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L₁-uniqueness of degenerate elliptic operators

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Robinson D., Sikora A.

Studia Mathematica

2011

tom 203

nr 1

79-103

Let Ω be an open subset of $ℝ^{d}$ with 0 ∈ Ω. Furthermore, let $H_{Ω} = -∑^{d}_{i,j=1} ∂_{i}c_{ij}∂_{j}$ be a second-order partial differential operator with domain $C_{c}^{∞}(Ω)$ where the coefficients $c_{ij} ∈ W^{1,∞}_{loc}(Ω̅)$ are real, $c_{ij} = c_{ji}$ and the coefficient matrix $C = (c_{ij})$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If $∫_{0}^{∞} ds s^{d/2}e^{-λμ(s)²} < ∞$ for some λ > 0 where $μ(s) = ∫_{0}^{s} dt c(t)^{-1/2}$ then we establish that $H_{Ω}$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent to the capacity of the boundary of Ω, measured with respect to $H_{Ω}$, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets A of the boundary of the set and the order of degeneracy of $H_{Ω}$ at A.

Ograniczanie wyników

1 Studia Mathematica

1 Robinson D.

1 Sikora A.

1 2011

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L₁-uniqueness of degenerate elliptic operators