From Wade's "Introduction to Analysis":
NOTE: Because French mathematicians (e.g., Borel, Jordan, and Lebesgue) did fundamental work on the connection between analysis and set theory, and ensemble is French for set, analysts frequently use E to represent a general set.
Most undergraduate analysis texts use $E$ for a subset of $\mathbb{R}$, but in complex analysis and PDE we tend to use $\Omega$ almost exclusively (at least in the texts I have read). Is there any rhyme or reason behind using $\Omega$, and does anyone know the history leading to the ubiquity of $\Omega$ to mean a region in $\mathbb{R}^n$ for PDE specifically?