The first question - the shape of the domain - is the one in which the power of the Lions-Lax-Milgram theorem can be seen.

The Lions-Lax-Milgram theorem can be applied by using the following result, the hypotheses of which are quite common and easy to verify in practical applications:

In mathematics, the Lions-Lax-Milgram theorem (or simply Lions' theorem) is a result in functional analysis with applications in the study of partial differential equations.