Harmonic functions are analytic

Question

Let $B_1\subset \mathbb{R}^N$ be the unitary ball centered in the origin, and let $u$ be an harmonic function in $B_1$, i.e. $$ -\Delta u = 0 \,\,\,\,\, \mbox{in}\,\,\, B_1. $$ How can i prove that $u$ is analytic? Moreover, how can be proved this results using some estimates on the high order derivative? I know that it must be a classic result, also in several variables, but i cannot find a good reference.

Answer 1

A typical approach would be the same as for proving that holomorphic functions are analytic. That is, represent $u$ in terms of its boundary values on some ball contained in the domain of $u$ (the Poisson formula does that). The Poisson kernel is real-analytic, since it is basically $(r^2-|x|^2)/|\zeta-x|^2$ where both numerator and denominator are polynomials of $x$ (the point $\zeta$ is a boundary point, $|\zeta|=r$). The power series converges when $|x|<r$, hence it can be integrated term by term on compact subsets of this ball. This gives a power series representation of $u$.