Extension of harmonic function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:31:08Z http://mathoverflow.net/feeds/question/101507 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101507/extension-of-harmonic-function Extension of harmonic function Slm2004 2012-07-06T15:12:35Z 2012-07-06T16:59:17Z <p>Suppose $u$ is a harmonic function of a domain $\Omega\subset \mathbb{R}^n$ and $u$ is continuous up to the boundary. If $\partial\Omega$ has an open smooth portion, can $u$ be extended to a harmonic function outside this smooth portion?</p> <p>I have a very vague claim that if this portion is analytic, then we can extend $u$ by schwarz reflection principle. But I don't know anything about the smooth case. Can anyone give me a hint?</p> http://mathoverflow.net/questions/101507/extension-of-harmonic-function/101515#101515 Answer by Pietro Majer for Extension of harmonic function Pietro Majer 2012-07-06T16:59:17Z 2012-07-06T16:59:17Z <p>Certainly not: think of $\Omega$ the unit disk, and $u$ the harmonic extension to $\Omega$ of any continuous, nowhere differentiable function on $\partial \Omega$.</p>