On the criticality and the principal eigenvalue of almost periodic elliptic operators
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917245588865024 |
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| author | Rossi, Luca |
| author_facet | Rossi, Luca |
| contents | We review the notion and the properties of the generalised \pe\ for elliptic operators in unbounded domains, and we relate it with the criticality theory. We focus on operators with almost periodic coefficients. We present a Liouville-type result in dimension $N\leq2$. Next, we show with a counter-example that criticality is not equivalent to the existence of an almost periodic principal eigenvalue, even for self-adjoint operators. Finally, we exhibit an almost periodic operator which is subcritical but which admits a critical limit operator. This is a manifestation of the instability character of the criticality property in the almost periodic setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03644 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the criticality and the principal eigenvalue of almost periodic elliptic operators Rossi, Luca Analysis of PDEs 35J15, 35P05 35B15 We review the notion and the properties of the generalised \pe\ for elliptic operators in unbounded domains, and we relate it with the criticality theory. We focus on operators with almost periodic coefficients. We present a Liouville-type result in dimension $N\leq2$. Next, we show with a counter-example that criticality is not equivalent to the existence of an almost periodic principal eigenvalue, even for self-adjoint operators. Finally, we exhibit an almost periodic operator which is subcritical but which admits a critical limit operator. This is a manifestation of the instability character of the criticality property in the almost periodic setting. |
| title | On the criticality and the principal eigenvalue of almost periodic elliptic operators |
| topic | Analysis of PDEs 35J15, 35P05 35B15 |
| url | https://arxiv.org/abs/2602.03644 |