arXiv 2206.033372022-06-07math.AP

当$p$变为$1$时，具有Robin边界条件的$p$ -拉普拉斯算子的解的行为

Behaviour of solutions to $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$

Francesco Della Pietra, Francescantonio Oliva, Sergio Segura de León

摘要：

本文研究了如下非齐次Robin边值问题解的渐近性质，即$p\to 1^{+}$：\begin{equation} \label{pbabstract} \tag{P} \left\{\begin{array}{ll} \displaystyle -Δ_p u_p = f & \text{in }Ω, \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot ν+λ|u_p|^{p-2}u_p = g& \text{on } \partialΩ, \end{array}\right. \end{equation}其中$Ω$是$\mathbb R^{N}$中具有充分光滑边界的有界区域，$ν$是其单位向外法向量，$Δ_p v$是具有$p>1$的$p$ -Laplacian算子.数据$f\in L^{N,\infty}(Ω)$（表示Marcinkiewicz空间）和$λ,g$是在具有$λ\ge0$的$\partialΩ$上定义的有界函数。我们找到一个阈值，在这个阈值之下，$p$族的解趋于0，在这个阈值之上，这个族就会爆炸。作为第二个兴趣，我们处理$1$ -拉普拉斯问题正式产生的$p\to 1^+$在\eqref{pbabstract}。

英文摘要：

We study the asymptotic behaviour, as $p\to 1^{+}$, of the solutions of the following inhomogeneous Robin boundary value problem: \begin{equation} \label{pbabstract} \tag{P} \left\{\begin{array}{ll} \displaystyle -Δ_p u_p = f & \text{in }Ω, \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot ν+λ|u_p|^{p-2}u_p = g& \text{on } \partialΩ, \end{array}\right. \end{equation} where $Ω$ is a bounded domain in $\mathbb R^{N}$ with sufficiently smooth boundary, $ν$ is its unit outward normal vector and $Δ_p v$ is the $p$-Laplacian operator with $p>1$. The data $f\in L^{N,\infty}(Ω)$ (which denotes the Marcinkiewicz space) and $λ,g$ are bounded functions defined on $\partialΩ$ with $λ\ge0$. We find the threshold below which the family of $p$--solutions goes to 0 and above which this family blows up. As a second interest we deal with the $1$-Laplacian problem formally arising by taking $p\to 1^+$ in \eqref{pbabstract}.

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。

摘要：

英文摘要：

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