= Zero Dirichlet solution of the modified Helmholtz equation
{title2=$\Delta u=\mu^2u,\ u|_{\partial\Omega}=0\ \Longrightarrow\ u=0$}
Let $\mu>0$ and let $u$ be a real smooth solution on a bounded regular domain with zero <Dirichlet boundary conditions>. <Green's first identity> gives
$$
\mu^2\int_\Omega u^2\,dV=\int_\Omega u\Delta u\,dV=-\int_\Omega|\nabla u|^2\,dV.
$$
The two sides have opposite signs unless both vanish, so $u=0$. This <energy method> also proves uniqueness for prescribed boundary data by applying it to the difference of two solutions. For complex solutions use the complex conjugate in the integral. This argument does not apply with the same sign to the <Helmholtz equation>, which can have nonzero zero-boundary <eigenfunctions>.
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