A sharp trace Adams' inequality in $\mathbb{R}^{4}$ and Existence of the extremals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910055359578112 |
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| author | Chen, Lu Lu, Guozhen Zhu, Maochun |
| author_facet | Chen, Lu Lu, Guozhen Zhu, Maochun |
| contents | Let $Ω\subseteq \mathbb{R}^{4}$ be a bounded domain with smooth boundary $\partialΩ$. In this paper, we establish the following sharp form of the trace Adams' inequality in $W^{2,2}(Ω)$ with
zero mean value and zero Neumann boundary condition: \begin{equation*} S(α)=\underset{\int_Ωudx=0,\frac{\partial u}{\partialν}|_{\partialΩ}=0,\VertΔu\Vert_{2}\leq{1}}{\underset {u\in{W^{2,2}(Ω)\setminus\{0\}}}{\sup}}\int_{\partial Ω} e^{αu^{2}}dσ<\infty \end{equation*} holds if and only if $ α\leq12π^2$.
Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of bi-harmonic equations on the half space $\mathbb{R}^4_{+}$. With this classification result, we can show that $S({12π^2})$ is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams-Onofri type inequality in general four dimensional bounded domains with smooth boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16347 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A sharp trace Adams' inequality in $\mathbb{R}^{4}$ and Existence of the extremals Chen, Lu Lu, Guozhen Zhu, Maochun Analysis of PDEs 35J60, 35B33, 46E30 Let $Ω\subseteq \mathbb{R}^{4}$ be a bounded domain with smooth boundary $\partialΩ$. In this paper, we establish the following sharp form of the trace Adams' inequality in $W^{2,2}(Ω)$ with zero mean value and zero Neumann boundary condition: \begin{equation*} S(α)=\underset{\int_Ωudx=0,\frac{\partial u}{\partialν}|_{\partialΩ}=0,\VertΔu\Vert_{2}\leq{1}}{\underset {u\in{W^{2,2}(Ω)\setminus\{0\}}}{\sup}}\int_{\partial Ω} e^{αu^{2}}dσ<\infty \end{equation*} holds if and only if $ α\leq12π^2$. Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of bi-harmonic equations on the half space $\mathbb{R}^4_{+}$. With this classification result, we can show that $S({12π^2})$ is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams-Onofri type inequality in general four dimensional bounded domains with smooth boundary. |
| title | A sharp trace Adams' inequality in $\mathbb{R}^{4}$ and Existence of the extremals |
| topic | Analysis of PDEs 35J60, 35B33, 46E30 |
| url | https://arxiv.org/abs/2308.16347 |