论文标题
连续函数晶格上的凸单单位元素半群
Convex monotone semigroups on lattices of continuous functions
论文作者
论文摘要
我们考虑在Banach晶格上凸出单调$ C_0 $ -Semigroups，这被认为是$σ$ -Dedekind完整Banach lattice的Riesz子空间。典型的示例包括所有有界均匀连续函数的空间，以及无穷大的所有连续函数的空间。我们表明，凸半群的经典发电机的域通常不是不变的。因此，我们提出了针对域的替代版本，例如单调域和Lipschitz集，为此我们在半群下证明了不变性。作为主要结果，我们根据发电机的扩展版本获得了半群的唯一性。结果用与汉密尔顿 - 雅各比 - 贝尔曼方程有关的几个示例进行了说明，包括移位半群的非线性版本和热量方程。特别是，我们确定了它们的对称Lipschitz集，这些集合是不变的，可以从微弱意义上理解发电机。
We consider convex monotone $C_0$-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a $σ$-Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton-Jacobi-Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow to understand the generators in a weak sense.
