A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912070101893120 |
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| author | Ciraolo, Giulio Li, Xiaoliang |
| author_facet | Ciraolo, Giulio Li, Xiaoliang |
| contents | In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-Δ_p \,u=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$, where $N\ge 2$, $1<p<+\infty$ and $f$ is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for $u$, which was established by Lions and Serra when $Ω$ is a ball. By exploiting a quantitative version of the Pólya-Szegö principle, we prove that the deviation of $u$ from its Schwarz symmetrization can be estimated in terms of the isoperimetric deficit of $Ω$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_09482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities Ciraolo, Giulio Li, Xiaoliang Analysis of PDEs In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-Δ_p \,u=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$, where $N\ge 2$, $1<p<+\infty$ and $f$ is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for $u$, which was established by Lions and Serra when $Ω$ is a ball. By exploiting a quantitative version of the Pólya-Szegö principle, we prove that the deviation of $u$ from its Schwarz symmetrization can be estimated in terms of the isoperimetric deficit of $Ω$. |
| title | A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.09482 |