$L_1$-uniqueness of degenerate elliptic operators

Derek W. Robinson¹ ; Adam Sikora²

¹ Centre for Mathematics and its Applications Australian National University Canberra, ACT 0200, Australia
² Department of Mathematics Macquarie University Sydney, NSW 2109, Australia

Studia Mathematica, Tome 203 (2011) no. 1, pp. 79-103

Résumé

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 $A$.

Détail

DOI : 10.4064/sm203-1-5

Keywords: omega subset mathbb omega furthermore omega sum partial partial second order partial differential operator domain infty omega where coefficients infty loc overline omega real coefficient matrix satisfies bounds leq omega int infty lambda infty lambda where int establish omega unique has unique extension which generates continuous semigroup only markov unique has unique extension which generates submarkovian semigroup moreover these uniqueness conditions equivalent capacity boundary omega measured respect omega being zero demonstrate capacity depends gross features hausdorff dimension subsets boundary set order degeneracy omega nbsp