English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We prove three theorems about the asymptotic behavior of solutions $u$ to thehomogeneous Dirichlet problem for the Laplace equation at boundary points withtangent cones. First, under very mild hypotheses, we show that the doublingindex of $u$ either has a unique finite limit, or goes to infinity; in otherwords, there is a well-defined order of vanishing. Second, under morequantitative hypotheses, we prove that if the order of vanishing of $u$ isfinite 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 thetangent cone. Finally, we construct a convex domain in three dimensions wheresuch an expansion fails at a boundary point, showing that some quantitativehypotheses are necessary in general. The assumptions in all of the results onlyinvolve regularity at a single point, and in particular are much weaker thanwhat is necessary for unique continuation, monotonicity of Almgren's frequency,Carleman estimates, or other related techniques.

Asymptotic expansions for harmonic functions at conical boundary points