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On self-adjointness of symmetric diffusion operators
Journal of Evolution Equations ( IF 1.1 ) Pub Date : 2020-04-03 , DOI: 10.1007/s00028-020-00572-3
Derek W. Robinson

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.



中文翻译:

对称扩散算子的自伴性

\(\ Omega \)\(\ mathbf {R} ^ d \)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\(C =(\,c_ {kl} \,)> 0 \)\(c_ {kl} = c_ { lk} \)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\(a \,I \)\(d_ \ Gamma \ rightarrow 0 \)。因此\(\ delta \ ge 0 \)措施的经营者和退化的顺序,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\(\ Omega \)\(C ^ 2 \)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\(\ Omega = \ mathbf {R} ^ d \反斜杠\ overline {\ Pi} \)\(\ Pi \)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。

更新日期:2020-04-03
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