We consider the null space (corresponding to the possible eigenvalue zero) of the linear indefinite elliptic PDE $\Delta u+k^2(x)u=0$ in $\Omega$ with $u=\partial_{\nu}u=0$ on the boundary. If the solution $u$ is analytic, then from the uniqueness of continuation $u\equiv 0$ in $\Omega.$ In other words, Neumann and Dirichlet eigenfunctions can not coincide. But what happens if the domain $\Omega$ and the coefficients $k(x)$ are not analytic so that $u$ is not likely analytic? Do we still have $u\equiv0$?
Can we approximate the domain and the coefficients by analytic ones? Then, we get also the solution approximated by analytic sequence of zero functions?
My motivation is to prove the uniqueness of the solution with the boundary condition $\partial_{\nu}u+\mathrm{i}\,k u=0.$ This is typical for the Helmholtz equation. The energy identity gives $$ \int_{\Omega}|\nabla u|^2-k^2(x)|u|^2+\int_{\partial\Omega}\mathrm{i}k|u|^2=0 $$ so $u=0$ on $\partial\Omega$ because $k(x)\in\mathbb{R}$ (we assume). And further $$ \int_{\Omega}\nabla u\cdot\nabla \bar{v}-k^2(x)u\cdot\bar{v}=\int_{\partial\Omega}\partial_{\nu}u\cdot\bar{v}=0, $$ for arbitrary test function $v$ so that $\partial_{\nu}u=0$ on $\Gamma.$ My goal is to prove $u\equiv0$ in $\Omega.$