I have been trying to find the meaning of "prescribing" a function on a curve. This is from some study notes on PDE's:
We can then index $Γ$ by a parameter $s: Γ := {(x, y) = (ϕ(s), ψ(s)) : s ∈ I ⊂ R}$. Here $ϕ$, $ψ$ are considered differentiable functions of s. Then assume $u = w(s)$ is prescribed on curve $Γ$.
It seems as if this should be an easy question to answer but a search on Google for its mathematical meaning comes up with nothing.
It is a kind of initial value for the PDE. It means that the values of the solution $u$ are given along the curve. The function $w$ is given, and you are looking for a function $u(x,y)$ which is a solution of the PDE and satisfies the initial condition $$ u(\phi(s),\psi(s))=w(s),\quad s\in I. $$