  We prove three theorems about the asymptotic behavior of solutions $u$ to the
homogeneous Dirichlet problem for the Laplace equation at boundary points with
tangent cones. First, under very mild hypotheses, we show that the doubling
index of $u$ either has a unique finite limit, or goes to infinity; in other
words, there is a well-defined order of vanishing. Second, under more
quantitative hypotheses, we prove that if the order of vanishing of $u$ is
finite at a boundary point $0$, then locally $u(x) = |x|^m \psi(x/|x|) +
o(|x|^m)$, where $|x|^m \psi(x/|x|)$ is a homogeneous harmonic function on the
tangent cone. Finally, we construct a convex domain in three dimensions where
such an expansion fails at a boundary point, showing that some quantitative
hypotheses are necessary in general. The assumptions in all of the results only
involve regularity at a single point, and in particular are much weaker than
what is necessary for unique continuation, monotonicity of Almgren's frequency,
Carleman estimates, or other related techniques.
