General theory on the variational form of a system of PDEs

variational formulation
1

I have been struggling to find the answers to the following general FEM questions for some time; I would very much appreciate some help!

From p. 260 of Langtangen & Mardal 2019, the variational form of a general system of coupled PDEs can be written as
flameshot__Apr25__15-30-41

I have the following 3 questions about this:

  1. Which test function, v^{(i)}, should be multiplied by which PDE since each PDE in a general coupled system of PDEs could contain every unknown, i.e. \mathcal{L}^{(0)} in the image above could just as easily be written as \mathcal{L}^{(i)} with i \neq 0?
  2. Am I correct in thinking that if a Dirichlet boundary condition is applied to a given unknown, u^{(i)}, then the test function corresponding to that subspace would be 0 on the boundary? Therefore, does the answer to question 1 depend on the boundary conditions that you want to apply since different surface integral terms will be 0 (as a result of being multiplied by a test function which is 0 on the boundary) depending on which test function you multiply each equation by?
  3. When you solve a system of coupled PDEs in FEniCS, for example 3 PDEs, you write each equation like L0 = …, L1 = …, L2 = … and then you combine these equations into a single system by L = L0 + L1 + L2, as in the Cahn-Hilliard demo. However, I am confused about whether the sign of each of L0, L1 and L2 matter? I was under the impression that you could multiply, for instance, L0 by negative 1 and still have a valid weak form for that PDE; therefore, would it be valid to multiply any of L0, L1 and L2 by -1 and still obtain the same system when using L = L0 + L1 + L2?

Many thanks in advance!