Lévy 符号收敛下半线性外部 Dirichlet 问题解的稳定性

中文摘要 AI 辅助

我们研究由可能非对称的 Lévy 算子驱动的半线性外部 Dirichlet 问题的稳定性。设 $L^n$ 和 $L$ 分别具有 Lévy-Khintchine 符号 $\psi_n$ 和 $\psi$。我们对算子的基本假设是 $\psi_n(\xi)\to\psi(\xi)$ 对所有 $\xi\in\mathbb R^d$ 成立。在区域的温和正则性假设和极限过程预解式的绝对连续性条件下，我们证明了解的点态和 $L^p$ 稳定性。不需要对称性、一致椭圆性、Lévy 核的可比性或公共能量空间；允许非零外部数据；且单调非线性在解变量上不受增长限制。因此，该框架涵盖局部、非局部和混合算子，特别是非局部到局部极限，以及分数阶、相对论性、各向异性和非对称稳定型算子。证明将 Lévy 过程的收敛性与退出时间和退出位置的详细分析以及具有变化终止时间的单调倒向随机微分方程的稳定性定理相结合。在对称情形下，我们还在 $C^0$ 区域上获得相应部分形式的 Mosco 收敛，并给出一个反例表明全空间 Mosco 收敛不一定能传递到任意开集上的部分形式。

英文摘要

We study stability of semilinear exterior Dirichlet problems driven by possibly nonsymmetric Levy operators. Let $L^n$ and $L$ have Lévy-Khintchine symbols $ψ_n$ and $ψ$, respectively. Our basic assumption on the operators is that $ψ_n(ξ)\toψ(ξ)$ for every $ξ\in\mathbb R^d$. Under mild regularity assumptions on the domain and an absolute-continuity condition for the resolvent of the limiting process, we prove pointwise and $L^p$-stability of solutions. No symmetry, uniform ellipticity, comparability of Lévy kernels, or common energy space is required; nonzero exterior data are allowed; and the monotone nonlinearity is subject to no growth restriction in the solution variable. The framework therefore covers local, nonlocal and mixed operators and, in particular, nonlocal-to-local limits, as well as fractional, relativistic, anisotropic and nonsymmetric stable-type operators. The proofs combine convergence of Lévy processes with a detailed analysis of exit times and exit positions and a stability theorem for monotone backward stochastic differential equations with varying terminal times. In the symmetric case, we also obtain Mosco convergence of the corresponding part forms on $C^0$ domains and give a counterexample showing that whole-space Mosco convergence need not pass to part forms on arbitrary open sets.

发表机构

• Faculty of Mathematics and Computer Science, Nicolaus Copernicus University（尼古拉·哥白尼大学数学与计算机科学学院）

机构由 AI 辅助整理，请以论文原文为准。