Evans PDE 讲义, 视频讲解, 练习参考解答 (更多见小锦教学微信公众号)

[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   . $$