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0
votes
0answers
16 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
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0answers
28 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
113 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
178 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 ...
