论文标题
部分可观测时空混沌系统的无模型预测
A comparison estimate for singular $p$-Laplace equations and its consequences
论文作者
论文摘要
比较估计值是研究准线性问题可能是椭圆形和抛物线方程的规则性问题的重要技术手段。在某些度量基准问题上，这种工具在Mingione，Duzaar-Mingione和Kuusi-Mingione等的许多论文中都不可或缺。但是，在强烈的奇异案例$ 1 <p \ leq \ frac {3n-2} {2n-1} $中，$ p $ - laplace类型椭圆方程与测量数据仍然不可用，其中$ n \ geq 2 $是环境空间的尺寸。在这项工作中，通过证明比较估计范围略大$ 1 <p <3/2 $，将在这项工作中完全解决此问题。应用程序包括庞贝拉型的“尖顶”类型不平等，解决方案及其衍生物的界限分别通过沃尔夫（Wolff）和里兹（Riesz）的潜力。还获得了一些全局式和加权估计值的有界域，这使我们能够处理梯度可能具有sublinear的化学型riccati型方程。
Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum problems to obtain pointwise bounds for solutions and their full or fractional derivatives in terms of appropriate linear or nonlinear potentials. However, a comparison estimate for $p$-Laplace type elliptic equations with measure data is still unavailable in the strongly singular case $1< p\leq \frac{3n-2}{2n-1}$, where $n\geq 2$ is the dimension of the ambient space. This issue will be completely resolved in this work by proving a comparison estimate in a slightly larger range $1<p<3/2$. Applications include a `sublinear' Poincaré type inequality, pointwise bounds for solutions and their derivatives by Wolff's and Riesz's potentials, respectively. Some global pointwise and weighted estimates are also obtained for bounded domains, which enable us to treat a quasilinear Riccati type equation with possibly sublinear growth in the gradient.
