The Dirichlet Problem For the Logarithmic p-Laplacian
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911037624680448 |
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| author | Dyda, Bartłomiej Jarohs, Sven Sk, Firoj |
| author_facet | Dyda, Bartłomiej Jarohs, Sven Sk, Firoj |
| contents | We introduce and study the logarithmic $p$-Laplacian $L_{Δ_p}$, which emerges from the formal derivative of the fractional $p$-Laplacian $(-Δ_p)^s$ at $s=0$. This operator is nonlocal, has logarithmic order, and is the nonlinear version of the newly developed logarithmic Laplacian operator. We present a variational framework to study the Dirichlet problems involving the $L_{Δ_p}$ in bounded domains. This allows us to investigate the connection between the first Dirichlet eigenvalue and eigenfunction of the fractional $p$-Laplacian and the logarithmic $p$-Laplacian. As a consequence, we deduce a Faber-Krahn inequality for the first Dirichlet eigenvalue of $L_{Δ_p}$. We discuss maximum and comparison principles for $L_{Δ_p}$ in bounded domains and demonstrate that the validity of these depends on the sign of the first Dirichlet eigenvalue of $L_{Δ_p}$. In addition, we prove that the first Dirichlet eigenfunction of $L_{Δ_p}$ is bounded. Furthermore, we establish a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic $p$-Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dirichlet Problem For the Logarithmic p-Laplacian Dyda, Bartłomiej Jarohs, Sven Sk, Firoj Analysis of PDEs We introduce and study the logarithmic $p$-Laplacian $L_{Δ_p}$, which emerges from the formal derivative of the fractional $p$-Laplacian $(-Δ_p)^s$ at $s=0$. This operator is nonlocal, has logarithmic order, and is the nonlinear version of the newly developed logarithmic Laplacian operator. We present a variational framework to study the Dirichlet problems involving the $L_{Δ_p}$ in bounded domains. This allows us to investigate the connection between the first Dirichlet eigenvalue and eigenfunction of the fractional $p$-Laplacian and the logarithmic $p$-Laplacian. As a consequence, we deduce a Faber-Krahn inequality for the first Dirichlet eigenvalue of $L_{Δ_p}$. We discuss maximum and comparison principles for $L_{Δ_p}$ in bounded domains and demonstrate that the validity of these depends on the sign of the first Dirichlet eigenvalue of $L_{Δ_p}$. In addition, we prove that the first Dirichlet eigenfunction of $L_{Δ_p}$ is bounded. Furthermore, we establish a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic $p$-Laplacian. |
| title | The Dirichlet Problem For the Logarithmic p-Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.11181 |