= Dirichlet energy dissipation for the heat equation
{c}
For a real smooth solution $w_t=\Delta w$ on a bounded regular domain with $w_t=0$ on its boundary, differentiation and <Green's first identity> yield
$$
\frac d{dt}\int_V|\nabla w|^2\,dV=-2\int_V(\Delta w)^2\,dV\leq0.
$$
The boundary term is zero because the boundary values are fixed in time, not because the normal derivative vanishes. Equality at a particular time is equivalent to $\Delta w=0$ throughout the domain at that time.
Back to article page
