I am interested in Theorems regarding existence, uniqueness, and regularity of solutions to linear 2nd order elliptic PDEs with periodic boundary conditions, e.g., $$\begin{cases} -u''+u=f & 0<x<1\\ u(0)=u(1) & \\ u'(0)=u'(1)\end{cases}$$ where $f\in L^2(0,1)$. The literature I can find is focused on numerical methods for such problems or analysis for much more complicated problems with periodic BCs. Is there a foundational paper or textbook that discusses this type of problem as Evans does for Dirichlet and Neumann boundary conditions?
2026-03-27 12:56:08.1774616168
Reference request: existence, uniqueness, and regularity of solutions to elliptic PDEs with periodic boundary conditions
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A nice treatment of Sobolev spaces and boundary value problems in one dimension can be found in Chapter 8 of the text:
Brezis, Haim, Functional analysis, Sobolev spaces and partial differential equations, Universitext. New York, NY: Springer (ISBN 978-0-387-70913-0/pbk; 978-0-387-70914-7/ebook). xiii, 599 p. (2011). ZBL1220.46002.
In particular Section 8.4 discusses boundary value problems, and Example 7 in said particular treats the exact problem you mention. The techniques developed in said text should also allow you to establish the well-posedness theory of similar equations.