通过积分恒等式得到的关于p-拉普拉斯算子的定量Gidas-Ni-Nirenberg型结果
A quantitative Gidas-Ni-Nirenberg-type result for the $p$-Laplacian via integral identities
AI总结：
本文首次利用积分恒等式证明了p-拉普拉斯算子（包括p=2的经典情形）的定量Gidas-Ni-Nirenberg型对称稳定性结果，并给出奇异集测度估计与一致梯度界。
摘要：
我们证明了一个涉及p-拉普拉斯算子的Gidas-Ni-Nirenberg型对称性结果的定量版本。这里的定量稳定性是通过积分恒等式实现的，其基础是J. Serra在2013年建立的刚性证明，该证明将P. L. Lions针对经典拉普拉斯算子在二维情形下提出的论证推广到了一般维数和p-拉普拉斯算子。关于经典Gidas-Ni-Nirenberg对称性定理（涉及经典拉普拉斯算子）的稳定性结果，曾由Rosset在1994年以及Ciraolo、Cozzi、Perugini、Pollastro在2024年通过移动平面法建立。据作者所知，本文首次给出了当p不等于2时涉及p-拉普拉斯算子的定量Gidas-Ni-Nirenberg型结果。即使对于经典拉普拉斯算子（即p=2的情形），这也是首次利用积分恒等式来实现Gidas-Ni-Nirenberg型结果的稳定性。此外，我们还得到了奇异集测度的定量估计和一个显式的一致梯度界。
英文摘要：
We prove a quantitative version of a Gidas-Ni-Nirenberg-type symmetry result involving the $p$-Laplacian. Quantitative stability is achieved here via integral identities based on the proof of rigidity established by J. Serra in 2013, which extended to general dimension and the $p$-Laplacian operator an argument proposed by P. L. Lions in dimension $2$ for the classical Laplacian. Stability results for the classical Gidas-Ni-Nirenberg symmetry theorem (involving the classical Laplacian) via the method of moving planes were established by Rosset in 1994 and by Ciraolo, Cozzi, Perugini, Pollastro in 2024. To the authors' knowledge, the present paper provides the first quantitative Gidas-Ni-Nirenberg-type result involving the $p$-Laplacian for $p \neq 2$. Even for the classical Laplacian (i.e., for $p=2$), this is the first time that integral identities are used to achieve stability for a Gidas-Ni-Nirenberg-type result. In passing, we obtain a quantitative estimate for the measure of the singular set and an explicit uniform gradient bound.
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