= 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 $|\nabla u|\leq Cs^{-1}\sup_{B_{cs}}|u|$. If $u$ extends continuously with value zero on that side, $s|\nabla u|\to0$ locally along it. This justifies vanishing boundary terms such as $u_x\sin(nx)$ in a distributional Fourier sine projection.
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