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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.
6
votes
Accepted
Finite speed of propagation of wave equation
The propagation speed is still finite because the following standard argument works independently of what happens at the boundary:
Assume that $u\in C^2$ solves the wave equation and $u(t=0,x)=0$ on …
- 24.2k
1
vote
existence of a special conformal mapping
This is not a complete answer, just as suggestion: move things over to the unit disk, with the Cayley transform
$$
\varphi(z)=\frac{z+i}{z-i} .
$$
Then $\Phi$ will be as desired precisely if $F=\varph …
- 24.2k
1
vote
How to find an ODE with prescribed terminal values?
In this generality, the answer is no, for the simple reason that a flow always has a positive derivative, so no $f$ with $f'=0$ somewhere can be realized in this way. (Such an $f$ could be fixed point …
- 24.2k
2
votes
Accepted
Universal constant for reverse inequality between first eigenvalues of Neumann and Dirichlet...
No. Consider a thin rectangle, with side lengths $a\ll 1$ and $1$. Then $\lambda_1^D=\pi^2(1+1/a^2)$ (eigenfunction $\sin \pi x\sin \pi y/a$), $\lambda_2^N = \pi^2$ (eigenfunction $\cos \pi x$), and n …
- 24.2k
3
votes
Concerning the decay of the ground state of certain Schrodinger operators
It is certainly not true in general that $\int_{V\le E_0}|\psi|^2$ will be close to $\|\psi\|^2$ for the ground state $\psi$. (What is true along these lines has been explained by Willie in his answer …
- 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
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
8
votes
Accepted
On the domain of functionals in measure with singular kernels
It is well known (to those who know it well) that the Hausdorff dimension of a set is closely related to its capacities. More precisely, if we define the capacitary dimension of a set $A\subset\mathbb …
- 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
2
votes
Pseudodifferential operators on spaces with boundary
The formula will work just fine (with no pseudo required), provided you interpret $\widehat{f}$ suitably. Since $f$ is only a function on a half space, we have to make an agreement on what exactly we …
- 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
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
2
votes
Limits for eigenvalues for the Dirichlet Laplacian
This is basically a comment on Dario's answer. I'm going to compare the Dirichlet problem on $B$ with the one on $B_0\equiv B\setminus\{ 0\}$ (though I'm not going to justify formally that this is wha …
- 24.2k
0
votes
Solution to inhomogenous PDE
Alternatively, the self-adjoint operator $-\Delta$ on the domain $H^2\subseteq L^2$ has spectrum $[0,\infty)$ in any dimension. Since $-1$ is not in the spectrum, $(-\Delta+1)^{-1}$ is bounded on $L^2 …
- 24.2k
3
votes
Accepted
Density argument with Schwartz functions?
We can do this by approximating the derivative of such an $f$ in $L^2$, as follows: Given $\epsilon>0$, take $L>0$ so large that $\int_{|x|>L}(x^2|f|^2+|f'|^2)<\epsilon$.
Now approximate $f'$ in $L^2 …
- 24.2k