0
votes
1answer
35 views

Show that $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$

Elias M. Stein said that by an application of Green's theorem the following equality holds $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$ where $\Delta _{S}$ is a ...
5
votes
0answers
79 views

the spectrum and determinant of the Laplacian on $S^3$

I came across the following statement in a paper: On $S^3$, the eigenvalues of the vector Laplacian on divergenceless vector fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell ...
2
votes
1answer
61 views

Application of Green's theorem to probability

I encountered this problem while reading a statistic text. Since I am not quite familar with the background knowledge. Wonder can someone help me to explain the details of the following proof? ...
0
votes
0answers
30 views

Bounding the partial derivatives of $\left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}$.

Let $\tau = (\tau_1,\ldots,\tau_n) \in \mathbf{R}^n$, $s\geq 0$, and define the function $f : \mathbf{R}^n \to \mathbf{R}$ by $$ f(\xi) = \left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}. $$ ...
0
votes
0answers
149 views

Higher dimensional integration by parts, specific case.

Suppose $\sigma$ is $C^\infty$ function on $(\mathbf{R}^n)^2\backslash \{(0,0)\}$ that is supported in the cube $[-1/4,1/4]^{2n}$ which satisfies $$ |\partial_{y_1}^{\alpha_1}\partial_{y_2}^{\alpha_2} ...
3
votes
1answer
196 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
0answers
265 views

Proving the maximum principle for harmonic real valued functions in $\mathbb{R}^n$

Assuming the principle is stated as such: Let $U\subset\mathbb{R}^n$ be a bounded domain and $u$ harmonic in $U$ such that $\sup_{x\in U}u(x)\leq A$ for some $A\in\mathbb{R}$. Then either $\forall ...