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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2411.16048 |
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| _version_ | 1866910741785739264 |
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| author | Wang, Wei Zhang, Zhifei |
| author_facet | Wang, Wei Zhang, Zhifei |
| contents | In this paper, we study the stationary solutions of semilinear elliptic equation with singular nonlinearity $$ Δu=u^{-p}+f,\,\,u\geq 0\text{ in }Ω\subset\mathbb{R}^n, $$ where $ n\geq 2 $, $ p>1 $, $ Ω$ is a bounded domain, and $ f\in L^q(Ω) $ with $ \frac{1}{2}+\frac{1}{2p}<\frac{q}{n} $. We establish a sharp estimate for the Minkowski content of the rupture set $ \{u=0\} $ and demonstrate that this set is $ (n-2) $-rectifiable. For this, we examine the stratification of the rupture set based on the symmetry properties of tangent functions, leading to the proof of $ k $-rectifiability for each $ k $-stratum. As a significant byproduct of our analysis, we improve the integrability of $ D^ju $ with $ j\in\mathbb{Z}_+ $ to the optimal Lorentz space $ L^{\frac{2(p+1)}{j(p+1)-2},\infty} $, under the assumption that $ D^{j-1}f $ is bounded. As an application of our results in the static case of the equation, for a class of suitable weak solutions to the three-dimensional evolutional problem $$ \partial_tu=Δu-u^{-p},\,\,u\geq 0\text{ in }(Ω\subset\mathbb{R}^3)\times(0,T), $$ where $ p>3 $ and $ T>0 $, we show that $ \{u(\cdot,t)=0\} $ is $ 1 $-rectifiable for a.e. $ t\in(0,T) $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16048 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fine structure of rupture set for semilinear elliptic equation with singular nonlinearity Wang, Wei Zhang, Zhifei Analysis of PDEs In this paper, we study the stationary solutions of semilinear elliptic equation with singular nonlinearity $$ Δu=u^{-p}+f,\,\,u\geq 0\text{ in }Ω\subset\mathbb{R}^n, $$ where $ n\geq 2 $, $ p>1 $, $ Ω$ is a bounded domain, and $ f\in L^q(Ω) $ with $ \frac{1}{2}+\frac{1}{2p}<\frac{q}{n} $. We establish a sharp estimate for the Minkowski content of the rupture set $ \{u=0\} $ and demonstrate that this set is $ (n-2) $-rectifiable. For this, we examine the stratification of the rupture set based on the symmetry properties of tangent functions, leading to the proof of $ k $-rectifiability for each $ k $-stratum. As a significant byproduct of our analysis, we improve the integrability of $ D^ju $ with $ j\in\mathbb{Z}_+ $ to the optimal Lorentz space $ L^{\frac{2(p+1)}{j(p+1)-2},\infty} $, under the assumption that $ D^{j-1}f $ is bounded. As an application of our results in the static case of the equation, for a class of suitable weak solutions to the three-dimensional evolutional problem $$ \partial_tu=Δu-u^{-p},\,\,u\geq 0\text{ in }(Ω\subset\mathbb{R}^3)\times(0,T), $$ where $ p>3 $ and $ T>0 $, we show that $ \{u(\cdot,t)=0\} $ is $ 1 $-rectifiable for a.e. $ t\in(0,T) $. |
| title | Fine structure of rupture set for semilinear elliptic equation with singular nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.16048 |