Questions tagged [differential-operators]
Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
515
questions
7
votes
2
answers
955
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Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
-1
votes
0
answers
22
views
Leibniz rule for poly-harmonic operators
It is well-known that, for two smooth functions $u,v$ defined in $\mathbb{R}^n$ ($n\geq 2$), the laplace operator shows $$\Delta (u v) = v\,\Delta u + 2\,\nabla u\,\nabla v+u\,\Delta v,$$what if the ...
4
votes
1
answer
206
views
Conormal distributions and the wave front set
Let $X$ be a smooth closed manifold and $Y$ a regular submanifold. For all conormal distributions at $Y$ on $X$, their wave front set is contained in the conormal bundle of $Y$. Is the reciprocal true?...
3
votes
3
answers
304
views
Generalized Fuchsian-type PDE?
Consider
$$
\big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0
$$
with the initial condition $A(x,0)=1$. In a small $t$...
4
votes
0
answers
412
views
A 4th-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$x^3 f_{xxxt}+ f =0$
Does anyone know if this type of PDE already appeared in the literature? ...
0
votes
0
answers
56
views
Computing the eta invariant of a rather contrived operator on the circle
For physical reasons, I am interested in computing the eta invariant of the following Hermitian operator acting on complex valued functions on the circle with circumference 1. I define the operator ...
3
votes
3
answers
613
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Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation
The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry,
$...
1
vote
1
answer
142
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Let $H$ be a Laplacian. Is the covariant derivative $\nabla$ such that $H+\operatorname{Tr}(\nabla^2)$ is of order zero unique?
Let $E$ be a vector bundle over a Riemannian manifold $M$. Furthermore let $H$ be a generalized Laplacian, i.e.
$$\forall f\in C^\infty(M):[[H,f],f]=-|df|^2.$$
Proposition 2.5 in Heat Kernels and ...
16
votes
1
answer
713
views
The determinant as a differential operator
According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
7
votes
2
answers
560
views
Line graphs called "graph derivatives": any intuition?
Short version: in several papers, line graphs (and closely related graphs) are called graph derivatives or derived graphs; is there any intuition for such terminologies, in connection with the ...
3
votes
1
answer
167
views
Precise definition of a linear total differential operator
In the works of A. M. Vinogradov on calculus on the infinite jet space, differential equations and "diffieties", a central notion is that of a $\mathcal C$-differential operator. If $\pi:Y\...
8
votes
0
answers
217
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 ...
4
votes
1
answer
292
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Some folklore about crystaline rings of differential operators
This question is a follow up to my previous question on rings of crystaline differential operators, to which I refer for the adequate definitions.
First, let's consider the case of an algebraically ...
3
votes
1
answer
192
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Relation between enveloping algebras and algebras of differential operators
I asked this question on math stack exchange about 3 years ago, but received no answer.
Our base field $\mathsf{k}$ will be algebraically closed of zero characteristic. Let $X$ be an smooth affine ...
2
votes
0
answers
62
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The $n$-th reproducing kernel of orthogonal polynomial
Let $N$ be a non negative integer. Define the sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ with respect to the inner product
$$
\langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)}
$$
...
0
votes
0
answers
47
views
Rotational invariance of Laplace-Beltrami eigenvalue problem on smooth manifolds
I am currently looking at the eigenvalue problems of the Laplace-Beltrami operator. Let $(M,g$) be a smooth and oriented Riemann manifold. I am investigating the eigenvalue problem of the Laplace-...
0
votes
0
answers
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A differential operator of differential operators
Consider a differential operator of the form $x^2-2\frac{\partial}{\partial x}$, where $x$ itself is the laplacian.
Does such an operator make sense? The motivation for this is that such an object is ...
7
votes
2
answers
570
views
What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
2
votes
1
answer
107
views
Can I characterize functions (in 2D), which will have compactly supported/support contained Poisson solution?
I have the problem of solving Poisson equation in 2D
$$
\Delta u = f
$$
Let's say for a moment I want to solve it on $\mathbb{R}^2$, for $f(x,y), x\in \mathbb{R}, y\in \mathbb{R}$.
I know however that ...
12
votes
2
answers
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What is the "correct" generalization of operator norms for nonlinear operators?
I have been recently wondering what is a (or even the) "correct" generalization of the notion of an operator norm to nonlinear operators?
Please excuse the naivete of my question; if you think that ...
5
votes
1
answer
979
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
vote
1
answer
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Is a differential operator $\frac{1}{2}\frac{d^{2}}{dx^{2}} - a x^{b} \frac{d}{dx}$ well-known?
I would like to know if the following differential operator on $(0,\infty)$ is well-known or derived from such one:
\begin{align}
L := \frac{1}{2}\frac{d^{2}}{dx^{2}} - a x^{b} \frac{d}{dx} \quad (a,b ...
3
votes
2
answers
123
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Lumer-Phillips-type theorem for non-autonomous evolutions
The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
1
vote
1
answer
330
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Reference request: inverse of differential operators
I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question).
As an example ...
4
votes
2
answers
689
views
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 ...
-4
votes
1
answer
116
views
Coordinate free computation of the second derivative of a functional [closed]
Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a ...
4
votes
1
answer
222
views
Diagonalizing selfadjoint operator on core domain
Let $A$ be a densely-defined, positive, self-adjoint operator with compact resolvent on a Hilbert space $H$. Then, $\text{Range}(1+A)=H$ and there is a basis for $H$ consisting of eigenvectors of $1+A$...
6
votes
1
answer
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Poisson structure on the cotangent bundle
Let $X$ be a smooth variety over $\mathbb{C}$ and $\mathscr{A}$ a sheaf of twisted differential operators on $X$. The latter comes equipped with a natural filtration and the associated graded algebra $...
2
votes
1
answer
175
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). ...
1
vote
1
answer
206
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Differential operators in $\Bbb R^n$
Put $P_j=\frac{\partial}{\partial \xi_j}$ et $Q_j=2 i \xi_j$ with$\xi=\left(\xi_1, \ldots, \xi_n\right)$ et $x=\left(x_1, \ldots, x_n\right)$. How to prove :
$\exp \left(\sum_{j=1}^n x_j P_j\right)(...
0
votes
0
answers
55
views
Determinant of 2D non-positive second order partial differential operator
If I have an ordinary second order differential operator the Gelfand-Yaglom method is often useful to calculate its (regularized) determinant. The great advantage is that one doesn't have to calculate ...
1
vote
1
answer
172
views
Pseudo-differential operators and differential operator
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 by ...
4
votes
1
answer
137
views
approximation of a Feller semi-group with the infinitesimal generator
Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator.
If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.
Is this formula always ...
4
votes
1
answer
166
views
Reference request on rings of crystalline differential operators
Let $\mathbb{k}$ be an algebraically closed field of positive characteristic, $X$ an affine smooth variety over it. Then the ring of crystalline differential operators on $X$ is generated by $\mathcal{...
2
votes
0
answers
49
views
A mapping property for fractional Laplace--Beltrami operator
Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
1
vote
0
answers
121
views
Ideals whose alebraic variety is a singleton
I do not work in algebra, so i apologize in advance if there are some unclear/wrong sentences. Let us consider the ring $\mathbb{C}[X_1,\ldots,X_q]$ of polynomials in $q$ variables. For an ideal $I$ ...
2
votes
0
answers
63
views
Künneth formula and continuity of the isomorphism
In the book Sheaf Theory, by Bredon (edition from 1997), Theorem 14.1, he writes a natural exact sequence, which, in some nice cases, leads to the Künneth formula. Do we have any reason to believe ...
4
votes
1
answer
108
views
Looking for a paper on (formally) self-adjoint differential operators
This is a long shot, but I've about lost my mind over this. About a year ago, I came across a paper published in the last 20-30 years (as it was neatly typeset in modern $\rm\LaTeX$ styles) that ...
10
votes
1
answer
709
views
Hodge decomposition in elliptic complexes
EDIT: In the book "Principles of Algebraic Geometry" by Griffiths and Harris the authors prove the Hodge decomposition for the Dolbeault operator $\bar\partial$ on differential forms on a ...
4
votes
1
answer
748
views
Infinitesimal generator of a Markov process
We define the infinitesimal generator of a Markov process as
$$Gf = \lim_{h\to 0} \frac{K_hf - f}{h}$$
where $f$ is a function on the state space, $K_h$ is a continuous time operator on the Markov ...
2
votes
1
answer
190
views
Linear elliptic equation
Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
1
vote
0
answers
791
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 every ...
5
votes
1
answer
559
views
differential operator power coefficients
Let $(F(x)\frac{d}{dx})^n=\sum_{i=1}^n H_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H_{n,i}(y_0, y_1, \ldots , y_n)$. What is known about it?
2
votes
1
answer
210
views
Hodge decomposition for non-elliptic complexes
It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/...
1
vote
1
answer
192
views
Solution of nonlinear differential equation $g = c_1 f^2 + c_2 (f')^2$ for function $f$
I would like to find an analytic solution (if possible) of the differential equation:
$g = c_1 f^2 + c_2 (f')^2$
Where $c_1$ and $c_2$ are constants, $g$ is a known function of $x$, $f$ is another ...
1
vote
0
answers
118
views
Maximal domain of an unbounded linear operator in a weighted Hilbert-space
Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
1
vote
0
answers
93
views
The asymptotic growth of codimension of range of polynomial differential equation on finite fields
Inspired by the seminal paper of Andre Weil on the number of solutions of equations on finite fields we would like to present the following question:
Let $P(x,y), Q(x,y)$ be two polynomials of ...
26
votes
2
answers
2k
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Codimension of the range of certain linear operators
Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map
$D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the differential operator corresponding ...
3
votes
1
answer
332
views
Existence of solution to linear inhomogeneous first order PDEs systems
Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response.
For $i=1,\ldots, r$, ...
5
votes
2
answers
361
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 (...