2020/06/23 by Derek W. Robinson, Robinson, Derek W.
Computer Science · Mathematics · #31C25 #47D07 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2006.13403
openalex publication_date 2020/06/23 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Let \Ω be a domain in Rid with boundary \Γ !, d_\Γ\nthe Euclidean distance to the boundary and H=- divv(C ,\∇) an elliptic\noperator with C=( ,ckl ,)>0 where ckl=clk are real, bounded,\nLipschitz functions. We assume that C\∼ c ,d_\Γ ,\δ as\nd_\Γ\→0 in the sense of asymptotic analysis where c is a strictly\npositive, bounded, Lipschitz function and \δ\≥0. We also assume that\nthere is an r>0 and a b\δ,r>0 such that the weighted Hardy\ninequality \
int_
Gamma
!
!r d_
Gamma
,
delta
,|
nabla
psi|2
geq\nb
delta,r
,2
int_
Gamma
!
!r d_
Gamma
,
delta-2
,|
psi|2 is\nvalid for all \ψ\∈ Cc^\∞(\Γ ! !r) where\n\Γ ! !r= x\∈\Ω: d_\Γ(x)<r . We then prove that the\ncondition (2-\δ)/2<b_\δ is sufficient for the essential\nself-adjointness of H on Cc^\∞(\Ω) with b_\δ the supremum\nover r of all possible b\δ,r in the Hardy inequality. This result\nextends all known results for domains with smooth boundaries and also gives\ninformation on self-adjointness for a large family of domains with rough, e.g. \nfractal, boundaries.\n