连续函数格上的凸单调半群
Convex monotone semigroups on lattices of continuous functions
AI总结：
本文研究Banach格上凸单调$C_0$-半群，提出单调定义域与Lipschitz集合等替代定义域并证明其不变性，进而通过扩展生成元获得半群唯一性，并应用于Hamilton-Jacobi-Bellman方程等例子。
摘要：
我们考虑Banach格上的凸单调$C_0$-半群，该Banach格被假定为$\sigma$-Dedekind完备Banach格的Riesz子空间。典型例子包括所有有界一致连续函数的空间以及所有在无穷远处消失的连续函数的空间。我们证明了凸半群经典生成元的定义域通常不具有不变性。因此，我们提出了定义域的替代版本，如单调定义域和Lipschitz集合，并证明了这些集合在半群下的不变性。作为主要结果，我们利用生成元的扩展版本获得了半群的唯一性。这些结果通过若干与Hamilton-Jacobi-Bellman方程相关的例子加以说明，包括平移半群和热方程的非线性版本。特别地，我们确定了它们的对称Lipschitz集合，这些集合具有不变性，并允许在弱意义下理解生成元。
英文摘要：
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.
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。
