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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.
15
votes
Accepted
$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$
That doesn't work because $H_0^1$ functions are small near the boundary, so testing against them won't detect bad behavior of $u$ near $\partial\Omega$.
For a concrete example, take $\Omega$ as the u …
- 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
8
votes
What is an "integrable hierarchy"? (to a mathematician)
One more remark perhaps as a supplement to the existing answers, to further motivate the term hierarchy.
The standard way to generate the common hierarchies (Toda, KdV are the most standard examples) …
- 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
7
votes
Accepted
Are there fundamental solutions of the laplacian that decay rapidly?
No. You want the Fourier transform to satisfy $-|\xi|^2 \widehat{E}(\xi)=1$, so $\widehat{E}=-1/|\xi|^2 + \widehat{F}$, with $\textrm{supp}\:\widehat{F}=\{0\}$, but this says that $\widehat{F}$ is a l …
- 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
$$
-\frac{d^2f}{dr^2} -\frac{2}{r} \frac{df}{dr} + w(r)f(r) = 0 ; \quad\q …
- 24.2k
6
votes
Accepted
Finite speed of propagation of wave equation
The propagation speed is still finite because the following standard argument works independently of what happens at the boundary:
Assume that $u\in C^2$ solves the wave equation and $u(t=0,x)=0$ on …
- 24.2k
5
votes
Accepted
Schrödinger operator with Coulomb potential
This has to be shown separately. There are potentials with this decay $V(x)=O(|x|^{-1})$ that have embedded (in the ac spectrum) eigenvalues. The most famous of these is the von Neumann-Wigner potenti …
- 24.2k
5
votes
Higher integrability for Sobolev functions
No. Consider a function $f\in L^1(\mathbb R)$, $f\ge 0$, with
$f(x) = 2^{n^2}$ on $2^{-n}<x<2^{-n}+2^{-n^2-n}$ and essentially $f=0$ otherwise.
Then $\int_{-r}^r f(x)\, dx \simeq \sum_{n\gtrsim (-\log …
- 24.2k
5
votes
The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
The Fourier transform of $|x|^b$, $b\notin\mathbb Z$, is the function $c|\xi|^{-1-b}$ away from $\xi =0$. See entry 313 of the table here and the discussion in the last column.
Moreover, the large $\x …
- 24.2k
5
votes
Accepted
Eigenvalues of a Schrödinger operator
This is a slightly expanded (and slightly more systematic) summary of my comments above. First of all, the equation
$$
-\varphi''+\frac{1}{r}\varphi' + (V+\frac{m}{r^2})\varphi=\lambda\varphi \quad\qu …
- 24.2k
4
votes
Accepted
Schrödinger eigenfunctions are bounded
In general, there won't be a uniform bound on all eigenfunctions simultaneously. If $[a,b]$ is a short interval with Dirichlet boundary conditions $y(a)=y(b)=0$ and constant potential $V=c$, then the …
- 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
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
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