Let \(\Omega \) be a domain in \(\mathbf{R}^d\) with boundary \(\Gamma \) and let \(d_\Gamma \) denote the Euclidean distance to \(\Gamma \). Further let \(H=-\,\mathrm{div}(C\nabla )\) where \(C=(\,c_{kl}\,)>0\) with \(c_{kl}=c_{lk}\) real, bounded, Lipschitz continuous functions and \(D(H)=C_c^\infty (\Omega )\). The matrix \(Cd_\Gamma ^{\,-\delta }\) is assumed to converge uniformly to a diagonal matrix \(a\,I\) as \(d_\Gamma \rightarrow 0\). Thus \(\delta \ge 0\) measures the order of degeneracy of the operator and a, a positive Lipschitz function, gives the boundary profile of the operator. In addition we place a mild restriction on the order of degeneracy of the derivatives of the coefficients at the boundary. Then we derive sufficient conditions for H to be essentially self-adjoint as an operator on \(L_2(\Omega )\) in three general cases. Specifically, if \(\Omega \) is a \(C^2\)-domain, or if \(\Omega =\mathbf{R}^d\backslash S\) where S is a countable set of positively separated points, or if \(\Omega =\mathbf{R}^d\backslash \overline{\Pi }\) with \(\Pi \) a convex set whose boundary has Hausdorff dimension \(d_H\in \{1,\ldots , d-1\}\) then the condition \(\delta >2-(d-d_H)/2\) is sufficient for essential self-adjointness. In particular \(\delta >3/2\) suffices for \(C^2\)-domains. Finally we prove that \(\delta \ge 3/2\) is necessary in the \(C^2\)-case.

令\（\ Omega \）为\（\ mathbf {R} ^ d \）中具有边界\（\ Gamma \）的域，令\（d_ \ Gamma \）表示到\（\ Gamma \）的欧几里得距离。进一步让\（H =-\，\ mathrm {div}（C \ nabla）\）其中\（C =（\，c_ {kl} \，）> 0 \）与\（c_ {kl} = c_ { lk} \）实有界Lipschitz连续函数和\（D（H）= C_c ^ \ infty（\ Omega）\）。假设矩阵\（Cd_ \ Gamma ^ {\，-\ delta} \）统一收敛为对角矩阵\（a \，I \）为\（d_ \ Gamma \ rightarrow 0 \）。因此\（\ delta \ ge 0 \）措施的经营者和退化的顺序一，积极李氏功能，使操作员的边界轮廓。此外，我们对边界处的系数导数的简并性顺序进行了适度的限制。然后，我们推导了足够的条件，使H在三种常见情况下作为\（L_2（\ Omega）\）上的算子基本上是自伴的。具体来说，如果\（\ Omega \）是\（C ^ 2 \）域，或者\（\ Omega = \ mathbf {R} ^ d \反斜杠S \），其中S是可计数的正分隔点集，或者\（\ Omega = \ mathbf {R} ^ d \反斜杠\ overline {\ Pi} \）与\（\ Pi \）凸集的边界具有Hausdorff维数\（d_H \ in \ {1，\ ldots，d-1 \} \）然后条件\（\ delta> 2-（d-d_H）/ 2 \）就足以自我陪伴。特别是\（\ delta> 3/2 \）足以满足\（C ^ 2 \）-域。最后，我们证明在（C ^ 2 \）情况下\（\ delta \ ge 3/2 \）是必要的。