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16 votes
2 answers
1k views

How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $饾惗^饾憳$ in an open set, and we usually use these identities to calculate ...
YuerWu's user avatar
  • 415
11 votes
2 answers
712 views

Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad ...
Bazin's user avatar
  • 16.2k
11 votes
1 answer
668 views

Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?

Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
Saal Hardali's user avatar
  • 7,789
11 votes
1 answer
1k views

Research topics in microlocal analysis

Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
Andrew's user avatar
  • 589
7 votes
1 answer
2k views

A good reference for the wave front set

Hello, I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's ...
Anand's user avatar
  • 1,649
6 votes
2 answers
371 views

Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side: $$ \Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial t}+\beta^2w\right)+\...
Mechanical engineer's user avatar
5 votes
1 answer
281 views

de Rham theorem for tempered distributions

I am wondering if the following statement holds. If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
Will Kwon's user avatar
  • 323
4 votes
3 answers
1k views

A space of distributions vanishing on the boundary

The revised question After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let $$\mathcal ...
Alex M.'s user avatar
  • 5,407
4 votes
1 answer
225 views

Asymptotics of integral representation of distribution

I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community. Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
Dispersion's user avatar
4 votes
1 answer
792 views

How to show that this limit converges in the distributional sense to a dirac delta function

Let $$\begin{eqnarray}\nonumber f(y, t) &=& \frac{C}{\sigma ^2 t} \left[\frac{(1-\alpha) (b-y)}{\alpha t^{\alpha}} \, _1F_1\left[\frac{\alpha+1}{2 \alpha};\frac{3}{2};-\frac{ (b-y)^2}{2 \...
Comic Book Guy's user avatar
4 votes
2 answers
368 views

Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)? To be more detailed: if I want to show that some ...
Alex M.'s user avatar
  • 5,407
4 votes
0 answers
143 views

If theorem valid for compactly supported distribution then is it also valid for tempered distribution?

I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution. For instance, Theorem: Any $A \in \Psi^{m}$ ...
Curious student's user avatar
3 votes
2 answers
382 views

Singular support: equivalent definition

Let $U\subset\mathbb{R}^{d}$ be an open set. The singular support of a distribution $u\in\mathcal{D}^{\prime}(U)$ is defined to be the compliment of the set of points, which have a neighbourhood in ...
B.Hueber's user avatar
  • 1,171
3 votes
1 answer
182 views

Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product

Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing). For any pair of ...
Isaac's user avatar
  • 3,477
3 votes
1 answer
348 views

Well-posedness of heat equation with distributional right hand side

The question is about well-posedness of heat equation $$ \frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T], $$ subjected to ...
Mechanical engineer's user avatar
2 votes
1 answer
374 views

Simplify proof for rapidly decaying functions

I want to show the following theorem in a lecture: Let $F \in C^{\infty}(\mathbb{C}^{k}, \mathbb{C})$ such that $F(0)=0.$ Let $G: \mathbb{R}^n \rightarrow \mathbb{C}^{k}$, $x \mapsto (f_1(x),..,f_k(...
Jonathan's user avatar
  • 181
2 votes
0 answers
52 views

On distributions and kernels

Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
G. Blaickner's user avatar
  • 1,429
2 votes
0 answers
43 views

Distributions and time-kernels

Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
G. Blaickner's user avatar
  • 1,429
2 votes
0 answers
126 views

Mixed partial derivatives of planar functions converging to delta distribution

Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
cts12's user avatar
  • 51
1 vote
0 answers
95 views

Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$. Let $...
Alex M.'s user avatar
  • 5,407
0 votes
1 answer
612 views

Calculating a distributional derivative

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
dcs24's user avatar
  • 213
0 votes
1 answer
445 views

Domain of the Stokes operator

Let $\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$) $\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$ $\mathcal H:=\overline{\mathfrak ...
0xbadf00d's user avatar
  • 167