Questions tagged [differential-operators]
Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
542 questions
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Differentially closed fields
Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$.
Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...
- 2,751
7
votes
0
answers
188
views
$\eta$ invariants of Pin+ manifolds $\mathbb{RP}^{8k}$
In general, I am curious about the 'quantization' of $\eta$-invariants on Pin+ manifold, i.e., what is the 'minimal unit' of $\eta$-invariants on a manifold with certain choice of Pin+ structure.
...
- 575
6
votes
3
answers
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Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?
The adjoint of the exterior derivarive is defined by
$\delta:=(-1)^k\ast^{-1}d\ast$,
but I need a way which avoids the Hodge $\ast$ operator.
Is there another definition?
For example, for ...
- 4,312
6
votes
1
answer
2k
views
Relation between harmonic vector field and harmonic 1-form
Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function
$$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^...
- 4,195
6
votes
1
answer
241
views
Self-adjointness and choosing appropriate function spaces
Consider the following operator on some (yet undecided) space $S$ of functions over $[0\:\:1]$
$$L(u)=\sin(x)u-x\dfrac{\partial u}{\partial x}$$
Now, its formal adjoint is
$L^*(v)=\sin(x)v+\dfrac{\...
- 318
6
votes
1
answer
2k
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is $\nabla \cdot ( c^2 \nabla)$ a Laplace-Beltrami operator?
Someone mentioned, in passing, to me that $u \mapsto \nabla \cdot ( c^2 \nabla u)$ is a Laplace-Beltrami operator. Does anyone have some insight into this? From my understanding, the Laplace-operator ...
- 155
6
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3
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681
views
Reference request: Long-term behaviour of the heat equation for bounded initial data
Let us consider the heat equation
\begin{align*}
\frac{\partial}{\partial t}u(t,x) & = \Delta u(t,x), \\
u(0,x) & = f(x)
\end{align*}
on the whole space $\mathbb{R^d}$. If $f \in L^p := L^...
- 12.6k
6
votes
1
answer
575
views
Spectrum of the complex harmonic oscilllator
Let
$$
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\...
- 521
6
votes
2
answers
934
views
Surface Laplace-Beltrami without coordinates, exterior calculus?
Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator $\...
- 3,059
6
votes
1
answer
1k
views
Laplace-Beltrami and the isometry group
H$\vphantom{a}$i. Consider the Laplacian on $\mathbb R^n$,
$$
\Delta=\partial_i^2
$$
It is easy to prove that the most general differential operator that commutes with rotations and translations is ...
6
votes
2
answers
448
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 (...
- 789
6
votes
1
answer
307
views
Hochschild cohomology and differential operators
The Hochschild-Kostant-Rosenberg theorem says, that for a commutative algebra $R$ over a field $k$ with certain smoothness and finiteness, we have an identification $\mathrm{HH}^\bullet(R)\cong \...
- 985
6
votes
1
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351
views
Fredholm theory of non elliptic operators
In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
6
votes
2
answers
2k
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The spectrum of the Hodge Laplacian on a Riemannian manifold
The Hodge Laplacian operator on differential forms on a (compact?) Riemannian manifold carries useful information about the topology of the manifold. In particular, the multiplicity of the zero ...
- 7,746
6
votes
1
answer
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Construction of non-split extension of simple modules of Lie algebras using linear differential operators
Consider the natural action of $W_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentially a $\mathfrak{sl}_2$-tuple ($\left[...
- 6,622
6
votes
1
answer
460
views
Abstract definition of differential operators
Let $E,F$ be complex vector bundles over some closed manifold $M$. We investigate operators $T:C^{\infty}(M,E) \to C^{\infty}(M,F)$ between smooth sections of these bundles. We say that such operator ...
- 243
6
votes
1
answer
173
views
local property of nonlocal differential operators
Motivation: In general, a nonlocal operator acting on the product of two functions doesn't have the product rule as the local operator does. However, the Hilbert transform on the real line has a very ...
- 78
6
votes
1
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653
views
Elliptic Operators on Vector Bundles
I know the kernel of an elliptic operator on a compact manifold has finite dimension. Is the kernel of an elliptic operator on sections of a vector bundle a finite dimensional space?
- 61
6
votes
1
answer
629
views
Index of a differential operator between trivial bundles.
Let $M$ be a closed parallelizable manifold and $D: \Gamma(E) \to \Gamma(F)$ an elliptic differential operator between trivial vector bundles $E,F \to M$. The Atiyah Singer index theorem implies that ...
- 3,244
6
votes
1
answer
642
views
Twisted differential operator, chiral differential operator, $???$ (continue the sequence)
Let $X$ be a smooth variety.
One can define the notion of a sheaf of twisted differential operators (TDO) on $X$. They "quantise" functions on $T^*X$. Examples include the usual sheaf of ...
- 5,701
6
votes
1
answer
382
views
Elliptic operator with finite spectrum?
Is it possible for a (non-symmetric) elliptic differential operator to have finite spectrum. If so, is there an explicit example?
- 494
6
votes
2
answers
1k
views
The adjoint operators as elliptic operators
Edit:
It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally ...
- 356
6
votes
1
answer
436
views
Definition of the nonlinear part of the drift in a (stochastic) Navier-Stokes equation
Let
$T>0$
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\overline{\mathcal V}^{\left\|\;\cdot\;\...
- 167
6
votes
3
answers
794
views
A possible trace (inequality) defined under negative Sobolev scale
Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space $H^{-s}(\partial \Omega)$? We usually encounter trace theorem as
$$Tr: H^{s}(\Omega) \...
- 61
6
votes
1
answer
156
views
Ordinary differential operators satisfying braid relation?
Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...
user74900
6
votes
0
answers
113
views
Schwartz kernel of spectral projection of Laplacian and integrated density of states
I'm reposting here a question I asked on MSE which did not receive an answer.
I am considering the Dirichlet Laplacian $\Delta$ on some smooth domain $U$. For now assume that $U$ is bounded, and later ...
- 251
6
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0
answers
124
views
An application of Leray-Schauder degree theory for Nirenberg problem on the 2-sphere
I'm studying the article "The scalar curvature equation on 2- and 3-spheres" by Chang, Gursky and Yang and I'm particulary interested in the 2-sphere case.
They prove that if $K:S^2\...
- 521
6
votes
0
answers
2k
views
On the definition of Gysin homomorphism
According to Lawson-Michelsohn's book (p239), the Gysin homomorphism for a continuous map between (compact) manifolds $f:Y\to X$ is defined by setting
$$
f_!=PD_X^{-1}\circ f_*\circ PD_Y
$$
where $PD$ ...
- 2,789
6
votes
0
answers
375
views
Is there an analog of the Levi–Civita connection for schemes?
Is there an analog of the Levi–Civita connection for schemes?
There exists algebraic de Rham theory, $n$-forms on vector bundles (algebraically), and familiar constructions from differential geometry....
- 327
6
votes
0
answers
116
views
Topological constraints for existing of certain differential operators on manifolds
At the beginning a word of warning: this would be rather vague question: vague as it is, I'm not requiring a precise answer, rather some intuitive explanation.
In the flat case $M=\mathbb{R}^n$ ...
- 9,330
6
votes
0
answers
141
views
Analytic continuation of $\mathfrak{so}(n)$ algebras to real $n$?
In a 1988 paper "The Lie algebras $\mathfrak{gl}(\lambda)$ and cohomologies of Lie algebras of differential operators", Feigin defined the analytic continuation of $\mathfrak{sl}(n)$ algebras (over $\...
- 679
6
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0
answers
306
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How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?
The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$
For $z=x+ i y \in \mathbb C$ ...
- 650
5
votes
1
answer
1k
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 $...
- 749
5
votes
1
answer
543
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 ...
- 1,144
5
votes
2
answers
305
views
Simplification of integral on the sphere
In the article: https://arxiv.org/abs/0906.3217 the authors prove in Lemma 1 a formula which helps compute more easily the integral of the Hessian of a function defined on $\Bbb{S}^2$. More precisely, ...
- 2,222
5
votes
2
answers
904
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 ...
- 5,405
5
votes
2
answers
977
views
Symbol of the Laplace-Beltrami on $\mathbb{S}^2$
This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e.
A differential operator $P=\sum_{|\...
- 329
5
votes
1
answer
932
views
Mellin transform between heat kernel and zeta-function
For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of square-...
- 617
5
votes
1
answer
584
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?
- 1,422
5
votes
2
answers
3k
views
Diagonalization of a matrix of differential operators
Dear community,
i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them.
To explain my question i will use an example:
Let $V^k$ be the ...
5
votes
2
answers
814
views
index of a family of Dirac operators in $K^1$
Suppose I have a family of Dirac operators over a compact base space B. From the paper of Atiyah and Singer about skew adjoint Fredholm operators we know that it has an index in $K^1(B)$.
Suppose ...
- 1,282
5
votes
2
answers
233
views
Analytic approximations of smooth vector fields
Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with
$$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$
on $\mathbb{R}^3$ for any $\alpha,K$.
Further, we ...
- 749
5
votes
1
answer
308
views
The division problem for tempered functions
It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...
- 193
5
votes
1
answer
227
views
Are all the mappings which satisfy this equation scaled isometries?
Let $M,N$ be smooth oriented $d$-dimensional Riemannian manifolds, $\, f:M \to N$ a smooth map. Let $\Omega^1(M,f^*TN)=\Gamma(T^*M \otimes f^*TN)$ be the space of $f^*TN$-valued one-forms.
Let $d$ ...
- 6,741
5
votes
2
answers
353
views
For which tempered distributions is the fractional derivative well-defined?
Let $\gamma \geq 0$ and consider the fractional derivative operator defined in Fourier domain by
$$\mathcal{F} \{\mathrm{D}^{\gamma} \varphi \} (\omega) = (\mathrm{i} \omega)^{\gamma} \mathcal{F}\{\...
- 2,306
5
votes
2
answers
247
views
Elements of graded algebra associated with the algebra of differential operators as smooth sections
Let $M$ be a compact manifold and $E$ a complex vector bundle. We will consider differential operators $P$ acting between $\Gamma^{\infty}(M,E)$. Let $\mathcal{P}$ be the algebra of all differential ...
- 9,330
5
votes
1
answer
449
views
Practical way to check whether a distribution is conormal
Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L_1 \...
- 1,329
5
votes
1
answer
557
views
Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?
Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$.
Is there an ...
- 2,399
5
votes
1
answer
618
views
Symbol map in Getzler calculus
I hope someone can help me, although this question is rather specific.
I am reading John Roe's chapter on Getzler symbols in "Elliptic operators, topology and asymptotic methods" to understand the ...
- 9,853
5
votes
2
answers
619
views
Is there a generalization of Floquet theory to elliptic functions?
Hi,
Consider a system of linear differential equations
$$
{d f \over dz} = A(z) f,
$$
where $A(z)$ is a matrix-function. If $z \in \mathbb{R}$ and the function is periodic $A(z) = A(z + T)$, ...
- 649