L₁-uniqueness of degenerate elliptic operators

Derek 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 -