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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.12637 |
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| _version_ | 1866912074652712960 |
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| author | Abatangelo, Laura Ferrero, Alberto Luzzini, Paolo |
| author_facet | Abatangelo, Laura Ferrero, Alberto Luzzini, Paolo |
| contents | The Grushin Laplacian $- Δ_α$ is a degenerate elliptic operator in $\mathbb{R}^{h+k}$ that degenerates on $\{0\} \times \mathbb{R}^k$. We consider weak solutions of $- Δ_αu= Vu$ in an open bounded connected domain $Ω$ with $V \in W^{1,σ}(Ω)$ and $σ> Q/2$, where $Q = h + (1+α)k$ is the so-called homogeneous dimension of $\mathbb{R}^{h+k}$. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of $Ω$. As an application we derive strong unique continuation properties for solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On solutions to a class of degenerate equations with the Grushin operator Abatangelo, Laura Ferrero, Alberto Luzzini, Paolo Analysis of PDEs 35H10, 35J70, 35B40, 35A16 The Grushin Laplacian $- Δ_α$ is a degenerate elliptic operator in $\mathbb{R}^{h+k}$ that degenerates on $\{0\} \times \mathbb{R}^k$. We consider weak solutions of $- Δ_αu= Vu$ in an open bounded connected domain $Ω$ with $V \in W^{1,σ}(Ω)$ and $σ> Q/2$, where $Q = h + (1+α)k$ is the so-called homogeneous dimension of $\mathbb{R}^{h+k}$. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of $Ω$. As an application we derive strong unique continuation properties for solutions. |
| title | On solutions to a class of degenerate equations with the Grushin operator |
| topic | Analysis of PDEs 35H10, 35J70, 35B40, 35A16 |
| url | https://arxiv.org/abs/2410.12637 |