Geodesic
The Multidimensional Weyl Theorem and Covering Families
Matematičeskie zametki, Tome 73 (2003) no. 1, pp. 38-48.

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The well-known theorem of Weyl about the essential self-adjointness of the Sturm–Liouville operator $Lu=-(p(x)u')'+q(x)u$ in $L_2(\mathbb R^1)$ with $D_L=C_0^\infty(\mathbb R^1)$, $p(x)>0$, and $q(x)\ge\operatorname{const}$ is generalized to second-order elliptic operators in $L_2(G)$ ($G\subseteq\mathbb R^n$). The multidimensional Weyl theorem is derived from a more general theorem; to state and prove the latter, a special covering family is constructed. The results obtained imply the known multidimensional analogs of the Weyl theorem and, unlike these analogs, apply to open proper subsets $G$ in $\mathbb R^n$ . 
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A. G. Brusentsev. The Multidimensional Weyl Theorem and Covering Families. Matematičeskie zametki, Tome 73 (2003) no. 1, pp. 38-48. http://geodesic.mathdoc.fr/item/MZM_2003_73_1_a3/

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