Questions tagged [differential-operators]
Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
503
questions
7
votes
2
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379
views
+50
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 ...
- 63
1
vote
1
answer
119
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 ...
- 21
1
vote
1
answer
181
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 ...
- 327
3
votes
2
answers
98
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, ...
- 193
-4
votes
1
answer
96
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 ...
- 1
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 $1+A$...
- 113
1
vote
1
answer
201
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)(...
- 415
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 ...
- 986
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 ...
- 253
4
votes
1
answer
129
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{...
- 1,927
2
votes
0
answers
38
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 ...
- 3,883
1
vote
0
answers
105
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 $I$ ...
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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 ...
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 ...
- 20.4k
4
votes
1
answer
101
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 ...
- 350
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/...
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1
vote
1
answer
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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 ...
- 63
1
vote
0
answers
90
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.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 63
2
votes
1
answer
97
views
Can I characterize functions (in 2D), which will have compactly supported/support contained Poisson solution?
I have the problem of solving Poisson equation in 2D
$$
\Delta u = f
$$
Let's say for a moment I want to solve it on $\mathbb{R}^2$, for $f(x,y), x\in \mathbb{R}, y\in \mathbb{R}$.
I know however that ...
- 151
1
vote
0
answers
93
views
The asymptotic growth of codimension of range of polynomial differential equation on finite fields
Inspired by the seminal paper of Andre Weil on the number of solutions of equations on finite fields we would like to present the following question:
Let $P(x,y), Q(x,y)$ be two polynomials of ...
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2
votes
1
answer
188
views
Linear elliptic equation
Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
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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$, ...
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 ...
- 413
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 ...
- 385
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 ...
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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{...
- 681
2
votes
1
answer
135
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 ...
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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}$$
for $Re(z)>0$ ?
Also can the Ricci flow be ...
- 377
4
votes
1
answer
119
views
Is the Sobolev space $H^1(\mathbb{R})$ contained in the domain of $(-\partial_x \alpha(x) \partial_x)^{1/2}$?
Let $\alpha(x) : \mathbb{R} \to (0,\infty)$ have bounded variation (BV) and suppose $\inf_{\mathbb{R}} \alpha > 0$. Consider the second order differential operator
$$H : =-\partial_x (\alpha(x) \...
- 447
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 ...
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 ...
- 318
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 ...
- 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 ...
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\...
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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 ...
- 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 ...
- 2,493
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_{\...
- 185
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 ...
- 168k
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$ ...
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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 ...
- 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 ...
- 1,034
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. ...
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?...
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^...
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}...
- 81
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^\...
- 303
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 $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$
...
- 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
$$
g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
- 1
0
votes
0
answers
74
views
Discontinuity of the Fourier transform of $ x \mapsto (1+ x^2)^{- \gamma/2}$ for $\gamma \leq 1$
Fix $\gamma > 0$. Let $\mathcal{F}$ be the Fourier transform and consider the function
$f(x) = (1+ x^2)^{- \gamma/2}$ for $x \in \mathbb{R}$. This function is in $\mathcal{S}'(\mathbb{R})$ and its ...
- 2,142
0
votes
0
answers
90
views
Positive definite matrix and Hörmander theory
Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$
Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set
$$
\varphi_{\...
- 101