We discuss Poincare type inequalities for functions with zero mean values on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. For some basic domains (rectangles, quadrilaterals, and right triangles) exact constants in these inequalities has been found in Nazarov and Repin (ArXiv Ser Math Anal, 2012, arXiv:1211.2224). We shortly discuss two examples, which show that the estimates can be helpful for quantitative analysis of PDEs. In the first example, we deduce estimates of modeling errors generated by simplification (coarsening) of a boundary value problem. The second example presents a new form of the functional type a posteriori estimate that provides fully guaranteed and computable bounds of approximation errors. Constants in Poincare type inequalities enter these estimates.
展开▼
机译：我们讨论在Lipschitz域的整个边界或边界的可测部分上均值为零的函数的Poincare型不等式。对于某些基本域（矩形，四边形和直角三角形），已在Nazarov和Repin中找到了这些不等式的精确常数（ArXiv Ser Math Anal，2012，arXiv：1211.2224）。我们不久将讨论两个示例，这些示例表明估计值可有助于对PDE进行定量分析。在第一个示例中，我们推导了通过简化（粗化）边值问题而生成的建模误差的估计。第二个示例介绍了一种新形式的函数类型后验估计，该函数提供了完全有保证且可计算的近似误差范围。 Poincare类型不等式的常数输入这些估计值。
展开▼
