree) and the de Rham dga on $\Omega_{\mathbb C}^\bullet(M)$ have isomorphic ...

**1**

vote

**0**answers

147 views

### Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.
Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf
Let ...

**3**

votes

**0**answers

57 views

### Boundedness Spectral Triple Axioms for de Rham Complex

In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...

**2**

votes

**0**answers

85 views

### Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...

**3**

votes

**1**answer

144 views

### Horizontal lift of differential operator

On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that
$X^{\mathrm{hor}}$ is a ...

**0**

votes

**0**answers

105 views

### Elliptic PDE-Fredholm PDE(Is there a contradictory situation)

Let $E$ be a smooth vector bundle on a closed manifold $M$. Assume that $D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a diff. operator which is a fredhoolm operator, in the algebraic ...

**2**

votes

**0**answers

126 views

### Is Laplacian a surjective operator?

For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image ...

**1**

vote

**0**answers

141 views

### The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...