Let $u$ be a harmonic function on some open set $\Omega\subset\mathbb{R}^{n}$ and $u\in L^{p}\left(\Omega\right)$. Is there any reference on extending $u$ to harmonic function on a larger open set $\Omega'$ and to be from $L^{p}\left(\Omega'\right)$?
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$\begingroup$ What about $\log r$ on $R^2\setminus\{0\}$? $\endgroup$– Otis ChodoshCommented Mar 1, 2014 at 17:41
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$\begingroup$ In general, a harmonic function in any $L^p$, even with $p=\infty$ does not extend to a larger open set. $\endgroup$– Alexandre EremenkoCommented Mar 1, 2014 at 17:54
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$\begingroup$ Where can I find that? I need only the case $1\leq p<\infty$. $\endgroup$– AlemCommented Mar 1, 2014 at 21:11
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1 Answer
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For every $n$ and for every regular region (in the sense of Dirichlet problem), it is easy to construct a harmonic function in $L^\infty$ that does not extend to any larger open set. Just solve the Dirichlet problem with a continuous nowhere differentiable boundary data.
If the boundary has irregular points, for example an isolated point, such extension is sometimes possible, depending on $n$ and $p$. One can make an exact statement about $n$ and $p$ for this kind of "removable singularity theorem", if this is what you really want. For example, if $p=\infty$, closed sets of zero capacity are removable, but closed sets of positive capacity are not.
answered Mar 1, 2014 at 18:03