Asymptotic expansions for harmonic functions at conical boundary points
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Dennis Kriventsov [1] ; Zongyuan Li [2]
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[1] Rutgers University
Rutgers University
City of New Brunswick, Estados Unidos
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[2] City University of Hong Kong
City University of Hong Kong
RAE de Hong Kong (China)
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 41, Nº 3, 2025, págs. 837-866
- Idioma: inglés
- DOI: 10.4171/RMI/1513
- Enlaces
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Resumen
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.
