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) = |x| mψ (x/|x|) + o (|x|m), where |x|^mψ (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.