On Page 341, after Theorem 5 (Higher boundary regularity) in Chapter 6.3 (Regularity), Evans states that
Remark. If $u$ is the unique solution of $$ \left\{\begin{aligned} L u=f & \text { in } U \\ u=0 & \text { on } \partial U \end{aligned}\right. $$ then estimate $$ \|u\|_{H^{m+2}(U)} \leq C\left(\|f\|_{H^{m}(U)}+\|u\|_{L^{2}(U)}\right), $$ simplifies to read $$ \|u\|_{H^{m+2}(U)} \leq C\|f\|_{H^{m}(U)} . $$
I don't understand why uniqueness of solution can eliminate $\|u\|_{L^{2}(U)}$ on the right hand side.