Questions tagged [differential-operators]

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

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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 ...
Davidi Cone's user avatar
3 votes
3 answers
305 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 $t$...
Math2024's user avatar
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? ...
Math2024's user avatar
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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 ...
Blind Miner's user avatar
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 ...
Filippo's user avatar
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16 votes
1 answer
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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 ...
Denis Serre's user avatar
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3 votes
1 answer
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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\...
Bence Racskó's user avatar
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 ...
jg1896's user avatar
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2 votes
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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 $$ \langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)} $$ ...
Karim's user avatar
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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 ...
jg1896's user avatar
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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-...
SebastianP's user avatar
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 ...
Puzzled's user avatar
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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 ...
diff-geo's user avatar
7 votes
2 answers
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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 ​​...
kumquat's user avatar
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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 ...
tnrtnr's user avatar
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1 answer
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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 ...
CBBAM's user avatar
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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, ...
Peter Wacken's user avatar
-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 ...
Gauge's user avatar
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4 votes
1 answer
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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 $1+A$...
Curious's user avatar
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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)(...
zoran  Vicovic's user avatar
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 ...
Fetchinson0234's user avatar
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 ...
Marco's user avatar
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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{...
jg1896's user avatar
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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 ...
Ali's user avatar
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1 vote
0 answers
122 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 $I$ ...
Pii_jhi's user avatar
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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 ...
Max Reinhold Jahnke's user avatar
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 ...
asv's user avatar
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4 votes
1 answer
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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 ...
Cameron Williams's user avatar
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/...
Arturo's user avatar
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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 ...
Alex's user avatar
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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{...
kumquat's user avatar
  • 63
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 ...
VojtaK's user avatar
  • 151
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 ...
Ali Taghavi's user avatar
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\...
Samir's user avatar
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3 votes
1 answer
333 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 $i=1,\ldots, r$, ...
A. J. Pan-Collantes's user avatar
5 votes
1 answer
203 views

Domains with discrete Laplace spectrum

Let $\Omega \subset \mathbb{R}^n$ be a domain. Assume that the Laplacian $-\Delta=-(\partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_n^2)$ has a discrete spectrum on $L^2(\Omega)$ (i.e., we are ...
mmen's user avatar
  • 433
3 votes
0 answers
94 views

Existence result for an operator obtained by integrating Laplace-Beltrami operator to normal direction in Fermi coordinate

I am going through some literature and encountered with some known facts about Fermi coordinate and Laplace-Beltrami operator. Let $u$ be a function on $\mathbb{R}^{n+1}$ and $\Gamma_0$ be a $0$ level ...
user494715's user avatar
3 votes
1 answer
409 views

On the domain of the Neumann Laplacian

Let $U$ be a bounded domain of $\mathbb{R}^d$, and write $m$ for the Lebesgue measure on $U$. For $k=1,2$, we denote by $H^k(U)$ the set of all locally $m$-integrable functions $u\colon U \to \mathbb{...
sharpe's user avatar
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2 votes
1 answer
140 views

On a core for Neumann Laplacian on $C(\overline{D})$

Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p_t\}_{t>0}$ denote ...
sharpe's user avatar
  • 701
2 votes
1 answer
201 views

The heat equation for complex time

Let $\Delta$ be a Laplacian or an elliptic operator over a manifold, can the heat equation be defined for complex time? Can we define: $$e^{-z \Delta}$$ for $Re(z)>0$ ? Also can the Ricci flow be ...
Antoine Balan's user avatar
4 votes
1 answer
184 views

Is the Sobolev space $H^1(\mathbb{R})$ contained in the domain of $(-\partial_x \alpha(x) \partial_x)^{1/2}$?

Let $\alpha(x) : \mathbb{R} \to (0,\infty)$ have bounded variation (BV) and suppose $\inf_{\mathbb{R}} \alpha > 0$. Consider the second order differential operator $$H : =-\partial_x (\alpha(x) \...
JZS's user avatar
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1 vote
0 answers
35 views

Approximating spectra of (finite rank pertubations of) Laurent operators by spectra of (pertubations of) periodic finite operators

A tridiagonal matrix is a matrix which only has elements on three diagonals. So for $\alpha, \beta, \gamma \in \mathbb{C}$ consider the bi-infinite tridiagonal Laurent operator $T$ with $\beta $ on ...
Frederik Ravn Klausen's user avatar
2 votes
0 answers
102 views

Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform

Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...
Mirar's user avatar
  • 350
1 vote
0 answers
93 views

Kernel representation of a power of (pseudo-)differential operator

Let $\mathcal{T}$ be a (pseudo-)differential operator that admits the following kernel representation: \begin{equation} \mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt. \end{equation} What can ...
Mirar's user avatar
  • 350
0 votes
0 answers
106 views

Elliptic partial differential equations with Robin boundary condition and domain of fractional power of Robin Laplacian operator

This question has been posted on Mathematics Stack Exchange but got no response, and so I put it here. When I read the paper "On the attractor for a semilinear wave equation with critical ...
monotone operator's user avatar
6 votes
1 answer
569 views

Spectrum of the complex harmonic oscilllator

Let $$ H_\lambda=-\frac{d^2}{dx^2}+\lambda^2 x^2,\quad\lambda>0. $$ It is known that the spectrum of $H_\lambda$ is the set $\{(2n-1)\lambda,n\in \Bbb N^*\}$. Now put $$ (U_\mu \phi)(x)= e^{\mu\...
zoran  Vicovic's user avatar
3 votes
0 answers
237 views

A generalization of Weierstrass transform

As stated in this article, the Weierstrass transform of $f(x)$ is defined as: \begin{equation} W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy \end{equation} which can be ...
Mirar's user avatar
  • 350
3 votes
1 answer
106 views

Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (say $D: H_0^k(\Omega) \to H_0^{k-1}(\Omega)$)

Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator between the ...
Overflowian's user avatar
  • 2,523
1 vote
0 answers
73 views

Highy non-linear PDE involving directional derivative

Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed \begin{equation}\label{ConvoDef} \left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
MrPie 's user avatar
  • 185
13 votes
1 answer
431 views

Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature?

The physicist Yoichiro Nambu introduced in a 1950 paper A Note on the Eigenvalue Problem in Crystal Statistics the notion of an "eigenoperator" (page 12, see Nambu and the Ising model for a ...
Carlo Beenakker's user avatar

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