Nonlinear Potential Theory and Harmonic Analysis

这个项目的主要目标是发展新的方法的非线性势理论，调和分析和偏微分方程适用于广泛的一类拟线性和完全非线性椭圆方程。解的存在性理解重整化或粘性的意义上，尖锐的积分和逐点估计的解决方案，和可移动的奇点将进行研究。这个项目的一个相关部分是关注非线性积分不等式的沃尔夫的潜力，以及不确定的重量出现在形式的有界性问题的线性偏微分算子的分布系数。拟议的研究将建立新的应用程序的各种问题的非线性科学，谐波分析，算子理论，数学物理，广泛应用于传热，流体流动，量子理论，控制理论和随机过程的研究。更广泛的影响将导致在应用数学，物理学，环境和技术科学，这可能是有用的教学和培训研究生，年轻的数学家和应用科学家的基本非线性现象的理解的进一步进步。

AbstractThe main goal of this project is to develop new methods of nonlinear potential theory, harmonic analysis, and partial differential equations applicable to a wide class of quasilinear and fully nonlinear elliptic equations. Existence of solutions understood in the renormalized or viscosity sense, sharp integral and pointwise estimates of solutions, and removable singularities will be studied. A related part of this project is concerned with nonlinear integral inequalities for Wolff's potentials, as well as inequalities with indefinite weights which appear in the form boundedness problem for linear partial differential operators with distributional coefficients. The proposed research will establish new applications to various problems of nonlinear science, harmonic analysis, operator theory, and mathematical physics which are widely used in the studies of heat transfer, fluid flow, quantum theory, control theory, and stochastic processes. The broader impact will result in further advances in understanding of basic nonlinear phenomena in applied mathematics, physics, environmental and technological sciences which may be useful in teaching and training of graduate students, younger mathematicians and applied scientists.