论文标题
$ p $ -laplacian具有罗宾边界条件的解决方案的行为为$ p $ $ 1 $
Behaviour of solutions to $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$
论文作者
论文摘要
我们研究了以下不均匀的罗宾边界价值问题的解决方案的渐近行为，为$ p \至1^{+} $：\ begin {equination} \ label {pbabstract} \ tag {p} \ left \ {\ begin {array} {ll} \ displaystyle-Δ_pu_p = f＆\ text {in}ω， \ displaystyle | \ nabla u_p |^{p-2} \ nabla u_p \ cdot ccdotν+λ| u_p |^{p-2} u_p = g＆\ g＆\ text {on} \partialΩ， \ end {array} \ right。 \ end {equation}其中$ω$是$ \ mathbb r^{n} $中有足够光滑边界的有界域，$ν$是其单位向外的正常向量，$δ_pv $是$ p $ -p $ -laplacian操作员，$ p> 1 $。 l^{n，\ infty}（ω）$（表示marcinkiewicz space）和$λ，g $的数据$ f \ in $ \partialΩ$，用$λ\ ge0 $定义。我们发现$ p $的家族的阈值 - 解决方案转到了这个家庭炸毁的0及以上。作为第二次利息，我们通过将$ p \ to $ p \ to 1^+$在\ eqref {pbabstract}中正式产生的$ 1 $ -laplacian问题处理。
We study the asymptotic behaviour, as $p\to 1^{+}$, of the solutions of the following inhomogeneous Robin boundary value problem: \begin{equation} \label{pbabstract} \tag{P} \left\{\begin{array}{ll} \displaystyle -Δ_p u_p = f & \text{in }Ω, \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot ν+λ|u_p|^{p-2}u_p = g& \text{on } \partialΩ, \end{array}\right. \end{equation} where $Ω$ is a bounded domain in $\mathbb R^{N}$ with sufficiently smooth boundary, $ν$ is its unit outward normal vector and $Δ_p v$ is the $p$-Laplacian operator with $p>1$. The data $f\in L^{N,\infty}(Ω)$ (which denotes the Marcinkiewicz space) and $λ,g$ are bounded functions defined on $\partialΩ$ with $λ\ge0$. We find the threshold below which the family of $p$--solutions goes to 0 and above which this family blows up. As a second interest we deal with the $1$-Laplacian problem formally arising by taking $p\to 1^+$ in \eqref{pbabstract}.
