Improved Friedrichs inequality for a subhomogeneous embedding
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866917336373526528 |
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| author | Bobkov, Vladimir Kolonitskii, Sergey |
| author_facet | Bobkov, Vladimir Kolonitskii, Sergey |
| contents | For a smooth bounded domain $Ω$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - λ_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(Ω)$, where $λ_1$ is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation $-Δ_p u = λ_1 \|u\|_q^{p-q} |u|^{q-2} u + f$ coupled with zero Dirichlet boundary conditions possesses a weak solution provided $f$ is orthogonal to the minimizer of $λ_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_14111 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Improved Friedrichs inequality for a subhomogeneous embedding Bobkov, Vladimir Kolonitskii, Sergey Analysis of PDEs Classical Analysis and ODEs Functional Analysis Spectral Theory 35J92, 35P30, 35A23, 47J10 For a smooth bounded domain $Ω$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - λ_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(Ω)$, where $λ_1$ is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation $-Δ_p u = λ_1 \|u\|_q^{p-q} |u|^{q-2} u + f$ coupled with zero Dirichlet boundary conditions possesses a weak solution provided $f$ is orthogonal to the minimizer of $λ_1$. |
| title | Improved Friedrichs inequality for a subhomogeneous embedding |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis Spectral Theory 35J92, 35P30, 35A23, 47J10 |
| url | https://arxiv.org/abs/2210.14111 |