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

Daomin Cao, Jie Wan

Abstract

We consider Green's function $ G_K $ of the elliptic operator in divergence form $ \mathcal{L}_K=-\text{div}(K(x)\nabla ) $ on a bounded smooth domain $ Ω\subseteq\mathbb{R}^n (n\geq 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\notin C^1(Ω\timesΩ) $ in general.

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

Abstract

We consider Green's function of the elliptic operator in divergence form on a bounded smooth domain with zero Dirichlet boundary condition, where is a smooth positively definite matrix-valued function on . We obtain a high-order asymptotic expansion of , which defines uniquely a regular part . Moreover, we prove that the associated Robin's function is smooth in , despite the regular part in general.
Paper Structure (3 sections, 15 theorems, 84 equations)

This paper contains 3 sections, 15 theorems, 84 equations.

Table of Contents

  1. Introduction and main results
  2. Odd dimensional case
  3. Even dimensional case: proof of Theorems \ref{['thm2']} and \ref{['thm3']}

Key Result

Theorem 1.2

Let $n\geq 3$ be odd and $\gamma\in(0,1)$ be an arbitrary constant. Then for any $l\in\mathbb{N}$, there exists a unique $\Phi_i\in E^{n+2(2i-1)}_{i+2+2(2i-1)}$ for $i=1,\cdots, n+l-2$ depending on $y\in\Omega$ and $H^l(x,y)=H^l_y(x)\in C\left (\Omega, C^{l,\gamma}\left (\overline{\Omega}\right )\ri

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 23 more