论文标题
尖锐的多维赫尔米特 - hadamard不平等
A sharp multidimensional Hermite-Hadamard inequality
论文作者
论文摘要
令$ω\ subset \ mathbb {r}^d $，$ d \ geq 2 $，为有界的凸域，$ f \ colonω\ to \ mathbb {r} $是一个非阴性的subharmonic函数。在本文中，我们证明了不等式\ [\ frac {1} {|ω|} \int_Ωf（x）\，dx \ leq \ leq \ frac {d} {| \partialΩ|} \ int _ {\ int _ {\partialΩ} \]同等地，结果可以说为圣经扭转函数梯度的结合。具体而言，如果$ω\ subset \ mathbb {r}^d $是一个有界的凸域，而$ u $是$-ΔU= 1 $的解决方案，则具有均匀的dirichlet边界条件，则\ [\ | \ | \ nabla u \ \ | _ {l^\^\ f^\ iffty（Ω） d \ frac {|ω|} {| \partialΩ|} \，。此外，这两个不等式在某种意义上都是敏锐的，如果常数$ d $被较小的东西取代，则存在不等式失败的凸形域。这是由于最近的最佳常数从上面界定的$ d^{3/2} $，这改善了这一结果。
Let $Ω\subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon Ω\to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|Ω|}\int_Ωf(x)\,dx \leq \frac{d}{|\partialΩ|}\int_{\partialΩ} f(x)\,dσ(x)\,. \] Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $Ω\subset \mathbb{R}^d$ is a bounded convex domain and $u$ is the solution of $-Δu =1$ with homogeneous Dirichlet boundary conditions, then \[ \|\nabla u\|_{L^\infty(Ω)} < d\frac{|Ω|}{|\partialΩ|}\,. \] Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al.
