Tagged Questions
7
votes
2answers
102 views
Mean Value Property of Harmonic Function on a Square
A friend of mine presented me the following problem a couple days ago:
Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
5
votes
0answers
74 views
Can we do some scaling argument in the presence of inhomogeneous norms?
Notation:
$B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$.
$\hat{f}$ stands for the Fourier transform of $f$.
Question. The following inequality holds true for all $f\in ...
4
votes
1answer
158 views
Laplace equation Dirichlet problem on punctured unit ball.
Let $\Omega = \{ x \in \mathbb{R}^n: 0<|x|<1 \}$ and consider the Dirichlet problem
\begin{align}
\Delta u &= 0 \\
u(0) &= 1 \\
u &= 0 ~~~\text{if} ~~|x|=1
\end{align}
By considering ...
0
votes
1answer
56 views
Simple Harmonic estimate
I know this is simple, and I see all the pieces of the puzzle are there, but I can't seem to get it.
Let $u$ be a solution of
$$\Delta u = f \;\;\; x \in B_4 $$
Then if we can bound
$$\int_{B_4} ...
0
votes
0answers
36 views
Strichartz estimates and operator from $L^{2}_{x}$ to $L^{6}_{x,t}$
I want to prove that the operator $T=| \nabla|^{1/6} e^{-t\partial ^{3}_{x}}\tilde{P}_{N}$ takes functions from $L^{2}_{x}$ to $L^{6}_{x,t}$. The hint is to first prove for Schwartz functions, and ...
2
votes
1answer
51 views
Alternate definitions of $C^{1,\alpha}(S^1)$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definitition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
3
votes
1answer
112 views
Harmonic function with condition on part of its boundary
Suppose $u$ is harmonic in the interior of the unit square $0 \leq x \leq 1$, $0\leq y\leq1$. Suppose furthermore that $u$ and its first derivatives continuously extend to the bottom side $0\leq x ...
1
vote
2answers
67 views
Estimate on a simple-looking integral arising from harmonic analysis/harmonic extensions
Let $z\in \mathbb{D}, t\in S^1, \beta\in \mathbb{R}$. I was dealing with the following integral arising from some other calculation regarding harmonic extension on $\mathbb{D}$:
...
3
votes
1answer
134 views
On the regularity of the Laplace equations and tensor products and such
To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
3
votes
1answer
155 views
Nirenberg-Gagliardo- Sobolev inequalities
I need a small help in understanding the following that how "Nirenberg -Gagliardo-Sobolev inequalities" were used. This is a part of the paper.
Denote
$$
H^1=W^{1, 2}(\Omega)\\
V_1=\{ f\in H^2 ...
1
vote
0answers
51 views
Inequality for harmonic extension : Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?
Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, is the following true ?
(Question 1) Let $p(z,t) = \frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ ...
2
votes
1answer
201 views
Properties of subharmonic functions
A function $f$ is called subharmonic if $f:U\rightarrow\mathbb R$ (with $U\subset\mathbb R^n$) is upper semi-continuous and $$\forall\space \mathbb B_r(x)\subset ...
1
vote
3answers
338 views
Solution of Laplace's equation in an annulus with constant Dirichlet conditions?
What's the solution to Laplace's equation $\nabla^2V=0 $ in the annulus with centre 0, inner radius 1, and outer radius 2, with boundary conditions $V=0$ on the inner boundary and $V=1$ on the outer ...
3
votes
0answers
89 views
Curvatures of contours of solutions of 3d Poisson's equation
Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation
$$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ...
82
votes
4answers
2k views
What do modern-day analysts actually do?
In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
10
votes
3answers
469 views
Do discontinuous harmonic functions exist?
A function, $u$, on $\mathbb R^n$ is normally said to be harmonic if $\Delta u=0$, where $\Delta$ is the Laplacian operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. So obviously, ...
3
votes
3answers
159 views
What is a “domain” in the maximum-minimum principle?
The maximum-minimum principle says that
A harmonic function on a domain cannot attain its maximum or its minimum unless it is constant.
Here is my question:
If we restrict our attention in
...
1
vote
3answers
157 views
Why is it useful to express PDE solutions as $L^2$-convergent series?
The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...
