  This paper characterizes the minimax linear estimator of the value of an
unknown function at a boundary point of its domain in a Gaussian white noise
model under the restriction that the first-order derivative of the unknown
function is Lipschitz continuous (the second-order H\"{o}lder class). The
result is then applied to construct the minimax optimal estimator for the
regression discontinuity design model, where the parameter of interest involves
function values at boundary points.
