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Results tagged with ap.analysis-of-pdes
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user 25510
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
Can I characterize functions (in 2D), which will have compactly supported/support contained ...
Since we are in the plane I use complex notation.
The general solution of your equation is the sum of the
potential and an arbitrary harmonic function:
$$u(z)=\frac{1}{2\pi}\int\int\log|z-\zeta| f(\ze …
- 91.8k
10
votes
Accepted
On critical points of harmonic functions
This is not true in dimension 2.
Function $f(z)=ze^z$ is entire, and $f'(z)=0$ at one point,
$z=-1$. It follows that the function
$$u(x,y)=\mathrm{Re}f(x+iy)=e^x(x\cos y-y\sin y)$$
is harmonic, has on …
- 91.8k
4
votes
Reference for harmonic functions in cylinders
First, some general background. For a bounded domain, the boundary value problem is solved by the Poisson formula, however an explicit form of the Poisson kernel for a cylinder of finite length is pro …
- 91.8k
5
votes
Heating a long cylinder: steady states
To separate contribution of the "ends" and the lateral surface, write $w=u+v$, where $u$ has zero
boundary conditions on the lateral surface, and $v$ is zero
on the ends.
The estimate $|u|\leq Ce^{-kt …
- 91.8k
3
votes
Accepted
Why we have $f=0$
Fourier transform of $f(x)=e^{-tx^2}$ is $\hat{f}(\xi)=ce^{-\xi^2/(4t)}$.
Multiplication of $f$ by a polynomial results in applying
a differential operator with constant coefficients to $\hat{f}$, and …
- 91.8k
5
votes
Accepted
Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?
The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal …
- 91.8k
8
votes
Does the pointwise mean value property imply harmonicity?
This question was addressed by Hansen and Nadirashvili in a series of papers, see, for example:
MR1315353
Hansen, W., Nadirashvili, N.,
On Veech's conjecture for harmonic functions.
Ann. Scuola Norm. …
- 91.8k
15
votes
Accepted
Existence of a smooth compactly supported function
The answer is yes if $\epsilon<1$, and no when $\epsilon\geq 1$.
This follows from Carleman's quasianalyticity criterion, see for example, Hormander, Analysis of linear partial differential operators, …
- 91.8k
3
votes
Plummer and Coulomb kernel for the Poisson equation
The radial Laplacian in spherical coordinates is
$$\Delta=\frac{d^2}{dr^2}+\frac{n-1}{r}\frac{d}{dr}.$$
By differentiating we obtain
$$\Delta p_\epsilon=-\epsilon^2n(n-2)(r^2+\epsilon^2)^{-1-n/2}.$$
T …
- 91.8k
9
votes
How to trap a particle without using potential field which is infinity at some point? (quant...
Trapping means that an eigenfunction has compact support. For locally bounded potential this is impossible, because of uniqueness theorem. Suppose for simplicity that our system is 1-dimensional. Then …
- 91.8k
2
votes
Accepted
If subharmonic functions converge weakly to a subharmonic limit, why do their smoothings con...
More detail on weak convergence of subharmonic functions, including a proof of this statement, can be found in his other book:
Hormander, Notions of convexity, Theorems 3.2.12 and 3.2.13.
- 91.8k
2
votes
Accepted
Positive subharmonic functions with constant integral blowing up at boundary
Let $\Omega$ be the unit ball, $B$ some smaller concentric ball, and
$u_n(x)=1$ for $|x|\leq 1-1/n$ and $u_n(x)=n(n-1)|x|+2n-n^2$ for $1-1/n\leq|x|\leq 1$.
Then your conditions 1,2,4 are satisfied exa …
- 91.8k
10
votes
What is an "exact solution" to a PDE?
There is no formal definition. This depends on context.
Those who say that "exact solution" means a "closed form solution" have to explain what a "closed form" is. A series whose coefficients are rati …
- 91.8k
6
votes
Accepted
What is the example of non-regular boundary point?
Example 1. In dimension 2, all isolated boundary points (punctures) are irregular.
Example 2. (Generalization) In dimension $n$ if you remove from a region $D$ a smooth
$n-2$ dimensional surface $S$, …
- 91.8k
4
votes
Accepted
Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero?
You have to say what is the meaning of $\Delta u$, and of $\int\Delta u$. For the integral to have a meaning, $u$ has to be a distribution and $\Delta u$ has to be a (signed, Radon) measure. Such dist …
- 91.8k