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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.06726 |
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| _version_ | 1866913493150597120 |
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| author | Fu, Haotong Wang, Wei Zhang, Zhifei |
| author_facet | Fu, Haotong Wang, Wei Zhang, Zhifei |
| contents | In this paper, we investigate the interior regularity theory for stationary solutions of the supercritical nonlinear elliptic equation $$ -Δu=|u|^{p-1}u\quad\text{in }Ω,\quad p>\frac{n+2}{n-2}, $$ where $ Ω\subset\mathbb{R}^n $ is a bounded domain with $ n\geq 3 $. Our primary focus is on the structure of stratification for the singular sets. We define the $ k $-th stratification $ S^k(u) $ of $ u $ based on the tangent functions and measures. We show that the Hausdorff dimension of $ S^k(u) $ is at most $ k $ and $ S^k(u) $ is $ k $-rectifiable, and establish estimates for volumes associated with points that have lower bounds on the regular scales. These estimates enable us to derive sharp interior estimates for the solutions. Specifically, if $ α_p=\frac{2(p+1)}{p-1} $ is not an integer, then for any $ j\in\mathbb{Z}_{\geq 0} $, we have $$ D^ju\in L_{\operatorname{loc}}^{q_j,\infty}(Ω), $$ which implies that for any $ Ω'\subset\subsetΩ$, $$ \sup\{λ>0:λ^{q_j}\mathcal{L}^n(\{x\inΩ':|D^ju(x)|>λ\})\}<+\infty, $$ where $ \mathcal{L}^n(\cdot) $ is the $ n $-dimensional Lebesgue measure, and $$ q_j=\frac{(p-1)(\lfloorα_p\rfloor+1)}{2+j(p-1)}, $$ with $ \lfloorα_p\rfloor $ being the integer part of $ α_p $. The proofs of these results rely on Reifenberg-type theorems developed by A. Naber and D. Valtorta to study the stratification of harmonic maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06726 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative stratification and sharp regularity estimates for supercritical semilinear elliptic equations Fu, Haotong Wang, Wei Zhang, Zhifei Analysis of PDEs In this paper, we investigate the interior regularity theory for stationary solutions of the supercritical nonlinear elliptic equation $$ -Δu=|u|^{p-1}u\quad\text{in }Ω,\quad p>\frac{n+2}{n-2}, $$ where $ Ω\subset\mathbb{R}^n $ is a bounded domain with $ n\geq 3 $. Our primary focus is on the structure of stratification for the singular sets. We define the $ k $-th stratification $ S^k(u) $ of $ u $ based on the tangent functions and measures. We show that the Hausdorff dimension of $ S^k(u) $ is at most $ k $ and $ S^k(u) $ is $ k $-rectifiable, and establish estimates for volumes associated with points that have lower bounds on the regular scales. These estimates enable us to derive sharp interior estimates for the solutions. Specifically, if $ α_p=\frac{2(p+1)}{p-1} $ is not an integer, then for any $ j\in\mathbb{Z}_{\geq 0} $, we have $$ D^ju\in L_{\operatorname{loc}}^{q_j,\infty}(Ω), $$ which implies that for any $ Ω'\subset\subsetΩ$, $$ \sup\{λ>0:λ^{q_j}\mathcal{L}^n(\{x\inΩ':|D^ju(x)|>λ\})\}<+\infty, $$ where $ \mathcal{L}^n(\cdot) $ is the $ n $-dimensional Lebesgue measure, and $$ q_j=\frac{(p-1)(\lfloorα_p\rfloor+1)}{2+j(p-1)}, $$ with $ \lfloorα_p\rfloor $ being the integer part of $ α_p $. The proofs of these results rely on Reifenberg-type theorems developed by A. Naber and D. Valtorta to study the stratification of harmonic maps. |
| title | Quantitative stratification and sharp regularity estimates for supercritical semilinear elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.06726 |