Prove uniqueness for the Dirichlet and Neumann problems for the reduced Helmholtz equation $\triangle u − ku = 0$ in a bounded planar domain $D$, where $k$ is a positive constant.
How can I prove this? I found that Green’s third identity to be useful in proving this but I am not sure how to apply it here.
Green's identity says that $$\int_D(f\Delta f+|\nabla f|^2)=\int_{\partial\Omega}f\frac{\partial f}{\partial \nu}.$$ Now take $f=u$ and by the assumption that $\Delta u-ku=0$ in $D$, we have $$\int_D(ku^2+|\nabla u|^2)=\int_{\partial\Omega}u\frac{\partial u}{\partial \nu}.$$ Dirichlet boundary condition says that $u=0$ on $\partial\Omega$, and Nerumann boundary condition says that $\frac{\partial u}{\partial \nu}=0$ on $\partial\Omega$. In either case, we have $$\int_{\partial\Omega}u\frac{\partial u}{\partial \nu}=0.$$ Combining all these, we have $$\int_D(ku^2+|\nabla u|^2)=0.$$ Since $k$ is positive by assumption, we have $u\equiv 0$.
