Wolff势及其在p-Laplacian型方程中的应用

现代调和分析的发展与偏微分方程解正则性质的研究密切相关，这为调和分析领域带来许多富有意义且亟待解决的关键问题.本项目将利用内插定理、Whitney分解、覆盖技术等调和分析工具，重点探究一般区域上Wolff势在不同函数空间中的有界性质，明晰其界与区域的几何性质之间的依赖关系.进一步在非光滑区域上建立具有不连续系数的p-Laplacian型稳态Stokes方程组、以及分数p-Laplacian椭圆方程这两类模型边值问题的解及相关梯度的整体点态与振荡型Wolff势估计.由此，非线性Wolff势成为重要纽带，将解的正则性问题转为其在给定函数空间中的有界性研究，统一构建这两类边值问题的解在各种函数空间中的整体正则性质.本项目的研究结果不仅丰富和发展了非线性势理论，还为建立非线性偏微分方程解的Calderón-Zygmund型估计提供了一种有效的分析方法，促进了调和分析与偏微分方程两个领域的交叉发展.

The development of modern harmonic analysis has a close connection with the studies on regularity theory of solutions to partial differential equations, which produces many meaningful and important issues needed to be solved in this field. By involving the interpolation theorem, the Whitney decomposition, covering techniques and other analytic tools, this project will focus on boundedness of the Wolff potential in various function spaces on general domains, and clarify the dependence between the boundedness and the geometric structure of domains. Furthermore, we shall establish global pointwise and oscillatory potential estimates to the boundary value problems of the p-Laplacian type stationary Stokes system with discontinuous coefficients, as well as nonlocal elliptic equations with the fractional p-Laplacian on non-smooth domains. In this project, the Wolff potential will build an important bridge to link the regularity of solutions and its boundedness in the given function spaces. This will help us to use a unified approach to obtain global regularity of solutions to two kinds of boundary value problems in different function spaces. The expected results will not only enrich and develop the nonlinear potential theory, but also provide an effective approach to establish the Calderón-Zygmund type estimates of solutions to nonlinear partial differential equations. This project will promote the cross-development of harmonic analysis and partial differential equations.