论文标题
平均曲率在混合耐硬-sobolev痕迹不等式中的作用
The role of the mean curvature in a mixed Hardy-Sobolev trace inequality
论文作者
论文摘要
Let $Ω$ be a smooth bounded domain of $\mathbb{R}^{N+1}$ of boundary $\partial Ω= Γ_1 \cup Γ_2$ and such that $\partial Ω\cap Γ_2$ is a neighborhood of $0$, $h \in \mathcal{C}^0(\partial Ω\cap Γ_2) $ and $s \在[0,1）$中。我们提议研究存在肯定解决方案的存在，以解决与混合边界条件\ begin {align} \ begin {case}Δu= 0＆\ qquad \ qquad \ textrm { ν} = h（x）u + \ frac {u^{q（s）-1}}} {d（x）^{s}}＆\ qquad \ qquad \ textrm {on}γ_2，\ end \ end {cases} \ end {cases} \ end end end eNd {align} wher Hardy-Sobolev跟踪指数和$ν$是$ \ partialω$的外部单位正常。特别是，当$ n \ geq 3 $且平均曲率以$ 0 $为$ 0 $的潜在$ H $以下时，我们证明了最小化器的存在。
Let $Ω$ be a smooth bounded domain of $\mathbb{R}^{N+1}$ of boundary $\partial Ω= Γ_1 \cup Γ_2$ and such that $\partial Ω\cap Γ_2$ is a neighborhood of $0$, $h \in \mathcal{C}^0(\partial Ω\cap Γ_2) $ and $s \in [0,1)$. We propose to study existence of positive solutions to the following Hardy-Sobolev trace problem with mixed boundaries conditions \begin{align} \begin{cases} Δu= 0& \qquad \textrm{ in } Ω\\\ u=0 & \qquad \textrm{ on } Γ_1 \\\ \frac{\partial u}{\partial ν}=h(x) u + \frac{u^{q(s)-1}}{d(x)^{s}} & \qquad \textrm{ on } Γ_2, \end{cases} \end{align} where $q(s):=\frac{2(N-s)}{N-1}$ is the critical Hardy-Sobolev trace exponent and $ν$ is the outer unit normal of $\partial Ω$. In particular, we prove existence of minimizers when $N \geq 3$ and the mean curvature is sufficiently below the potential $h$ at $0$.
