Nonlinear Potential Theory and Harmonic Analysis

本项目的主要目标是发展适用于一大类拟线性和完全非线性椭圆型方程的非线性位势理论、调和分析和偏微分方程组的新方法。我们将研究在重整化或粘性意义下了解的解的存在性，解的精确积分和逐点估计，以及可移除奇点。这个项目的一个相关部分是关于Wolff势的非线性积分不等式，以及在具有分配系数的线性偏微分算子的形式有界性问题中出现的具有不定权的不等式。这项研究将为广泛应用于传热学、流体流动、量子理论、控制论和随机过程研究中的非线性科学、调和分析、算子论和数学物理的各种问题建立新的应用。更广泛的影响将进一步促进对应用数学、物理、环境和技术科学中基本非线性现象的理解，这可能有助于研究生、年轻数学家和应用科学家的教学和培训。

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.