\(\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand\maps\colon
\newcommand\by\times
\newcommand{\anda}{\text{ and }}
\newcommand{\ifa}{\text{ if }}
\newcommand{\st}{\text{ s.t. }}
\newcommand{\asa}{\text{ as }}
\newcommand{\ina}{\text{ in }}
\newcommand{\ona}{\text{ on }}
\newcommand{\ata}{\text{ at }}
\newcommand{\wherea}{\text{ where }}
\newcommand{\witha}{\text{ with }}
\newcommand{\fora}{\text{ for }}
\newcommand{\ora}{\text{ or }}
\newcommand{\abs}[2][]{#1\lvert #2 #1\rvert}
\newcommand{\n}[2][]{#1\lVert #2 #1\rVert}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\trace}{\operatorname{trace}}
\newcommand{\without}{\setminus}
\newcommand{\dashint}{\hbox{\(-\)}\mspace{-19.25mu}\int}
\newcommand{\tdashint}{\raise 0.3ex \hbox{\(\scriptstyle-\)}\mspace{-14mu}\int}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Chapter1Maximum principles for ordinary differential equations
As we will see in this unit, maximum principles are an incredibly powerful tool for the rigorous study of both elliptic and parabolic partial differential equations. In this first chapter, as a sort of prologue, we explore maximum principles in the much simpler setting of second-order ordinary differential equations. In addition to their independent interest, these results and their proofs provide us with useful intuition for the general case. Functions of a single variable are also a fruitful source of examples and counterexamples.
