Questions tagged [differential-operators]

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

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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{\...
abx's user avatar
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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 ...
truebaran's user avatar
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3 votes
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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{...
Junhyeong Kim's user avatar
3 votes
1 answer
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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 ...
S.Surace's user avatar
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4 votes
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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 ...
Elliott's user avatar
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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 ...
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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 ...
Tobias Diez's user avatar
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$f_n$ 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 ...
Saj_Eda's user avatar
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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(...
Saj_Eda's user avatar
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2 answers
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Differentiability of the Moreau envelope

I've recently come across many results discussing the differentiation of the Moreau envelope defined by \begin{equation} e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(h), \end{equation} where $f$ is ...
ABIM's user avatar
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3 votes
1 answer
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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_{...
Asaf Shachar's user avatar
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7 votes
1 answer
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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 ...
SashaKolpakov's user avatar
3 votes
1 answer
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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 $D$}...
Harish's user avatar
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6 votes
1 answer
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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 ...
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3 votes
1 answer
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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 ...
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2 votes
0 answers
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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, \...
Hheepp's user avatar
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1 vote
0 answers
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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 \...
Ali Taghavi's user avatar
4 votes
0 answers
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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 $-\...
Alex M.'s user avatar
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5 votes
1 answer
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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 ...
Alex M.'s user avatar
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4 votes
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174 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 $\...
freeRmodule's user avatar
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5 votes
2 answers
237 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 ...
truebaran's user avatar
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3 votes
1 answer
69 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 ...
Akoben's user avatar
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0 answers
178 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 ...
freeRmodule's user avatar
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3 votes
1 answer
151 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 ...
emma's user avatar
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3 votes
1 answer
214 views

An elliptic operator whose corresponding 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 ...
Ali Taghavi's user avatar
2 votes
0 answers
97 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) \...
Harto Saarinen's user avatar
1 vote
0 answers
183 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(...
DLIN's user avatar
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3 votes
0 answers
79 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 > ...
Pan's user avatar
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5 votes
1 answer
297 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 ...
Noether's user avatar
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2 votes
1 answer
118 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 $f$ ...
Fedor Goncharov's user avatar
2 votes
1 answer
128 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 ...
Dale Wintermann's user avatar
1 vote
0 answers
59 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 ...
Ali Taghavi's user avatar
7 votes
1 answer
573 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{...
truebaran's user avatar
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2 votes
1 answer
393 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, ...
Peter's user avatar
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2 votes
2 answers
436 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 ...
Ali Taghavi's user avatar
8 votes
3 answers
777 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'...
Saal Hardali's user avatar
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5 votes
1 answer
168 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 ...
kt77's user avatar
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1 vote
0 answers
203 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 ...
Peter's user avatar
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3 votes
0 answers
536 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 ...
Math's user avatar
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3 votes
0 answers
87 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_{...
C. Dubussy's user avatar
3 votes
0 answers
108 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 ...
B K's user avatar
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5 votes
2 answers
270 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, ...
Beni Bogosel's user avatar
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8 votes
0 answers
451 views

Measuring the non-commutativity of the codifferential and pullbacks

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\N}{\...
Asaf Shachar's user avatar
  • 6,611
1 vote
1 answer
119 views

Invariance of the space of harmonic functions under derivation associated to a non-vanishing vector field

Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of ...
Ali Taghavi's user avatar
6 votes
1 answer
162 views

local property of nonlocal differential operators

Motivation: In general, a nonlocal operator acting on the product of two functions doesn't have the product rule as the local operator does. However, the Hilbert transform on the real line has a very ...
Ivy Hsu's user avatar
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4 votes
0 answers
99 views

Reference Request: De Rham isomorphism with Hilbert space coefficients

Let $M$ be a smooth, closed manifold, equipped with a smooth (finite) triangulation $K$. Further, let $H$ be a Hilbert space, $G := \pi_1(M)$ and let $\rho: G \to GL(H)$ be a representation (with $GL(...
H1ghfiv3's user avatar
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1 vote
1 answer
154 views

The index of certain differential operator on tori

Assume that $J$ is an almost complex structure on torus $\mathbb{T}^2$. Let $X$ be a non vanishing vector field on the torus. Let $g$ be a Riemannian metric with corresponding $LC$ connection $...
Ali Taghavi's user avatar
0 votes
0 answers
255 views

Gradient of the trace of the logarithm of a product

Suppose $G$ and $A$ are full rank matrices. Is there a closed-form solution for $$\nabla_G \mbox{Tr} (A \log GG^\top)$$ when $A$ is a PSD matrix?
Soheil Feizi's user avatar
2 votes
1 answer
146 views

Realization of symbol of Laplace operator via certain integral

Is there an elliptic operator $D$ on $C^{\infty}(S^2)$ whose principal symbol is not identical to thats of Laplacian but it satisfies $\int_{S^2} fDf =\int_{S^2} f\Delta (f)$ for all $f\in C^{\infty}(...
Ali Taghavi's user avatar
3 votes
0 answers
179 views

Prove a certain function maps to upper half plane

Suppose $M$ is a bounded self-adjoint operator on space of complex valued functions on the real line $S_1=L^2(\mathbb{R},a(x)dx)$, where $a(x)$ is a nice real positive analytical function ( I have in ...
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