arXiv 0912.45362009-12-22math.AP

退化椭圆算子的Markov唯一性

Markov uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora

备注：: 24 pages
发表信息：: Ann. Scuola Norm. Pisa Cl. Sci. X (2011) 1-28

摘要：

设$Ω$是$\Ri^d$的开子集，$H_Ω=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$是$L_2(Ω)$上的二阶偏微分算子，其定义域为$C_c^\infty(Ω)$，其中系数$c_{ij}\in W^{1,\infty}(Ω)$是实对称的，$C=(c_{ij})$是$Ω$上的严格正定矩阵.特别地，$H_Ω$是局部强椭圆的。我们分析了$H_Ω$的次马尔可夫扩张，即生成次马尔可夫半群的自伴扩张。我们的主要结果建立了$H_Ω$是马尔可夫唯一的，即它有一个唯一的亚马尔可夫扩张，当且仅当$\capp_Ω(\partialΩ)=0$其中$\capp_Ω(\partialΩ)$是$Ω$的边界关于$H_Ω$测量的容量。第二个主要结果证明了$H_Ω$的Markov唯一性等价于由$H_Ω$的Friedrich扩张生成的半群是保守的.

英文摘要：

Let $Ω$ be an open subset of $\Ri^d$ and $H_Ω=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(Ω)$ with domain $C_c^\infty(Ω)$ where the coefficients $c_{ij}\in W^{1,\infty}(Ω)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $Ω$. In particular, $H_Ω$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_Ω$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_Ω$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if $\capp_Ω(\partialΩ)=0$ where $\capp_Ω(\partialΩ)$ is the capacity of the boundary of $Ω$ measured with respect to $H_Ω$. The second main result establishes that Markov uniqueness of $H_Ω$ is equivalent to the semigroup generated by the Friedrichs extension of $H_Ω$ being conservative.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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