Helmholtz equation

Remarks

Example: in $\R^n$, for a unit vector $\omega$, the plane wave $u(x) = e^{i k \, \omega \cdot x}$ is a solution, since $\Delta u = (ik)^2 \abs{\omega}^2 u = -k^2 u$. Non-example: $u(x) = e^{k x_1}$ is not a solution, since $\Delta u + k^2 u = 2k^2 u \ne 0$; it solves instead $\Delta u - k^2 u = 0$, the modified Helmholtz equation. If $v(x, t) = u(x) e^{-i \omega t}$ solves the wave equation $\partial_t^2 v = c^2 \Delta v$, then $u$ solves the Helmholtz equation with $k = \omega / c$. On a bounded domain with zero boundary values, the equation has a nonzero solution exactly when $k^2$ is an eigenvalue of $-\Delta$ with that boundary condition. Some authors allow complex $k$.