Is there a difference between referring to the boundary of a domain $\Omega$ as $\Gamma$ or $\partial \Omega$ ? Or is this just preference or synonyms of the same thing? From my experience, they seem to be used arbitrarily, but I feel like I might be overlooking something.
Thank you
There is no real difference. About the only difference is that if you've introduced $\Omega$, then you can refer to $\partial \Omega$ without explaining what it is, since $\partial$ is the standard notation for the boundary mapping. By contrast if you suddenly start using $\Gamma$ to refer to the boundary without defining it, you may cause confusion.
It depends on the definition of $\Gamma$ in your context. At least in finite element literature $\Gamma$ is often defined to be the whole boundary of the domain $\Omega$, in other words the same as $\partial\Omega$, but not always. Scott and Brenner, for instance, occasionally defines $\Gamma$ as a part of the boundary where Dirichlet boundary conditions are applied. The remaining part of the boundary, where for instance Neumann conditions applies, may then be referred to as $\partial\Omega\setminus\Gamma$.
