Nonlinear potential theory, harmonic analysis and integral inequalities

该项目的主要目标是开发适用于具有奇异系数和数据的拟线性和完全非线性偏微分方程和不等式的势理论和调和分析方法。这包括非线性源项的椭圆和抛物问题的粘性解的存在性和正则性，积分算子的Neumann级数的核与拟度量核的尖锐估计，绿色函数和分数Schrodinger算子的条件规范，以及Trudinger和Wang的Hessian Sobolev和Poincare不等式的扩展。在这个项目中使用的工具将是二进模型，离散Carleson措施，沃尔夫的潜力，最大和奇异积分算子的非齐次空间，非标准的对偶，和加权范数不等式与不定权重，沿着现代偏微分方程理论的技术。这项研究集中在几个不同的数学分支之间的接口，将推进应用到当前感兴趣的微分方程领域。以及积分方程、非线性势理论、共形几何和数学物理。特别是，我们的研究的奇异性的解决方案将是重要的挑战性的问题，其中存在竞争的非线性微分算子和源项，奇异性的系数，数据和边界的域需要开发新的分析技术。 它们将导致对理论物理和应用科学中出现的基本非线性现象的进一步理解。

The main goal of this project is to develop methods of potential theory and harmonic analysis applicable to quasilinear and fully nonlinear partial differential equations and inequalities with singular coefficients and data. This includes existence and regularity of viscosity solutions to elliptic and parabolic problems with nonlinear source terms, sharp estimates of kernels of Neumann series of integral operators with quasimetric kernels, Green's functions and the conditional gauge associated with fractional Schrodinger operators, and extensions of Hessian Sobolev and Poincare inequalities of Trudinger and Wang. Among the tools employed in this project will be dyadic models, discrete Carleson measures, Wolff's potentials, maximal and singular integral operators on non-homogeneous spaces, non-standard duality, and weighted norm inequalities with indefinite weights, along with techniques of modern PDE theory.This research focusing on the interface between several diverse branches of mathematics will advance applications to areas of current interest in differential and integral equations, nonlinear potential theory, conformal geometry, and mathematical physics. In particular, our studies of singularities of solutions will be important to challenging problems where the presence of competing nonlinearities in differential operators and source terms, singularities of the coefficients, data and boundaries of the domains require the development of new analytic techniques. They will result in further understanding of fundamental nonlinear phenomena arising in theoretical physics and applied sciences.