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Quantitative unique continuation for Robin boundary value problems on $C^{1,1}$ domains

In this paper, we prove two unique continuation results for second order elliptic equations with Robin boundary conditions on $C^{1,1}$ domains. The first one is a sharp vanishing order estimate of Robin problems with Lipschitz coefficients and differentiable, sign-changing potentials. This generalizes the result for the "Robin eigenfunctions" in [26], which deals with the case with constant potentials. The second result is a unique continuation result from the boundary -- any non-trivial solution cannot vanish at infinite order from the boundary or vanish on an open subset on the boundary. Such result generalizes the one in [1] for the Laplace equation on $C^{1,1}$ domains with zero Neumann boundary conditions.