Questions tagged [differential-operators]

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

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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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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
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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, ...
Luke's user avatar
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-4 votes
1 answer
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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
211 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 $1+A$...
Curious's user avatar
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1 answer
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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)(...
zoran  Vicovic's user avatar
0 votes
0 answers
53 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
114 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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1 answer
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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
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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
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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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2 votes
0 answers
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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
613 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
193 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 answer
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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
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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
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2 votes
1 answer
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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
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1 vote
0 answers
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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
188 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
225 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
160 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
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0 votes
0 answers
83 views

Operators decomposition in pseudo-Hilbert space

Let $(H,g)$ be a pseudo-Hilbert space, i.e. $H$ is an infinite dimensional vector space endowed with an indefinite symmetric product $g$. Suppose we have a linear operator $D:H\to H$ and let $D^*$ be ...
John117's user avatar
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3 votes
0 answers
78 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
282 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
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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
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2 votes
1 answer
150 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
119 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
28 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
100 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
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1 vote
0 answers
81 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
  • 318
0 votes
0 answers
76 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
540 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
203 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
  • 318
3 votes
1 answer
102 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
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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
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13 votes
1 answer
406 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
0 votes
1 answer
75 views

Rotation of the coordinate system for multi-index differentiations

Let $\mathbf{f} = (f_1,\dotsc, f_n)$ be a $C^l$-map from an open subset $U$ of $\mathbb R^d$ to $\mathbb R^n$, and let $x_0 \in U$ be such that $\mathbb R^n$ is spanned by partial derivatives of $f$ ...
taylor's user avatar
  • 425
0 votes
1 answer
103 views

Explicit solution of the Lamé equation for n=1

The Jacobi form of Lamé equation is given by \begin{equation} \left(\frac{d^2 }{du^2} - n(n+1)k^2 \operatorname{sn}^{2}(u)-E\right)\Psi (u) = 0, \end{equation} where $k\in(0, 1)$ is parameter ...
dannyt's user avatar
  • 51
5 votes
1 answer
451 views

The principal symbol as an element in the K-theory

This line The symbol may naturally be thought of as an element in the K-theory of X appears in the nLab page on principal symbols for differential operators. What does this mean? Are they talking ...
Jake Wetlock's user avatar
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3 votes
0 answers
163 views

Shift Operators and the Weyl Algebra

I have a question about the action of a shift operator $E$ on polynomials $Ep(x) = p(x+1)$ in the context of linear differential operators in one variable with polynomial coefficients, i.e. ...
the_sandcastler's user avatar
4 votes
0 answers
118 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?...
Jesús A. Álvarez López's user avatar
0 votes
0 answers
187 views

A question about second fundamental form of Riemannian isometric embedding

I have got a question unsolved for some time. I do not know whether it is trivial or not: **I omit a very important fact: The metric at point p is second-order flat, i.e. $d_p \phi(-,v) = 0$ and $d_p^...
threeautumn's user avatar
1 vote
0 answers
21 views

Regularity of solutions of a 2nd order singular integro-differential operator

I have trouble finding the regularity of the solutions to a particular equation. I define $$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}...
BlueCharlie's user avatar
1 vote
0 answers
81 views

The module generated by kernel of an elliptic differential operator

Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\...
Ali Taghavi's user avatar
2 votes
0 answers
153 views

Compactness of a nonlinear operator

Let $H^{1}_{0}(0;\pi)=\{f\in L^{2}(0; \pi): f^{\prime}\in L^{2}(0; \pi)\ \text{and}\ f(0)=f(\pi)=0 \} .$ equipped with the following norm $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$ ...
Jaouad's user avatar
  • 31
0 votes
0 answers
72 views

Linear dependence of the derivatives of a vector valued function

Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function $$ g:\mathbb{R}^5\rightarrow\mathbb{R}^5 $$ given by $$ g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
BlaCa's user avatar
  • 1
0 votes
0 answers
74 views

Discontinuity of the Fourier transform of $ x \mapsto (1+ x^2)^{- \gamma/2}$ for $\gamma \leq 1$

Fix $\gamma > 0$. Let $\mathcal{F}$ be the Fourier transform and consider the function $f(x) = (1+ x^2)^{- \gamma/2}$ for $x \in \mathbb{R}$. This function is in $\mathcal{S}'(\mathbb{R})$ and its ...
Goulifet's user avatar
  • 2,142
0 votes
0 answers
90 views

Positive definite matrix and Hörmander theory

Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$ Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set $$ \varphi_{\...
Lucas De Souza's user avatar

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