# Questions tagged [differential-operators]

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

**8**

votes

**1**answer

136 views

### Spectrum of a first-order elliptic differential operator

Suppose that I have a first-order elliptic differential operator $A: \mathrm{dom}(A) \subset L^2(E) \to L^2(E)$, where $(E,h^E) \to M$ is a hermitian vector bundle and $M$ is a compact manifold.
I ...

**2**

votes

**0**answers

95 views

### Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...

**0**

votes

**0**answers

49 views

### Convolution with an analytic semigroup

Let $e^{At}$ denote an analytic semigroup on Hilbert space $X$ generate by $A:D(A)\to X$. Also, let $f\in L^1(0,\tau;X)$. I want to show that the convolution
$$ g(t)=\int_0^t e^{A(t-s)}f(s)ds$$
...

**2**

votes

**0**answers

124 views

### Existence of a certain kind of compact spin manifold with boundary

For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...

**4**

votes

**1**answer

205 views

### Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$

Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (...

**3**

votes

**2**answers

201 views

### Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability

We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...

**6**

votes

**3**answers

141 views

### Surjectivity of differential operators with constant coefficients

I would like a proof or a reference (or a counter-example...) for the following fact. Let $P\in \mathbb{C}[x_1,\ldots ,x_n]$ and $D\in \mathbb{C}[\frac{\partial }{\partial x_1} ,\ldots ,\frac{\...

**3**

votes

**0**answers

64 views

### Dixmier traces, Wodzicki residue and residues of zeta functions

Let $M$ be an $n$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $P$ of order $-n$. Here are some facts which I had learned so far:
1. There exists a density defined ...

**3**

votes

**1**answer

152 views

### Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...

**3**

votes

**1**answer

180 views

### Is there a vector field such that one differential form is the Lie derivative of the other?

I'm looking for a reference or answer for the following question:
Let $M$ be an (compact and orientable, if it helps) smooth manifold and $\nu$ and $\mu$ two differential forms. I'm looking for ...

**4**

votes

**0**answers

261 views

### Spectral Gap of Elliptic Operator

Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...

**3**

votes

**0**answers

106 views

### Differential operators on a compact Lie group associated to bracket-generating sets

Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...

**0**

votes

**0**answers

29 views

### Relative compactness of differential operator

Let $\Omega$ be $\mathbb R^n$ or a complete (unbounded) open manifold, and $f$ be a smooth function on $\Omega$.
We consider a self-adjoint 2nd. elliptic operator $H$ on $L^2$ space(to simplify the ...

**5**

votes

**0**answers

113 views

### Extension of elliptic complex to an exact sequence

This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator.
Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...

**0**

votes

**1**answer

104 views

### $f_n$ is bounded in $C(0,T;H^2(0,L))$ so is $f_n^p$?

Let $1<p<\infty$, and $f_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we ...

**0**

votes

**0**answers

25 views

### Eigenfunction of Distorted Laplacian on Smooth compact domain

Setup
Suppose that $D$ is a compact star-shaped domain in $\mathbb{R}^d$ which is diffeomorphic to a closed $d$-dimensional ball in $\mathbb{R}^d$. Let $a(t,x)>0$ be a class $\mathscr{C}^{\infty}(\...

**4**

votes

**0**answers

63 views

### Superposition operator from Sobolev space to Lebesgue space

Let $\Omega$ be a bounded, connected set in $\mathbb{R}^2$ with smooth boundary. I am wondering under what conditions on the real function $f(x):\mathbb{R}\to \mathbb{R}$ the superposition operator $F(...

**1**

vote

**1**answer

51 views

### Reference Request: Differentiability of Moreau Envelope

I've recently come across many results discussing the differentiation of the Moreau envelope defined by
$$
e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(z)
,
$$
where $f$ is a convex functional on a ...

**3**

votes

**1**answer

150 views

### Does the space of harmonic forms change continuously with the metric?

Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{...

**7**

votes

**1**answer

231 views

### Atiyah-Patodi-Singer for manifolds with cusps

Dear Colleagues and Friends,
Please let me know if you are aware of any references to the following question.
The classical result of Atiyah, Patodi and Singer tells us that if $W$ is a compact ...

**3**

votes

**1**answer

74 views

### PDE satisfied by projection of a function onto a subspace

Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE
$$
\begin{cases}
-\Delta_p u=f\;\text{in $D$}...

**6**

votes

**1**answer

123 views

### Ordinary differential operators satisfying braid relation?

Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...

**3**

votes

**1**answer

114 views

### Spectral decomposition of a specific operator

To understand a crucial example in representation theory, I need the explicit spectral decomposition of the differential operator
$$
Df(x)=(1+x^2)f''(x)+2xf'(x)
$$ on $L^2({\mathbb R})$. I'm not an ...

**2**

votes

**0**answers

152 views

### A question about whether an operator can be lipschitz or not

Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$.
Now define the operator $ \mathcal{A} : C^{\sigma, \sigma/2}(X) \to C^{\sigma, \...

**1**

vote

**0**answers

132 views

### The comparison of certain modules arising from the Cauchy-Riemann differential operator

Let $\Gamma=C^{\infty}(\mathbb{R}^2)$ be the space of all smooth complex valued functions on the plane. We define the following Cauchy Riemann differential operator $D$ on $\Gamma$:
$$D:\Gamma \...

**5**

votes

**0**answers

68 views

### What is the generator of the heat semigroup on non-complete manifolds?

If $M$ is a complete Riemannian manifold, it possesses a unique self-adjoint positive operator $-\Delta$ on $L^2(M)$. If $M$ is not complete, though, it is known that the Laplace-Beltrami operator $-\...

**6**

votes

**1**answer

64 views

### Gaussian bounds for the heat kernel of regular domains in Riemannian manifolds

In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same ...

**4**

votes

**0**answers

67 views

### $G$-Invariant Differential Operators

Let $G$ be a complex algebraic group, $K$ a closed subgroup so that $X=G/K$ is a homogeneous space.
Let $\mathcal{D}(X)$ denote the algebra of differential operators on $X$. The group $G$ acts on $\...

**5**

votes

**2**answers

192 views

### Elements of graded algebra associated with the algebra of differential operators as smooth sections

Let $M$ be a compact manifold and $E$ a complex vector bundle. We will consider differential operators $P$ acting between $\Gamma^{\infty}(M,E)$. Let $\mathcal{P}$ be the algebra of all differential ...

**3**

votes

**1**answer

57 views

### Simultaneous diagonalization on spaces with constant curvature

I have an operator $C$ that I wish to diagonalize on a Riemmanian manifold $M$ with constant curvature $\Lambda$
$$C = A + B$$
Now I know that these operators $A$ and $B$ commute in flat space, but on ...

**4**

votes

**0**answers

110 views

### Distributions Supported at a Point of a Variety

Let $X$ be a smooth algebraic variety over $\mathbb{C}$, let $a\in X$ be a closed point, and $R=\mathcal{O}_{X,a}$ the local ring at $a$. Write $\mathfrak{m}$ for the maximal ideal of $R$.
Define ...

**2**

votes

**1**answer

104 views

### Differential operators and rules Ore polynomial

(I have posed this question over at math.se but since there were no answers I hope it's okay to post here.)
When dealing with (nonlinear) dynamical systems, one often deals with state space ...

**4**

votes

**0**answers

72 views

### An elliptic operator whose coreresponding symbol Hamiltonian vector field has an isolated periodic orbit

Let $D$ be a differential operator on the space of smooth functions on a manifold $M$. The symbol of $D$ can be considered as a Hamiltonian on the cotangent bundle $T^*M$. We call ...

**2**

votes

**0**answers

73 views

### Ergodic type ODE problem

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...

**1**

vote

**0**answers

109 views

### Perturbation of Elliptic operator

Let $\Omega$ be an open region or a non-compact complete manifold, $L$ be an elliptic operator with possibly non vanishing zero-order term, e.g. $-\Delta+q$. Suppose $W$ is an operator such that $W(...

**3**

votes

**0**answers

54 views

### Generalized viscosity sub(super)solution and it's convolution

Suppose that $\Gamma \subsetneq \mathbb{R}^n$ is an open symmetric convex cone containing positive orthant.
Note that $\Gamma \subset \left\{x=(x_1,...,x_n) \in \mathbb{R}^n | \sum_{i=1}^{n} x_i > ...

**5**

votes

**1**answer

126 views

### The division problem for tempered functions

It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...

**2**

votes

**1**answer

64 views

### Positive form for a homogeneous elliptic pde

I have a pde of the following form:
\begin{align}
&P(x,D)u = f \text{ on } \Omega, \\
&P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha},
\end{align}
where one can assume that $f$ ...

**2**

votes

**1**answer

91 views

### Decomposition of the spectrum of an unbounded opeator [closed]

The Wikipedia article on spectral decomposition, see here
https://en.wikipedia.org/wiki/Self-adjoint_operator
says the following:
A self-adjoint operator A on $H$ has pure point spectrum if and ...

**1**

vote

**0**answers

48 views

### Ellipticity of certain differential operator associated to a pair of vector field via curvature tensor

What is a precise example of the following situation:
A compact Riemanian manifold $M$ admits two vector field $X,Y$ such that the the operator $$Z\mapsto R(X,Y)Z$$
Would be an elliptic operator and ...

**4**

votes

**1**answer

210 views

### Universal enveloping algebra and the algebra of invariant differential operators

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Then $\mathfrak{g}$ may be interpreted as the Lie algebra of right (equivalently left) invariant vector fields. Let $\mathcal{U}(\mathfrak{...

**2**

votes

**1**answer

359 views

### Unusual problem of calculus-of-variations. Attempt 2

I already tried to ask the same question here Unusual problem of calculus-of-variations, but I did a huge mistake and have no time to understand it. In the past post, I tried to simplify the problem, ...

**2**

votes

**2**answers

280 views

### A possible dynamical approach to the “Invariant Subspace Problem”

In the literature, is there any paper or research investigating the invariant subspace problem with consideration of differential operators acting on an appropriate Sobolev space?In particular is ...

**8**

votes

**3**answers

321 views

### What does the flow of the principal symbol of the differential operator tell us about the PDE?

Disclaimer: Let me apologize in advance for asking this slightly vague question
Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...

**4**

votes

**1**answer

95 views

### Reference for Weyl's law for higher order operators on closed Riemannian manifolds

I am looking at page 32 (beginning of Chapter 5) here. We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading ...

**1**

vote

**0**answers

186 views

### Unusual problem of calculus-of-variations

I have a domain $D:\quad$ $-1<x<1$ and $-f(x)<y<f(x)$.\
There is function $u(x,y)$ which is obey the equation $\Delta u(x,y)=0$, $\forall (x,y)\in D$ with the Dirichlet boundary condition ...

**3**

votes

**0**answers

250 views

### Propagation of Singularities

I'm following the book "Elementary Introduction To The Theory Of Pseudodifferential Operators" by X. S. Raymond and the Joshi Lectures Notes - https://arxiv.org/pdf/math/9906155.pdf - to prove the ...

**3**

votes

**0**answers

61 views

### Transfer modules and Weyl algebra

Let $V$ be a $\mathbb{C}$-vectorial space of dimension $n$ and $V^*$ the complex dual space.
I would like to understand the following isomorphism $$D_{V^* \leftarrow V \times V^*} \overset{L}\otimes_{...

**3**

votes

**0**answers

84 views

### Is the square root of curl^2-1/2 a natural (Dirac-)operator?

In current computations on a particular $3$-dimensional Riemannian manifold, a first order differential operator $D:\Gamma^\infty(TM,M)\to \Gamma^\infty(TM,M)$ acting on vector fiels shows up, with ...

**4**

votes

**2**answers

213 views

### Simplification of integral on the sphere

In the article: https://arxiv.org/abs/0906.3217 the authors prove in Lemma 1 a formula which helps compute more easily the integral of the Hessian of a function defined on $\Bbb{S}^2$. More precisely, ...