 Solution

Combined interior and boundary elliptic regularity for the Dirichlet Laplacian on a smooth bounded domain states that, for every integer $k\ge 0$,when $\Delta u=f$ and $u$ has zero boundary trace. More generally one first has an additional $\Vert u{\Vert }_{L^2}$ term, which uniqueness and the Poincare inequality remove here.

For the shifted equation, write $\Delta u=f+u$. The weak estimate gives $u\in H^1$. Applying the displayed estimate first with an $L^2$ right side gives $u\in H^2$. Repeating,until $u\in H^{k+2}$. The lower-order term is controlled at each stage, yieldingThis is boundary elliptic regularity for the shifted Dirichlet Laplacian.