= Dirichlet energy dissipation with a dynamic heat boundary condition
{c}
{title2=$E'=-2\int_Vw_t^2-2\int_{\partial V}\alpha(\partial_nw)^2$}
For a smooth <heat equation> solution $w_t=\Delta w$ with $w_t+\alpha\partial_nw=0$ on the <boundary> and nonnegative $\alpha$, <Green's first identity> gives the displayed dissipation identity for $E=\int_V|\nabla w|^2$. The outward <normal derivative> is used. Writing the <boundary> loss as $\alpha(\partial_nw)^2$ avoids division by $\alpha$ and remains valid where $\alpha=0$. A stationary solution is harmonic with $\alpha\partial_nw=0$; it need not be a particular fixed-data <Dirichlet problem> solution.
Back to article page
