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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.15486 |
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| _version_ | 1866913998315716608 |
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| author | Piscitelli, Gianpaolo |
| author_facet | Piscitelli, Gianpaolo |
| contents | In this paper, we study the optimal constant in the nonlocal nonlinear Poincaré-Wirtinger inequality in $(a,b)\subset\mathbb R$: \begin{equation*} λ_α(p,q,r){\left(\int_{a}^{b}|u|^{q}dx\right)^\frac pq}\le{\int_{a}^{b}|u'|^{p}dx+α\left|\int_{a}^{b}|u|^{r-2}u\, dx\right|^{\frac p{r-1}}}, \end{equation*}where $α\in\mathbb R$, $p,q,r >1$ such that $\frac{2p}{p+2}\le q\le p$ and $\frac q2+1\le r \le q+\frac q p$. This problem admits a variational characterization in the nonlocal setting, as the associated Euler-Lagrange equation involves an integral term depending on the unknown function over the entire interval of definition.
We prove the existence of a critical value $α_C=α_C (p,q,r)$ such that the minimizers are even and have constant sign for $α\leα_{C}$, while they are odd for $α\geq α_{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15486 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetry results for a nonlocal nonlinear Poincaré-Wirtinger inequality Piscitelli, Gianpaolo Analysis of PDEs In this paper, we study the optimal constant in the nonlocal nonlinear Poincaré-Wirtinger inequality in $(a,b)\subset\mathbb R$: \begin{equation*} λ_α(p,q,r){\left(\int_{a}^{b}|u|^{q}dx\right)^\frac pq}\le{\int_{a}^{b}|u'|^{p}dx+α\left|\int_{a}^{b}|u|^{r-2}u\, dx\right|^{\frac p{r-1}}}, \end{equation*}where $α\in\mathbb R$, $p,q,r >1$ such that $\frac{2p}{p+2}\le q\le p$ and $\frac q2+1\le r \le q+\frac q p$. This problem admits a variational characterization in the nonlocal setting, as the associated Euler-Lagrange equation involves an integral term depending on the unknown function over the entire interval of definition. We prove the existence of a critical value $α_C=α_C (p,q,r)$ such that the minimizers are even and have constant sign for $α\leα_{C}$, while they are odd for $α\geq α_{C}$. |
| title | Symmetry results for a nonlocal nonlinear Poincaré-Wirtinger inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.15486 |