On some Sobolev and Pólya-Szegö type inequalities with weights and applications
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| Format: | Preprint |
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2024
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| _version_ | 1866915850769924096 |
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| author | Giang, Trung Hieu Tri, Nguyen Minh Tuan, Dang Anh |
| author_facet | Giang, Trung Hieu Tri, Nguyen Minh Tuan, Dang Anh |
| contents | We are motivated by studying a boundary-value problem for a class of semilinear degenerate elliptic equations
\begin{align}\tag{P}\label{P}
\begin{cases}
- Δ_x u - |x|^{2α} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u) & \textrm{in } Ω,
u = 0 & \textrm{on } \partial Ω,
\end{cases}
\end{align}
where $x = (x_1, x_2) \in \mathbb{R}^2$, $Ω$ is a bounded smooth domain in $\mathbb{R}^3$, $(0,0,0) \in Ω$, and $α> 0$.
In this paper, we will study this problem by establishing embedding theorems for weighted Sobolev spaces. To this end, we need a new Pólya-Szegö type inequality, which can be obtained by studying an isoperimetric problem for the corresponding weighted area. Our results then extend the existing ones in \cite{nga, Luyen2} to the three-dimensional context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some Sobolev and Pólya-Szegö type inequalities with weights and applications Giang, Trung Hieu Tri, Nguyen Minh Tuan, Dang Anh Analysis of PDEs Functional Analysis 28A20, 46E30, 46E35, 51M16, 35J70 We are motivated by studying a boundary-value problem for a class of semilinear degenerate elliptic equations \begin{align}\tag{P}\label{P} \begin{cases} - Δ_x u - |x|^{2α} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u) & \textrm{in } Ω, u = 0 & \textrm{on } \partial Ω, \end{cases} \end{align} where $x = (x_1, x_2) \in \mathbb{R}^2$, $Ω$ is a bounded smooth domain in $\mathbb{R}^3$, $(0,0,0) \in Ω$, and $α> 0$. In this paper, we will study this problem by establishing embedding theorems for weighted Sobolev spaces. To this end, we need a new Pólya-Szegö type inequality, which can be obtained by studying an isoperimetric problem for the corresponding weighted area. Our results then extend the existing ones in \cite{nga, Luyen2} to the three-dimensional context. |
| title | On some Sobolev and Pólya-Szegö type inequalities with weights and applications |
| topic | Analysis of PDEs Functional Analysis 28A20, 46E30, 46E35, 51M16, 35J70 |
| url | https://arxiv.org/abs/2412.15490 |