Absence of $L^p$ spectrum for asymptotically flat diffusions in region with cavities
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909689344688128 |
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| author | Banerjee, Agnid Garofalo, Nicola |
| author_facet | Banerjee, Agnid Garofalo, Nicola |
| contents | We study solutions to variable-coefficient elliptic equations of the form $-\D(A(x) \nabla u) = κu$, $κ>0$, in an exterior domain $\Om\subset \Rn$, where $A(x)$ is uniformly elliptic and asymptotically flat. Extending Rellich's classical $L^2$ result for the Laplacian, we show that if $u\in L^p(\Om)$ for some $0<p<\frac{2n}{n-1}$, then $u\equiv 0$. The proof uses new monotonicity formulas based on weighted energies and vector fields adapted to the geometry of $A(x)$. Our results highlight a sharper integrability threshold in the variable-coefficient setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10728 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Absence of $L^p$ spectrum for asymptotically flat diffusions in region with cavities Banerjee, Agnid Garofalo, Nicola Analysis of PDEs 35P15, 35Q40, 47A75 We study solutions to variable-coefficient elliptic equations of the form $-\D(A(x) \nabla u) = κu$, $κ>0$, in an exterior domain $\Om\subset \Rn$, where $A(x)$ is uniformly elliptic and asymptotically flat. Extending Rellich's classical $L^2$ result for the Laplacian, we show that if $u\in L^p(\Om)$ for some $0<p<\frac{2n}{n-1}$, then $u\equiv 0$. The proof uses new monotonicity formulas based on weighted energies and vector fields adapted to the geometry of $A(x)$. Our results highlight a sharper integrability threshold in the variable-coefficient setting. |
| title | Absence of $L^p$ spectrum for asymptotically flat diffusions in region with cavities |
| topic | Analysis of PDEs 35P15, 35Q40, 47A75 |
| url | https://arxiv.org/abs/2507.10728 |