## $$L_{1}$$-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

### Keywords:

$$L_1$$-uniqueness; Markov uniqueness; capacity
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