Expansion of Green's function and regularity of Robin's function for elliptic operators in divergence form

2024/01/21 by Cao, Daomin, Wan, Jie · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.11486

Abstract

We consider Green's function G_K of the elliptic operator in divergence form L_K=-div(K(x)∇ ) on a bounded smooth domain Ω⊆ℝⁿ (n≥ 2) with zero Dirichlet boundary condition, where K is a smooth positively definite matrix-valued function on Ω. We obtain a high-order asymptotic expansion of G_K(x, y) , which defines uniquely a regular part H_K(x, y) . Moreover, we prove that the associated Robin's function R_K(x) = H_K(x, x) is smooth in Ω, despite the regular part H_K∉ C¹(Ω×Ω) in general.

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