When attending talks in PDE I often heard "existence proof follow from Gilbarg and Trudinger..."
Could anyone tell me rough what is the existence theorem for elliptic PDE roughly about? (My knowledge in elliptic PDE is 0)
2026-03-27 06:15:05.1774592105
Existence theorem in Gilbarg and Trudinger
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You have a PDE system like
$\begin{array} -\Delta u=0\;\; on \;\;\; \Omega\\ u=0\;\; in \;\;\; \partial \Omega \end{array} $
then you try to show that there exist a solution for the problem, that means, to find a function $u \in C^2(\Omega)\cap C^1(\bar\Omega)$ that satisfies the conditions
$\begin{array} -\Delta u=0\;\; on \;\;\; \Omega\\ u=0\;\; in \;\;\; \partial \Omega \end{array} $
You do not need to show who is the function, but even though you can show that it exists.