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具有Robin或Dirichlet边界条件的挠函数

On the torsion function with Robin or Dirichlet boundary conditions

M. van den Berg, D. Bucur

备注：
19 pages
发表信息：
Journal of Functional Analysis 266 (2014) 1647--1666
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摘要：

对于$p\in (1,+\infty)$和$b \in (0, +\infty]$的$p$ -扭函数与罗宾边界条件相关联的任意开集$\Om \subset \R^m$满足形式方程$-Δ_p =1$在$\Om$和$|\nabla u|^{p-2} \frac{\partial u}{\partial n} + b|u|^{p-2} u =0$上$\partial \Om$。我们得到的$L^\infty$规范的$u$ { \it仅}在底部的频谱（罗宾$p$-Laplacian），$b$和以下两个极端情况下的空间的维数：线性框架（对应于$p=2$）和任意$b>0$，和非线性框架（对应于任意$p>1$）和Dirichlet边界条件（$b=+\infty$）。在一般情况下，$p\not=2, p \in (1, +\infty)$和$b>0$我们的界限也涉及$\Om$的勒贝格测度。

英文摘要：

For $p\in (1,+\infty)$ and $b \in (0, +\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set $\Om \subset \R^m$ satisfies formally the equation $-Δ_p =1$ in $\Om$ and $|\nabla u|^{p-2} \frac{\partial u}{\partial n} + b|u|^{p-2} u =0$ on $\partial \Om$. We obtain bounds of the $L^\infty$ norm of $u$ {\it only} in terms of the bottom of the spectrum (of the Robin $p$-Laplacian), $b$ and the dimension of the space in the following two extremal cases: the linear framework (corresponding to $p=2$) and arbitrary $b>0$, and the non-linear framework (corresponding to arbitrary $p>1$) and Dirichlet boundary conditions ($b=+\infty$). In the general case, $p\not=2, p \in (1, +\infty)$ and $b>0$ our bounds involve also the Lebesgue measure of $\Om$.

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