The author defines $L_{\gamma}^{-1}$ in the Theorem 4.(the first step in the first picture ) .
Proof .1. Choose $\mu=\lambda$ as in theorem 3 and define the bilinear form $$B_{\gamma}[u,v]:=B[u,v]+\gamma (u,v),$$ corresponding to the operator $L_{\gamma}u=Lu+\gamma u$. Then for each $g\in L^2(U)$ there exists a unique function $u\in H_0^1(U)$ solving $$B_{\gamma}[u,v]=g(u,v) \tag{13}$$ for all $v\in H_0^1(U)$. Let us write $$u=L_{\gamma}^{-1}g$$ whenever (13) holds…
Though $L_{\gamma}^{-1}$ well-defined, is it a linear operator and how to figure it out?Could someone gives me some details, thank you!
I post the theorem 4 by the first picture below.
