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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.
3
votes
radial limits of subharmonic functions
The answer is no, even in dimension $2$. Consider a simply connected region $D$ in the unit disc which approached the boundary with infinite spiraling (that is such region where a continuous branch of …
- 91.8k
1
vote
Accepted
The integrability of fundamental solution of laplace equation follows from integrability of f ?
Your question text suggests that yo know how to do (1). (Interchange the order of integration).
Now (2) is easy: outside the support of $f$ we have $\Delta u=0$, that is $u$ is harmonic outside
the su …
- 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
2
votes
The PDE $u_t=u_{xx}-u_{yy}$: The simplest linear second-order PDE that isn't elliptic, parab...
Exponential solutions are easy to find. Just plug $\exp(ax+by+ct)$, and you will see that every
$(a,b,c)$ that satisfies $c=a^2-b^2$ gives you a solution. This polynomial is called the symbol
of the d …
- 91.8k
2
votes
A question on the integrability of eigenfunctions of the Laplacian
The answer is no if $k$ is independent on $\lambda$. Take a $2$-dimensional torus with a flat metric, which is a very nice manifold. Eigenfunctions can be found explicitly, and they are analytic. Ther …
- 91.8k
4
votes
PDEs involving measures; where to begin?
The place to start is:
Hormander, Analysis of Linear partial differential operators, vols. I-II
(if your coefficients are constant), and further volumes for non-constant
coefficients.
- 91.8k
3
votes
Dirichlet energy of a harmonic function bounded above by the energy of the boundary function?
This is correct (if $f$ is real, of course). Use the Green formula to transform the LHS:
$$\int_\Omega|\nabla u|^2dxdy=R\int_0^{2\pi}uu_rd\theta,$$
where $R$ is the radius of the disc.
Now expand
$$u …
- 91.8k
3
votes
Integrability of the Poisson integral
Equation $\Delta u=0$ is called the Laplace equation, btw.
Edit. The answer to your question is no.
Consider $f(z)=1/(z+i)$. On the real line it belongs to $L^p$ with any $p>1$.
In the upper half-pl …
- 91.8k
2
votes
Calculation of a complex integrand. A question from the book PDE by A. Friedman
I am not sure what $(\psi,M)$ type is, and you cannot assume that all readers of this site
have this 1969 book next to them but usually such things hold when the spectrum is bounded, and the
curve $\G …
- 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
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
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
How to find the eigenvalues equation of this PDE problem
Your coefficients are piecewise constant, which allows you to write the general solution of your differential equation. First write the general solutions on each interval where they are constant. This …
- 91.8k
1
vote
Dirichlet Problem Solvable when every component of the complement of the domain consists of ...
If they are talking about the Laplace operator, this statement is true only in dimension 2. And this is only sufficient, not necessary.
In general, for solvability of the classical Dirichlet problem, …
- 91.8k
2
votes
Harmonic function defined on a cone
If I understand correctly what a "harmonic function on the cone" is, there is no difference
between the cone and the disc for the Dirichlet problem.
This cone is a surface equipped with a Riemannian …
- 91.8k