On self-adjointness of symmetric diffusion operators

2019/11/08 by Robinson, Derek W
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1911.03018

Abstract

Let Ω be a domain in \Ri^d with boundary Γ and let d_Γ denote the Euclidean distance to Γ. Further let H=-\divv(C∇) where C=( c_kl )>0 with c_kl=c_lk are real, bounded, Lipschitz continuous functions and D(H)=C_c^∞(Ω). Assume also that there is a δ≥0 such that ‖C/d_Γ ^δ-aI‖→ 0 as d_Γ→0 with δ≥0 where a is a bounded Lipschitz continuous function with a≥μ>0 on a boundary layer Γ _r=\x∈Ω: d_Γ(x)2-(d-d_H)/2 is sufficient for H to be essentially self-adjoint as an operator on L₂(Ω). In particular δ>3/2 suffices for C²-domains. Finally we prove that δ≥ 3/2 is necessary in the C²-case.

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