混合边界条件下混合椭圆问题的尖锐特征值定理
A sharp eigenvalue theorem for mixed elliptic problems under mixed boundary conditions
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中文总结 AI 辅助
本文针对混合边界条件下含经典与分数阶拉普拉斯算子的混合椭圆问题，建立其正弱解存在性的特征值刻画定理，拓展了经典椭圆框架的相关结果。
中文摘要 AI 辅助
本文研究一类同时包含局部与非局部算子的特征值问题，即经典拉普拉斯算子（Laplacian）与分数阶拉普拉斯算子（fractional Laplacian），这类问题处于混合边界条件下。更确切地说，我们考虑如下问题：\begin{equation}\label{1}\left\{\begin{aligned}\mathcal{L}u &= \lambda f(u), \quad u>0 &&\text{在 }\Omega\text{ 内},\u&=0 &&\text{在 }U^c\text{ 内},\mathcal{N}_s(u)&=0 &&\text{在 }\mathcal{N}\text{ 上},\frac{\partial u}{\partial\nu}&=0 &&\text{在 }\partial\Omega\cap\overline{\mathcal{N}}\text{ 上},\end{aligned}\right.\tag{$P_\lambda$}\end{equation}其中 \\( U=\Omega\cup\mathcal{N}\cup \bigl(\partial\Omega\cap\overline{\mathcal{N}}\bigr) \\)，\\(\Omega\subseteq\mathbb{R}^n\\) 是具有光滑边界的有界开集，\\(\lambda>0\\) 是实参数，\\(f\\) 是满足 \\(f(0)=0\\) 的连续函数，且 \\( \mathcal{L}=-\Delta+(-\Delta)^s \\)，\\(s\in(0,1)\\)。我们建立了问题 \\((P_{\lambda})\\) 正弱解存在性的刻画定理，受 Molica Bisci 和 Rădulescu 提出的经典椭圆框架启发，得到了混合局部-非局部算子在混合边界条件下的对应刻画结果。
英文摘要
In this paper, we study a class of eigenvalue problems involving both local and nonlocal operators, namely the classical Laplacian and the fractional Laplacian, under mixed boundary conditions. More precisely, we consider the problem \begin{equation}\label{1} \left\{ \begin{aligned} \mathcal{L}u &= λf(u), \quad u>0 &&\text{in }Ω,\\ u&=0 &&\text{in }U^c,\\ \mathcal{N}_s(u)&=0 &&\text{in }\mathcal{N},\\ \frac{\partial u}{\partialν}&=0 &&\text{on }\partialΩ\cap\overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_λ$} \end{equation} where \( U=Ω\cup\mathcal{N}\cup \bigl(\partialΩ\cap\overline{\mathcal{N}}\bigr), \) \(Ω\subseteq\mathbb{R}^n\) is a bounded open set with smooth boundary, \(λ>0\) is a real parameter, \(f\) is continuous function with \(f(0)=0\), and \[ \mathcal{L}=-Δ+(-Δ)^s, \qquad s\in(0,1). \] We establish a characterization theorem for the existence of positive weak solutions to problem (P_λ). Motivated by the classical elliptic framework developed by Molica Bisci and Rădulescu \cite{MolicaBisciRadulescu2017}, we establish a corresponding characterization result for mixed local-nonlocal operators under mixed boundary conditions.
