Questions tagged [differential-operators]
Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
560 questions
0
votes
0
answers
52
views
Integrals over low-dimensional moving manifolds
In the context of certain stochastic interacting particle systems, I got into the following problem from differential geometry.
Setup.
Consider the two-dimensional torus $\mathbb T = \mathbb R^2/\...
- 201
2
votes
0
answers
85
views
Tychonoff counterexample for the fractional heat equation
The celebrated 1935 counterexample by Tychonoff shows that, with $\psi(t)=H(t) e^{-t^{-\nu}} $, with $\nu>1$ and $H=\mathbf 1_{\mathbb R_+}$,
the smooth function $u$ of two real variables given by
$...
- 16.7k
3
votes
1
answer
147
views
Smoothness of the differential on the group of diffeomorphisms over a compact Riemannian manifold
Let $(M, \langle \cdot, \cdot \rangle)$ be a closed Riemannian manifold (compact without boundary) and consider $\mathcal{D}^s$ the group of diffeomorphisms from $M$ to $M$ of class $H^s$ ($s$-th ...
- 165
1
vote
1
answer
202
views
An exact sequence of jet space associated to a vector bundle
Suppose $L$ be a line bundle over Riemann surface $X$. Then show that $ 0 \longrightarrow J^2(L) \longrightarrow J^1(J^1(L)) \longrightarrow L\otimes K_X \longrightarrow 0 ,$ where $J^k(L)$ is the $k$-...
- 39
2
votes
3
answers
315
views
Reference for relation between $ \left( \frac{1}{x} D \right)^{n+1} $ and the Bessel coefficients
Background
In the OEIS sequence A001498, the coefficients of the Bessel polynomials are described. They adhere to the formula $$ a(n, k) = \frac{(n+k)!}{\left(2^k \ (n-k)! \ k! \right)} \tag{1} \label{...
- 5,065
8
votes
1
answer
478
views
Tomonaga-Schwinger evolution equation: Rigorous setting?
Is there any rigorous setting for formulating and studying Tomonaga-Schwinger equations?
In quantum field theory (QFT), in particular in quantum electrodynamics (QED), the dynamic evolution in time is ...
- 332
3
votes
0
answers
105
views
Length of a formally exact compatibility complex for linear differential operator
Let $\Delta:\Gamma(E_0)\rightarrow\Gamma(E_1)$ be a linear differential operator (LDO) between vector bundles $E_0,E_1\rightarrow M$ (everything is assumed smooth by default, $\dim M=m$, and LDOs are ...
- 3,432
5
votes
0
answers
153
views
On Schuricht and Schönherr approach for proving the divergence theorem on general Borel sets
This question stems from a ZBmath search I did yesterday evening, and it is somewhat related to the following MathOverflow question: "On which regions can Green's theorem not be applied? ".
...
- 6,830
0
votes
0
answers
62
views
Commutator estimates for non-local operator
I am looking for a commutator estimate for a non-local operator: namely, for functions $f,g :\mathbb{R}^2\rightarrow \mathbb{R}$ (nice enough functions) can we find a commutator estimate of the form
$$...
- 101
1
vote
0
answers
129
views
Norm of the resolvent of the Laplacian acting on $L^p$
Let $M$ be a closed Riemannian manifold and let $\Delta$ be the positive Laplacian acting on functions on $M$. If $\lambda$ is not an eigenvalue of $\Delta$, we have the resolvent operator $R_\lambda=(...
- 374
-4
votes
1
answer
213
views
$L^p$ operator norm decay
Assume $f$ and $g$ are smooth compactly supported functions on the real line and $A$ and $B$ are linear self-adjoint operators on $L^2(\mathbb{R}^n)$. If the operators $A$ and $B$ commute is it ...
- 1
0
votes
0
answers
39
views
Decay of fundamental solution of differential operator
How can we find the decay of the fundamental solution of the 1D operator $L=D_x^{\alpha}+1$ for a real number $2>\alpha\geq 1$ where the differential operator $D_x^{\alpha}$ is defined by the ...
- 101
2
votes
0
answers
97
views
Inequality of Peetre for a positive operator
How can one prove the interpolation inequality of Peetre for a positive operator, stated as:
$$
A^s \leq C A^t + C' I, \quad \text{for } 0 \leq s \leq t,
$$
where ( A ) is a positive self adjoint ...
- 571
1
vote
0
answers
86
views
Estimation of Schatten norm
Let $L$ be the Laplacian operator on the Heisenberg group $\mathbb{H}^n$. The Fourier transform on this group is defined as follows: for $f \in L^1(\mathbb{H}^n)$, the Fourier transform $\widehat{f}(\...
- 571
1
vote
0
answers
86
views
Proof of codifferential identity [closed]
Let $(M,g)$ be a Riemannian manifold, $\omega \in \Omega^p(M)$ is a $p$-form and $D$ a linear connection in $T(M)$ such that $D(g)=0$ and is symmetric (this gives us the formula $D\wedge\omega=d\omega$...
- 111