论文标题
最佳的莫雷·索伯夫（Morrey-Sobolev）不等式
An optimal pointwise Morrey-Sobolev inequality
论文作者
论文摘要
令$ω$为$ \ mathbb {r}^{n}，$ $ $ n \ geq1。$对于每个$ p> n $的$ $ \ mathbb {r}^{n}。 v \ right \ vert _ {l ^{p}（ω）}，\ quad \ forall \，（x，v）\ in \ In \overlineΩ\ times w_ {0}％{0}％ ^{1，p}（p}（ω）。 \]我们表明，$ s_ {p} \在c_ {0}^{0,1-（n/p）}（n/p）}（\overlineΩ）$，以及$ s_ {p} $将距离转移到边界的距离函数，$ p \ rightArrow \ rightarrow \ rightarrow \ infty。 观点。
Let $Ω$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq1.$ For each $p>N$ we study the optimal function $s=s_{p}$ in the pointwise inequality \[ \left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert _{L^{p}(Ω)},\quad\forall\,(x,v)\in\overlineΩ\times W_{0}% ^{1,p}(Ω). \] We show that $s_{p}\in C_{0}^{0,1-(N/p)}(\overlineΩ)$ and that $s_{p}$ converges pointwise to the distance function to the boundary, as $p\rightarrow\infty.$ Moreover, we prove that if $Ω$ is convex, then $s_{p}$ is concave and has a unique maximum point.
