Let $U\subset \mathbb{R}^n$ be a bounded open set. Suppose we know that the continuous function $u$ is a weak solution of $$ Lu = 0 $$ on $U$, where $L$ is a second order linear differential operator with smooth coefficients on $U$. Also assume that $V\subset U$ is open and dense in $U$. If we know that $u$ is smooth on $V$, is it true that $u$ is smooth on $U$ as well? If not, what conditions do we need for this to be true?
Any reference to a text that talks about this will be much appreciated too.