Asymptotic expansions for harmonic functions at conical boundary points

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 $u(x)=\mathrm{\mid }x{\mathrm{\mid }}^m\psi (x\mathrm{/}\mathrm{\mid }x\mathrm{\mid })+o(\mathrm{\mid }x{\mathrm{\mid }}^m)$, where $\mathrm{\mid }x{\mathrm{\mid }}^m\psi (x\mathrm{/}\mathrm{\mid }x\mathrm{\mid })$ 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.