# Elliptic theory (existence and regularity) for pde of complex functions

I'm aware that this is a rather general question, but I only need some hint to literature.

The setup. I'm studying the existence and regularity of weak solutions to linear elliptic pde of the form $$\begin{cases} Lu=f & \text{in } U \\ u=g & \text{on } \partial U \end{cases}$$ Here $U \subset \mathbb{R}^N$ is open and bounded; $f : U \to \mathbb{R}$ is given; $L$ denotes a second-order elliptic operator and $u: \overline{U} \to \mathbb{R}$ is the unknown.

The problem. All books I read (Gilbarg+Trudinger, Evans, Taylor, Giaquinta) deal with real functions $u$. But there are a lot of equations (e.g. the Gross-Pitaevskii equation) that allow for complex valued solutions.

Why does none of the authors deal with solutions of complex range (i.e. with linear systems of pde)? None of them even mention such equations (except in the quasi-linear case).

Is that because the generalization to linear systems is trivial? Is it because it is too hard? Can someone point me to some literature that deals with this kind of stuff?

• I seem to remember my elliptic PDE lecturer saying something like "all of this works the same for complex values but I'll just do the real case"... I agree that most texts do not even address it at all. I believe most of the theory generalises with identical arguments - maybe try working through some of them yourself. Personally I've never had the need to. – Anthony Carapetis Aug 19 '13 at 7:56