The principal eigenvalue of a mixed local and nonlocal operator with drift
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| Format: | Preprint |
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2024
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| author | Cowan, Craig Smaily, Mohammad El Feulefack, Pierre Aime |
| author_facet | Cowan, Craig Smaily, Mohammad El Feulefack, Pierre Aime |
| contents | We study the eigenvalue problem involving the mixed local-nonlocal operator $ L:= -Δ+(-Δ)^{s}+q\cdot\nabla$~ in a bounded domain $Ω\subset\R^N,$ where a Dirichlet condition is posed on $\R^N\setminusΩ.$ The field $q$ stands for a drift or advection in the medium. We prove the existence of a principal eigenvalue and a principal eigenfunction for $s\in (0,1/2]$. Moreover, we prove $C^{2,α}$ regularity, up to the boundary, of the solution to the problem $Lu=f$ when coupled with a Dirichlet condition and $0<s<1/2$. To prove the regularity and the existence of a principal eigenvalue, we use a continuation argument, Krein-Rutman theorem as well as a Hopf Lemma and a maximum principle for the operator $L,$ which we derive in this paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_19577 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The principal eigenvalue of a mixed local and nonlocal operator with drift Cowan, Craig Smaily, Mohammad El Feulefack, Pierre Aime Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35A09, 35B50, 35B65, 35R11, 35J67, 47A75 We study the eigenvalue problem involving the mixed local-nonlocal operator $ L:= -Δ+(-Δ)^{s}+q\cdot\nabla$~ in a bounded domain $Ω\subset\R^N,$ where a Dirichlet condition is posed on $\R^N\setminusΩ.$ The field $q$ stands for a drift or advection in the medium. We prove the existence of a principal eigenvalue and a principal eigenfunction for $s\in (0,1/2]$. Moreover, we prove $C^{2,α}$ regularity, up to the boundary, of the solution to the problem $Lu=f$ when coupled with a Dirichlet condition and $0<s<1/2$. To prove the regularity and the existence of a principal eigenvalue, we use a continuation argument, Krein-Rutman theorem as well as a Hopf Lemma and a maximum principle for the operator $L,$ which we derive in this paper. |
| title | The principal eigenvalue of a mixed local and nonlocal operator with drift |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35A09, 35B50, 35B65, 35R11, 35J67, 47A75 |
| url | https://arxiv.org/abs/2406.19577 |