[Evans PDE Problem 6.6.1] Laplace's equation with potential function
Consider Laplace's equation with potential function :
and the divergence structure equation:where the function is postive.
(1)、 Show that if solves and also solves , then solves for .
(2)、 Conversely, show that if solves , then solves for some potential .
Proof.
(1)、 Assume . Then
(2)、 Assume
Then
Choosing , then . $$