How to show the Poynting vector of a standing electromagnetic wave integrated over a cavity in a perfect conductor is zero? Let me define what I mean exactly. Take a harmonic wave at one particular frequency. The electromagnetic wave equation becomes a Helmholtz equation. We have now an eigenvalue problem in an simply connected compact domain $D$ in $R^3$ with piecewise smooth boundary for the Laplacian operator with boundary value specified by the perfect conductor boundary, i.e., the tangential $\vec E$ field and normal $\vec B$ field vanish on the boundary. Poynting vector $\vec S=\vec E\times \vec B$. Now show $\int_{D} \vec S dV=0$. Note now there is no time dependence in this formulation.
The integral can be interpreted as the total linear momentum of the electromagnetic field in the cavity. The total momentum should be invariant for a standing wave confined in a closed domain. But how does one prove that for the field?
One may use the surface boundary integral of the Maxwell stress-energy tensor. One may be tempted to find a local argument where the stress tensor vanishes at every point at all time. It is not true. This can be seen in the case of rectangular cavity. Can one use, say, a conformal mapping in 3 dimension, to map an arbitrary geometry of the cavity to a rectangle and prove it on the rectangle?