Difference between revisions of "Aufgaben:Problem 11"
From Ferienserie MMP2
(Created page with "==Problem== Let \(u\) be a harmonic function on \(\mathbb{R}^3\). Assusme there exists \(C>0\), independent of \(x\), such that \(|u(x)| \leq C(1+|x|)\) on \(\mathbb{R}^3\). S...") |
|||
| Line 6: | Line 6: | ||
<li>\(u\) is a linear function, i.e. \(\exists a_0, a_1, a_2, a_3 \in \mathbb{R}\) such that \(u(x) = a_0 + a_1 x_1 + a_2 x_2 + a_3 x_3.\)</li> | <li>\(u\) is a linear function, i.e. \(\exists a_0, a_1, a_2, a_3 \in \mathbb{R}\) such that \(u(x) = a_0 + a_1 x_1 + a_2 x_2 + a_3 x_3.\)</li> | ||
</ol> | </ol> | ||
| + | |||
| + | ==Proof Sketch== | ||
| + | Proof in 2 dimensions: http://www.math.columbia.edu/~savin/c12d.pdf | ||
Revision as of 09:26, 9 June 2015
Problem
Let \(u\) be a harmonic function on \(\mathbb{R}^3\). Assusme there exists \(C>0\), independent of \(x\), such that \(|u(x)| \leq C(1+|x|)\) on \(\mathbb{R}^3\). Show that then
- \(\partial_i u\) is constant on \(\mathbb{R}^3\), \(\forall i = 1,2,3\).
- \(u\) is a linear function, i.e. \(\exists a_0, a_1, a_2, a_3 \in \mathbb{R}\) such that \(u(x) = a_0 + a_1 x_1 + a_2 x_2 + a_3 x_3.\)
Proof Sketch
Proof in 2 dimensions: http://www.math.columbia.edu/~savin/c12d.pdf
