Hello StackExchange writers,
I am currently working on Chapter 8 Problem 28 of Marsden/Tromba's Vector Calculus 5/E. The question asks for the following:
Suppose $D$ is the disk $\{(x,y)|x^2+y^2 < 1\}$ and $C$ is the circle $\{(x,y)|x^2+y^2 = 1\}$. Show that if $f$ is a continuous real-valued function on $C$, then there is a continuous function $u$ on $D \cup C$ that agrees with $f$ on $C$ and is harmonic on $D$.
I attempted to construct a harmonic function u by taking the second partial derivatives so that I can obtain $u_{xx}$ and $u_{yy}$ but I am not sure where to start.
Many thanks in advance.