Nonlinear Sobolev Inequalities, Potential Theory and Harmonic Analysis

本数学研究项目发展了势理论和调和分析方法，这些方法适用于研究几何积分不等式、存在性和正则性问题，以及具有奇异系数和奇异数据的重要的全非线性和拟线性方程和不等式解的估计。对于积分算子、格林函数和分数薛定谔算子相关的条件规范，将得到Trudinger和Wang的Hessian Sobolev和Poincare不等式的重要推广，Wolff势解的估计，以及Neumann级数核的尖锐估计。所使用的工具将包括并矢模型，非齐次空间上的最大和奇异积分算子，非标准对偶性，以及具有不定权重的加权范数不等式，以及来自偏微分方程理论的各种技术。该数学研究项目将加深对应用科学和工程中由非线性定律和非局部力控制的复杂系统的基本非线性和分形现象的理解，在非牛顿流体、传热、弹性、控制理论的数学建模方面具有潜在的应用价值。所采用的技术将连接几个数学领域，如微分几何、数学物理和共形几何。由此产生的协同效应将丰富分析工具和可用技术的武器库，并将吸引更多的学生和年轻研究人员进入这个充满活力的数学领域及其应用。

This mathematics research project develops methods of potential theory and harmonic analysis that are applicable to the study of geometric integral inequalities, existence and regularity problems, and estimates of solutions for important classes of fully nonlinear and quasilinear equations and inequalities with singular coefficients and singular data. Significant extensions of Hessian Sobolev and Poincare inequalities of Trudinger and Wang, estimates of solutions in terms of Wolff's potentials, as well as sharp estimates of kernels of Neumann series will be obtained for integral operators, Green's functions and the conditional gauge associated with fractional Schrodinger operators. Among the tools employed will be dyadic models, maximal and singular integral operators on non-homogeneous spaces, non-standard duality, and weighted norm inequalities with indefinite weights, along with diverse techniques from the theory of partial differential equations.This mathematics research project will bring a deeper understanding of fundamental nonlinear and fractal phenomena for complex systems governed by nonlinear laws and nonlocal forces arising in applied sciences and engineering, with potential applications to mathematical modeling of non-Newtonian fluids, heat transfer, elasticity, control theory. The techniques to be employed will bridge several areas of mathematics, such as differential geometry, mathematical physics and conformal geometry. The resulting synergies will enrich the arsenal of analytical tools and available techniques, and will attract more students and young researchers to this dynamic area of mathematics and its applications.