Construct harmonic function on noncompact manifold

Question

$M$ is a non-compact Riemannian manifold, $p \in M$. Consider Dirichlet problems: $\Delta u = 0$ in ${B_p}\left( i \right)$ ($i = 1,2, \dots $), $u{|_{\partial {B_p}\left( i \right)}} = {f_i}$, ${f_i} \in {W^{1,2}}\left( {{B_p}\left( i \right)} \right)$. Then each of the functions has a unique weak solution ${u_i} \in {W^{1,2}}\left( {{B_p}\left( i \right)} \right)$. Suppose ${f_i}$ are uniformly bounded. Then can we find a subsequence of ${u_i}$ converging? And converging to a harmonic function defined on whole $M$.

Answer 1

To avoid techcnical details about the 'order' of regularity of the solution of your Dirichlet problem, i suppose to work with precompact open set $ D_i $ with smooth boundary instead of the geodesic balls $ B_i(p) $ that in general are not smooth. Note that this choice can always be done. Therefore consider the problem

$$ \Delta u = 0 \;\; on \; D_i \; \; \; \; u|_{\partial D_i} = f_i \; \; \; (1) $$

and suppose $ f_i $ smooth on $ \partial D_i $. By hypothesis $ ||f_i||_{\infty,\partial D_i} < C $, where $ C $ is independent on $ i $. By maximum principle $ ||f_i||_{\infty,D_i} < C $. Now let $ V \subset\subset U $ and let $ (U,\psi) $ be a local coordinate neighbourhood with compact closure. Let $ i_0 $ such that $ \forall i \geq i_0 $

$$ U \subset D_i $$

Let $ u_i $ be the solution of (1), let $ \tilde{u}_i= u_i \circ \psi^{-1} $ and let $ \tilde{U}= \psi(U) $. Let $ \tilde{L} $ be the laplace operator in local coordinates. Then $ \tilde{L}\tilde{u}_i =0 $ on $ \tilde{U} $ and $ ||\tilde{u}_i||_{\infty,\tilde{U}} < C $.

By Schauder Estimates we have that

$$ ||D^{\alpha}\tilde{u}_i||_{\infty,\tilde{V}} < C $$

for every multindex $ \alpha $. Therefore $ \{D^{\alpha}\tilde{u}_i: i \geq i_0 \} $ is uniformly bounded and equicontinous. By a repeated application of Ascoli's theorem we easily conclude that there exists $ \tilde{u} \in C^{2}(\tilde{V}) $ such that

$$ D^{\alpha}\tilde{u}_i \rightarrow D^{\alpha}\tilde{u} $$

uniformly on $ \tilde{V} $ for every $ |\alpha| \leq 2 $. Then $ \tilde{L}\tilde{u}=0 $ on $ \tilde{V} $. By regularity we have automatically that $ \tilde{u} \in C^{\infty} (\tilde{V}) $.Therefore $ \Delta u = 0 $ on $ V $ where $ u = \tilde{u} \circ \psi $.

Using a cover on $ M $ by local charts and repeating this argument for every open set of this covering we obtain a global harmonic function on $ M $.