I'd like to know more about,

$$\lambda u = \nabla ^2 u$$

for unbounded domains (particularly the exterior of a disk in $\mathbb{R}^2$ or ball in $\mathbb{R}^3$), but have had a hard time finding references online. Can anyone suggest a website or book that covers this in depth? i.e. How to obtain solutions, proofs, important corollaries/theorems, etc.

  • $\begingroup$ Have you tried separation of variablaes? $\endgroup$ – DisintegratingByParts Oct 2 '18 at 22:02
  • $\begingroup$ Facepalm... will try that first.... although, the reason I bring it up is because I recall that unbounded domains can dramatically alter the nature of the eigenvalues. But first things first. $\endgroup$ – Eric Oct 2 '18 at 22:25
  • 1
    $\begingroup$ It's natural to restrict to $L^2$ solutions on the exterior domain. $\endgroup$ – DisintegratingByParts Oct 4 '18 at 2:36

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