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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.03168 |
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| _version_ | 1866915092704002048 |
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| author | Guo, Yanyan Luo, Huxiao Ruf, Bernhard |
| author_facet | Guo, Yanyan Luo, Huxiao Ruf, Bernhard |
| contents | We prove the following Limiting Bliss inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{β\left(\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,β), \ \hbox{ for } β\le 1 \end{equation} The inequalities are optimal with respect to $β\le 1$; there is compactness for $β<1$, and along the infinitesimal Moser sequence for $β= 1$.
Moreover, we show that the improved inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{\left(\log\frac{e}{s}+γ\log\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,γ) \end{equation} hold for $γ\leq1$, and for $γ=1$ the inequalities are critical with loss of compactness. The inequalities are optimal: no further improvement in the coefficient of the exponent is possible.
The second result extends the result in [J. M. do Ó, B. Ruf and P. Ubilla, A critical Moser type inequality with loss of compactness due to infinitesimal shocks, Calc. Var. Partial Differential Equations 62 (2023)] from $N=2$ to general dimensions $N\geq2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_03168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On inequalities of Bliss-Moser type with loss of compactness in $\mathbb{R}^N$ Guo, Yanyan Luo, Huxiao Ruf, Bernhard Analysis of PDEs We prove the following Limiting Bliss inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{β\left(\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,β), \ \hbox{ for } β\le 1 \end{equation} The inequalities are optimal with respect to $β\le 1$; there is compactness for $β<1$, and along the infinitesimal Moser sequence for $β= 1$. Moreover, we show that the improved inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{\left(\log\frac{e}{s}+γ\log\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,γ) \end{equation} hold for $γ\leq1$, and for $γ=1$ the inequalities are critical with loss of compactness. The inequalities are optimal: no further improvement in the coefficient of the exponent is possible. The second result extends the result in [J. M. do Ó, B. Ruf and P. Ubilla, A critical Moser type inequality with loss of compactness due to infinitesimal shocks, Calc. Var. Partial Differential Equations 62 (2023)] from $N=2$ to general dimensions $N\geq2$. |
| title | On inequalities of Bliss-Moser type with loss of compactness in $\mathbb{R}^N$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.03168 |