I've been trying to understand the problem below for ages and despite looking at many papers that all cite the problem, none fully explain how to go from the first PDE to the second in any more detail than just saying "thus it satisfies the adjoint equation". (One paper this can be found in is "Identifying the volatility of underlying assets from option prices" - Lishang & Youshan)
If anyone can please give a detailed explanation as to how this is done I'd be truly grateful. (And yes, I am familiar with the idea of adjoint operators and the idea that $\mathscr{L}^*$ satisfies $\langle\mathscr{L}u,v\rangle=\langle u,\mathscr{L}^*v\rangle$, just not on the actual algebra of the whole derivation.)
Let $\xi$ satisfy $$\xi_\tau=a(y)(\xi_{yy}-\xi_y)-(r-q)\xi_y+(h-a)(V_{yy}-V_y),$$ $$\xi|_{\tau=0}=0.$$
From previous work we have $$\int_\mathbb{R}(V(y,\tau^*)-V^*(y))\xi(y,\tau^*)\>dy+N\int_\mathbb{R}\nabla a\nabla(h-a)\>dy\geq0.$$
$N,h$ constants. Also note that the $*$ in the case of $V(y,\tau^*)$ and $V^*(y)$ does not refer to ajoints, just "market data" in the context of the problem.
Question:
Denoting $\mathscr{L}\xi=\xi_\tau-a(y)(\xi_{yy}-\xi_y)+(r-q)\xi_y$, along with the information above, how do you show (rather than simply verify) that the adjoint/ dual problem is given by $$\mathscr{L}^*\phi=-\phi_\tau-(a\phi)_{yy}-(a\phi)_y-(r-q)\phi_y=0,$$ $$\phi|_{\tau=\tau^*}=V(y,\tau^*)-V^*(y),$$ Where $\mathscr{L}^*$ is the adjoint operator of $\mathscr{L}$.