# L₁-uniqueness of degenerate elliptic operators

Derek W. Robinson; Adam Sikora

Studia Mathematica (2011)

- Volume: 203, Issue: 1, page 79-103
- ISSN: 0039-3223

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topDerek W. Robinson, and Adam Sikora. "L₁-uniqueness of degenerate elliptic operators." Studia Mathematica 203.1 (2011): 79-103. <http://eudml.org/doc/286338>.

@article{DerekW2011,

abstract = {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.},

author = {Derek W. Robinson, Adam Sikora},

journal = {Studia Mathematica},

keywords = {-uniqueness; Markov uniqueness; capacity},

language = {eng},

number = {1},

pages = {79-103},

title = {L₁-uniqueness of degenerate elliptic operators},

url = {http://eudml.org/doc/286338},

volume = {203},

year = {2011},

}

TY - JOUR

AU - Derek W. Robinson

AU - Adam Sikora

TI - L₁-uniqueness of degenerate elliptic operators

JO - Studia Mathematica

PY - 2011

VL - 203

IS - 1

SP - 79

EP - 103

AB - 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.

LA - eng

KW - -uniqueness; Markov uniqueness; capacity

UR - http://eudml.org/doc/286338

ER -

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