# Existence theorem in Gilbarg and Trudinger

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)

• The best way to get an answer to this question (IMHO) is take one or two afternoons time, sit down with the book in your library and have a look at what the chapter titles and main theorems in the chapters look like. The book is a collection of existence and regularity results for linear, quasiliniar and to some extent, fully nonlinear second order elliptic PDE being rather comprehensive with respect to the time it was published. Sep 9 '14 at 14:35
• i don't disagree with what you said. I know that book but my work is in parabolic pde and I really can't make anymore time to read elliptic. But I am always curious about them, and trying to find some common among parabolic and elliptic problems Sep 9 '14 at 14:43

$\begin{array} -\Delta u=0\;\; on \;\;\; \Omega\\ u=0\;\; in \;\;\; \partial \Omega \end{array}$
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
$\begin{array} -\Delta u=0\;\; on \;\;\; \Omega\\ u=0\;\; in \;\;\; \partial \Omega \end{array}$