区域分数p-拉普拉斯算子狄利克雷问题的最优稳定性
Optimal stability of Dirichlet problem for the regional fractional $p$-Laplacian
AI总结：
针对区域分数p-拉普拉斯算子(-Δ)^s_{p,Ω}的狄利克雷边值问题及相关归一化狄利克雷本征对，证明了当s→1⁻时解的最优稳定性。
AI中文摘要：
我们针对区域（分数）p-拉普拉斯算子(-Δ)^s_{p,Ω}的狄利克雷边值问题建立了最优稳定性，其中0<s≤1，1/s<p<∞，Ω是R^d中的有界 Lipschitz 区域。具体而言，若u_s属于W^{s,p}(Ω)，满足在Ω内(-Δ)^s_{p,Ω}u_s = f_s，在∂Ω上u_s = g_s，那么在数据f_s和g_s满足适当条件时，我们证明当s→1⁻时，||u_s - u₁||_{W^{s,p}(Ω)}→0。我们还得到了与(-Δ)^s_{p,Ω}相关的归一化狄利克雷本征对(λ_s,φ_s)的类似最优稳定性结果。
英文摘要：
We establish the optimal stability of Dirichlet boundary value problem for the regional (fractional) $p$-Laplacian $(-Δ)^s_{p,Ω}$ with $0<s\leq 1$, $\frac{1}{s}<p<\infty$ and $Ω\subset \mathbb{R}^d$ bounded Lipschitz. More precisely, if $u_s \in W^{s,p}(Ω)$ satisfies $(-Δ)^s_{p,Ω} u_s = f_s$ in $Ω$ and $u_s = g_s$ on $\partial Ω$, then under appropriate condition on the date $f_s$ and $g_s$ we show that $\|u_s - u_1\|_{W^{s,p}(Ω)} \to 0 \quad \text{as } s \to 1^-.$ We also obtain an analogous optimal stability of the normalized Dirichlet eigenpairs $(λ_s,φ_s)$ associated with $(-Δ)^s_{p,Ω}$.
