Questions tagged [differential-operators]
Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
8
votes
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
3answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
$$
\begin{cases}
-\Delta_p u=f\;\text{in $D$}...
6
votes
1answer
123 views
Ordinary differential operators satisfying braid relation?
Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...
3
votes
1answer
114 views
Spectral decomposition of a specific operator
To understand a crucial example in representation theory, I need the explicit spectral decomposition of the differential operator
$$
Df(x)=(1+x^2)f''(x)+2xf'(x)
$$ on $L^2({\mathbb R})$. I'm not an ...
2
votes
0answers
152 views
A question about whether an operator can be lipschitz or not
Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$.
Now define the operator $ \mathcal{A} : C^{\sigma, \sigma/2}(X) \to C^{\sigma, \...
1
vote
0answers
132 views
The comparison of certain modules arising from the Cauchy-Riemann differential operator
Let $\Gamma=C^{\infty}(\mathbb{R}^2)$ be the space of all smooth complex valued functions on the plane. We define the following Cauchy Riemann differential operator $D$ on $\Gamma$:
$$D:\Gamma \...
5
votes
0answers
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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 $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
1
vote
0answers
109 views
Perturbation of Elliptic operator
Let $\Omega$ be an open region or a non-compact complete manifold, $L$ be an elliptic operator with possibly non vanishing zero-order term, e.g. $-\Delta+q$. Suppose $W$ is an operator such that $W(...
3
votes
0answers
54 views
Generalized viscosity sub(super)solution and it's convolution
Suppose that $\Gamma \subsetneq \mathbb{R}^n$ is an open symmetric convex cone containing positive orthant.
Note that $\Gamma \subset \left\{x=(x_1,...,x_n) \in \mathbb{R}^n | \sum_{i=1}^{n} x_i > ...
5
votes
1answer
126 views
The division problem for tempered functions
It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...
2
votes
1answer
64 views
Positive form for a homogeneous elliptic pde
I have a pde of the following form:
\begin{align}
&P(x,D)u = f \text{ on } \Omega, \\
&P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha},
\end{align}
where one can assume that $f$ ...
2
votes
1answer
91 views
Decomposition of the spectrum of an unbounded opeator [closed]
The Wikipedia article on spectral decomposition, see here
https://en.wikipedia.org/wiki/Self-adjoint_operator
says the following:
A self-adjoint operator A on $H$ has pure point spectrum if and ...
1
vote
0answers
48 views
Ellipticity of certain differential operator associated to a pair of vector field via curvature tensor
What is a precise example of the following situation:
A compact Riemanian manifold $M$ admits two vector field $X,Y$ such that the the operator $$Z\mapsto R(X,Y)Z$$
Would be an elliptic operator and ...
4
votes
1answer
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
1answer
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
2answers
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
3answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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, ...
