Interior gradient estimate near a zero Dirichlet side

For a smooth uniformly elliptic homogeneous equation away from other boundaries, an interior gradient estimate on a ball of radius comparable to the distance $s$ from a flat side gives $|\nabla u|\le C{s}^{-1}{\sup }_{B_{cs}}|u|$. If $u$ extends continuously with value zero on that side, $s|\nabla u|\to 0$ locally along it. This justifies vanishing boundary terms such as $u_x\sin (nx)$ in a distributional Fourier sine projection.