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Results tagged with ap.analysis-of-pdes
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user 48839
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
Accepted
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
We can make more concrete what I suggested in my comment. Take $f(x)=x^{1/2p'}e^{-ix}$ on $1\le x\le 1/(10\epsilon)$ and $f=0$ otherwise. Then the LHS of (*) equals
$$
\int_1^{1+\epsilon} |G(t)|^2\, \ …
- 24.2k
1
vote
PDE for the probability of Brownian motion staying in an area (reference request)
I think your idea ("PS") works fine, at least when $A$ has finite measure and is a moderately reasonable set (let's say open) and probably in general with more effort. It does seem to get a bit techni …
- 24.2k
4
votes
Accepted
Schroedinger operator in 2 dimensions with singular potential
Taking advantage of the spherical symmetry to decompose this into a sum of one-dimensional problems sounds like the right approach. I will probably just be redoing what Reed-Simon had in mind here.
Th …
- 24.2k
5
votes
Accepted
Eigenvalues of a Schrödinger operator
This is a slightly expanded (and slightly more systematic) summary of my comments above. First of all, the equation
$$
-\varphi''+\frac{1}{r}\varphi' + (V+\frac{m}{r^2})\varphi=\lambda\varphi \quad\qu …
- 24.2k
7
votes
Accepted
Regularity of solution of $(-\Delta + w)f = 0$
As discussed in the comments, I interpret the question as asking about the asymptotics of $f'(r)$, $r\to 0+$, for solutions of
$$
-\frac{d^2f}{dr^2} -\frac{2}{r} \frac{df}{dr} + w(r)f(r) = 0 ; \quad\q …
- 24.2k
5
votes
The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
The Fourier transform of $|x|^b$, $b\notin\mathbb Z$, is the function $c|\xi|^{-1-b}$ away from $\xi =0$. See entry 313 of the table here and the discussion in the last column.
Moreover, the large $\x …
- 24.2k
13
votes
Are all positive eigenfunctions principal eigenfunctions?
You'll learn a lot more from Jochen's answer, but maybe I'll point out anyway that there is a very simple argument for this: The eigenfunction $u_0$ of the smallest eigenvalue is positive (see below), …
- 24.2k
5
votes
Higher integrability for Sobolev functions
No. Consider a function $f\in L^1(\mathbb R)$, $f\ge 0$, with
$f(x) = 2^{n^2}$ on $2^{-n}<x<2^{-n}+2^{-n^2-n}$ and essentially $f=0$ otherwise.
Then $\int_{-r}^r f(x)\, dx \simeq \sum_{n\gtrsim (-\log …
- 24.2k
3
votes
Sobolev space is spanned by distributions supported on half-lines?
First of all, as explained in my comment, this is the same as asking if the Hilbert transform is bounded on $L^2(\mathbb R, w\, dx)$, with $w(x)=(1+|x|)^{2s}$. Or, to state this one more time, this re …
- 24.2k
4
votes
Accepted
de Rham theorem for tempered distributions
This works. As explained in my comment, we need only show that if $p\in\mathcal D'(\mathbb R^n)$ and $\nabla p\in\mathcal S'$ (vector valued), then $p\in\mathcal S'$.
The condition for a function $\va …
- 24.2k
4
votes
Vacuum region with positive measure for the Schrödinger equation
This is only a very partial answer. In dimension $d=1$, the Paley-Wiener argument you refer to in your comment shows that $\psi(x,t)$ can not be zero on an open set: If $\psi(x,t)=0$ for $0\le x\le a$ …
- 24.2k
1
vote
Accepted
Convolution mollification of $H^s$ functions uniformly in the unit ball of this sobolev space
This works for all $s>0$.
If you take Fourier transforms (and write $\widehat{\varphi}=\psi$), then you are asking if
$$
\lim_{\epsilon\to 0}\sup_{\|(1+|t|^s)\widehat{u}\|=1}\|\widehat{u}(\psi(\epsilo …
- 24.2k
3
votes
Accepted
Laplace eigenfunction on a polygonal domain symmetric about an axis
There is no uniform bound, independent of the domain. Consider a rectangle with side lengths $\pi, \pi/K$, with $K\gg 1$. The eigenvalues are $m^2+n^2K^2$, $m,n\ge 1$. In particular, $\lambda_2$ is ob …
- 24.2k
1
vote
Accepted
$L^2$ bound and Sobolev spaces
This works for $\beta\ge\alpha$. In terms of the Fourier transform $g=\widehat{f}$, what you're trying to establish becomes
$$
h^{-2\alpha}\int |g(t)|^2 \left| e^{ith}-1\right|^2 \, dt \lesssim
\int | …
- 24.2k
1
vote
Localization of solutions for time-dependent Schroedinger equation
Here are some remarks to put Q2 into context, though I'm not answering the actual question. Basically, I'm going to give a quick summary of my 2007 paper Finite propagation speed and kernel estimates …
- 24.2k