AI 中文总结
研究加权图上拉普拉斯算子和多孔介质型算子的增生性与 m - 增生性，给出极大算子相关性质的条件及等价关系，探讨极小算子情况，还涉及不同\(p\)值下\(\ell^p\)空间的相关结论以及与其他概念的联系。
AI 中文摘要
我们研究加权图上拉普拉斯算子和多孔介质型算子的增生性和 m - 增生性。特别地，给出了一些蕴含极大算子这些性质的条件，并研究这些算子与各种限制何时一致。对于\(\ell^1\)上的多孔介质型算子以及\(p\in[1,\infty)\)时\(\ell^p\)上的拉普拉斯算子，证明存在定义域的稠密子集使极大算子是 m - 增生的，进而得出平移算子的增生性、m - 增生性和单射性等价。在图的附加条件下，证明极大算子在整个定义域而非仅稠密子集上是 m - 增生的。还研究了极小算子，表明其为 m - 增生当且仅当极小与极大算子一致且极大算子是增生的，并给出了蕴含这种一致性的条件。此外，对于\(\ell^p\)上的极小拉普拉斯算子，证明增生性和 m - 增生性不等价。对于\(\ell^2\)情形，给出与马尔可夫唯一性和本质自伴性的联系。对于\(\ell^\infty\)情形，则建立了无穷远处的随机完备性、\(\ell^\infty\)上极大拉普拉斯算子的 m - 增生性以及\(\ell^1\)上极小拉普拉斯算子的 m - 增生性之间的等价关系。
英文摘要
We study the accretivity and m-accretivity of Laplacian and porous medium-type operators on weighted graphs. In particular, we give several conditions that imply these properties for maximal operators and investigate when these operators agree with various restrictions. For porous medium-type operators on $\ell^1$ and for Laplacians on $\ell^p$ for $p \in [1,\infty)$, we show that there always exists a dense subset of the domain on which the maximal operator is m-accretive. As a consequence, we establish that accretivity, m-accretivity and injectivity of the shifted operator are all equivalent for these maximal operators. Under additional conditions on the graph, we then prove that the maximal operators are m-accretive on the entire domain, not just a dense subset. We also investigate minimal operators and show that they are m-accretive if and only if the minimal and maximal operators agree and the maximal operator is accretive. We then give some conditions that imply this agreement. Furthermore, for the minimal Laplacian on $\ell^p$, we show that accretivity and m-accretivity are not equivalent. For the $\ell^2$ case, we give connections to Markov uniqueness and essential self-adjointness. For the $\ell^\infty$ case, we establish the equivalence of stochastic completeness at infinity, m-accretivity for the maximal Laplacian on $\ell^\infty$, and m-accretivity of the minimal Laplacian on $\ell^1$.
Commentsv2: further weakened the assumptions on the nonlinearity ϕ
