Questions tagged [differential-operators]

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

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7 votes
2 answers
955 views

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 views

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 views

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 views

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 views

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 views

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
65 views

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
3k views

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
127 views

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 views

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 views

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
1k views

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 views

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 views

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 (...

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