Let $D$ be a bounded, connected set in $R^d$ and $∂D$ its boundary.
- Suppose $u$ is defined and continuous on $\bar{D}$, and $u$ is harmonic in $D$. Show that $u$ attains its maximum over $\bar{D}$ on $\partial D$.
- If $v$ is another function, harmonic in $D$ and continuous on $\bar{D}$, and $v=u$ on $\partial D$, then $v=u$ on $D$ as well.
For the second question : let $h=v-u$, by the first question $h$ attains its maximum over $\partial D$, or $h=0$ in $\partial D$, then by the maximum principle $h=0$ on $D$, i.e $u=v$ on $D$.
For the first one, I din't know what to do ! so any help is appreciated !
Source : Ex p.242 Brownian motion and stochastic calculus by Karatzas and Shreve.