$L_1$-uniqueness of degenerate elliptic operators

Volume 203 / 2011

Derek W. Robinson, Adam Sikora Studia Mathematica 203 (2011), 79-103 MSC: 47B25, 47D07, 35J70. DOI: 10.4064/sm203-1-5


Let $\Omega$ be an open subset of ${\mathbb R}^d$ with $0\in \Omega$. Furthermore, let $H_\Omega=-\sum^d_{i,j=1}\partial_ic_{ij}\partial_j$ be a second-order partial differential operator with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\overline\Omega)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0< C(x)\leq c(|x|) I$ for all $x\in \Omega$. If \[ \int^\infty_0ds\,s^{d/2}e^{-\lambda\mu(s)^2}< \infty \] for some $\lambda>0$ where $\mu(s)=\int^s_0dt\,c(t)^{-1/2}$ then we establish that $H_\Omega$ is $L_1$-unique, i.e. it has a unique $L_1$-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique $L_2$-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent to the capacity of the boundary of $\Omega$, measured with respect to $H_\Omega$, 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 $H_\Omega$ at $A$.


  • Derek W. RobinsonCentre for Mathematics and its Applications
    Australian National University
    Canberra, ACT 0200, Australia
  • Adam SikoraDepartment of Mathematics
    Macquarie University
    Sydney, NSW 2109, Australia

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