18.2.1.6 Solution stanexl-b5

Question:

• Show that if $u$ is a harmonic function in a finite domain, and positive on the boundary, then it is positive everywhere in the domain.
• Show by example that this does not need to be true for an infinite domain.
• Let $u$, $v$, and $w$ be harmonic functions. Show that if $u$ $\raisebox{-.3pt}{$\leqslant$}$ ${v}$ $\raisebox{-.3pt}{$\leqslant$}$ ${w}$ on the boundary of a finite domain, then $u$ $\raisebox{-.3pt}{$\leqslant$}$ ${v}$ $\raisebox{-.3pt}{$\leqslant$}$ ${w}$ everywhere inside the domain.

Answer:

• Use the minimum principle. The minimum of $u$ must be on the boundary. So can $u$ be negative or zero inside the region?
• Take the domain, for example, to be $y$ $\raisebox{.3pt}{$>$}$ 0, the boundary to be the $x$ axis, and the boundary condition on the $x$-axis to be $u$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1. Now consider the solution $u$ $\vphantom0\raisebox{1.5pt}{$=$}$ $1-y$. Verify that it satisfies the Laplace equation, and that it is positive on the boundary. Verify that it is negative within the region.
• Look at the difference $v-u$. Explain that if $u$ and $v$ are harmonic functions, then so is their difference. Now apply the earlier result about harmonic functions that are posirtive on the boundary.