带Robin边界条件的p-Laplacian的Serrin型超定问题
Serrin-type overdetermined problem for the $p$-Laplacian with Robin boundary conditions
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中文总结 AI 辅助
本文研究带Robin边界条件的p-Laplacian的Serrin型超定问题，通过新积分恒等式证明解的对称性，并得到类似Serrin和Alexandrov定理的刚性结果。
中文摘要 AI 辅助
设$\Omega$为$\mathbb{R}^N$（$N \geq 2$）中开有界连通子集，属于$C^{2,\alpha}$类，其中$\alpha \in (0,1)$。设$p \geq 2$且$\beta>0$。我们证明了在Robin边界条件下，满足由形状导数论证自然产生的超定条件以及对$\beta$和$\partial\Omega$主曲率最小值的额外条件的$p$-扭转问题的解的对称性。证明基于一些新的积分恒等式，这些恒等式涉及将$p$-Laplacian的线性化算子应用于标准$P$-函数。此外，我们还证明了其他一些具有Serrin定理和Alexandrov定理精神的刚性结果。
英文摘要
Let $Ω$ be an open bounded connected subset of $\mathbb{R}^N$, $N \geq 2$, of class $C^{2,α}$, for $α\in (0,1)$. Let $p \geq 2$ and $β>0$. We prove the symmetry of the solution of the $p$-torsion problem with Robin boundary condition subject to the natural overdetermined condition coming from a shape derivative argument and to an extra condition on $β$ and on the minimum of the principal curvatures of $\partialΩ$. The proof is based on some new integral identities, involving the linearized operator of the $p$-Laplacian applied to the standard $P$-function. In passing we prove some other rigidity results in the spirit of Serrin's and Alexandrov's Theorems.
发表机构
- Scuola Superiore Meridionale（南方高等学院）
- Università degli Studi di Verona（维罗纳大学）
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