Sharp Liouville Theorems
Consider the equation div(φ2∇σ)=0{\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in ℝN{\mathbb{R}^{N}}, where φ>0{\varphi>0}. Berestycki, Caffarelli and Nirenberg proved in [H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded dom...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2021-02-01
|
| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2020-2111 |
| _version_ | 1811280135643463680 |
|---|---|
| author | Villegas Salvador |
| author_facet | Villegas Salvador |
| author_sort | Villegas Salvador |
| collection | DOAJ |
| description | Consider the equation div(φ2∇σ)=0{\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in ℝN{\mathbb{R}^{N}}, where φ>0{\varphi>0}. Berestycki, Caffarelli and Nirenberg proved in [H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 69–94] that if there exists C>0{C>0} such that ∫BR(φσ)2≤CR2\int_{B_{R}}(\varphi\sigma)^{2}\leq CR^{2} for every R≥1{R\geq 1}, then σ is necessarily constant. In this paper, we provide necessary and sufficient conditions on 0<Ψ∈C([1,∞)){0<\Psi\in C([1,\infty))} for which this result remains true if we replace CR2{CR^{2}} by Ψ(R){\Psi(R)} in any dimension N. In the case of the convexity of Ψ for large R>1{R>1} and Ψ′>0{\Psi^{\prime}>0}, this condition is equivalent to ∫1∞1Ψ′=∞.\int_{1}^{\infty}\frac{1}{\Psi^{\prime}}=\infty. |
| first_indexed | 2024-04-13T01:08:03Z |
| format | Article |
| id | doaj.art-c5ad475d3fe745de8bef02db17ba1223 |
| institution | Directory Open Access Journal |
| issn | 1536-1365 2169-0375 |
| language | English |
| last_indexed | 2024-04-13T01:08:03Z |
| publishDate | 2021-02-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-c5ad475d3fe745de8bef02db17ba12232022-12-22T03:09:16ZengDe GruyterAdvanced Nonlinear Studies1536-13652169-03752021-02-012119510510.1515/ans-2020-2111Sharp Liouville TheoremsVillegas Salvador0Departamento de Análisis Matemático, Universidad de Granada, 18071Granada, SpainConsider the equation div(φ2∇σ)=0{\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in ℝN{\mathbb{R}^{N}}, where φ>0{\varphi>0}. Berestycki, Caffarelli and Nirenberg proved in [H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 69–94] that if there exists C>0{C>0} such that ∫BR(φσ)2≤CR2\int_{B_{R}}(\varphi\sigma)^{2}\leq CR^{2} for every R≥1{R\geq 1}, then σ is necessarily constant. In this paper, we provide necessary and sufficient conditions on 0<Ψ∈C([1,∞)){0<\Psi\in C([1,\infty))} for which this result remains true if we replace CR2{CR^{2}} by Ψ(R){\Psi(R)} in any dimension N. In the case of the convexity of Ψ for large R>1{R>1} and Ψ′>0{\Psi^{\prime}>0}, this condition is equivalent to ∫1∞1Ψ′=∞.\int_{1}^{\infty}\frac{1}{\Psi^{\prime}}=\infty.https://doi.org/10.1515/ans-2020-2111liouville theoremsde giorgi’s conjecture35b08 35b35 35b53 35j91 |
| spellingShingle | Villegas Salvador Sharp Liouville Theorems Advanced Nonlinear Studies liouville theorems de giorgi’s conjecture 35b08 35b35 35b53 35j91 |
| title | Sharp Liouville Theorems |
| title_full | Sharp Liouville Theorems |
| title_fullStr | Sharp Liouville Theorems |
| title_full_unstemmed | Sharp Liouville Theorems |
| title_short | Sharp Liouville Theorems |
| title_sort | sharp liouville theorems |
| topic | liouville theorems de giorgi’s conjecture 35b08 35b35 35b53 35j91 |
| url | https://doi.org/10.1515/ans-2020-2111 |
| work_keys_str_mv | AT villegassalvador sharpliouvilletheorems |