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\(L_{1}\)-uniqueness of degenerate elliptic operators. (English) Zbl 1222.47033

In the interesting paper under review, the authors establish \(L_{1}\)-uniqueness of degenerate elliptic operators. Precisely, let \(\Omega\) be an open subset of \({\mathbb R}^d\) containing the origin and 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 real coefficients \(c_{ij}\in W^{1,\infty}_{\text{loc}}(\overline\Omega)\) and the symmetric coefficient matrix \(C=(c_{ij})\) satisfies bounds \(0<C(x)\leq c(|x|) I\) for all \(x\in \Omega\). Assuming that
\[ \int^\infty_0ds\,s^{d/2}e^{-\lambda\mu(s)^2}<\infty \]
for some \(\lambda>0,\) where \(\mu(s)=\int^s_0dt\,c(t)^{-1/2},\) the authors 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., \(H_\Omega\) 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. It is also proved 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 \(A.\)

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47D07 Markov semigroups and applications to diffusion processes
35J70 Degenerate elliptic equations
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