Superlinear Equations and Weighted Norm Inequalities

Verbistky将系统地研究一类重要的超线性积分和偏微分方程 这是非线性分析中的模型问题， 在应用中常见。Verbitsky将结合联合收割机的分析技术，包括离散（小波型）分解的线性和非线性算子，加权范数不等式与迭代权重 以及对嵌入常数、相关准度量的良好控制 利用绿色核、精化3G-不等式和非线性位势 理论一个预期的新发展是， 低阶项和数据处的任意非负系数 可以在没有任何先验规则性假设的情况下处理。基本 这种方法是一个新的函数空间的介绍 本质上与非线性问题相关， 并建立了基本的嵌入定理。 解及其梯度的精确逐点估计 在应用问题中是非常有价值的。 拟议的研究开发了新的应用现代方法的谐波分析和潜在的理论，以前被广泛用于 线性分析非线性技术导致更精确的数学模型和数值估计的微分和积分方程的奇异系数， 数据它们描述了许多环境和技术现象 具有非线性源，并广泛用于热的研究 传输，流体流动，控制理论和随机过程。

Verbistky will investigate systematically an important class of superlinear integral and partial differential equations which serve as model problems in nonlinear analysis and also are common in applications. Verbitsky will combine a number of analytic techniques including discrete (wavelet-type) decompositions of linear and nonlinear operators, weighted norm inequalities with iterated weights and good control of the imbedding constants, quasi-metrics associated with Green's kernels, refined 3G-inequalities and nonlinear potential theory. An expected new development is that arbitrary nonnegative coefficients at the lower order terms and data can be handled without any a priori regularity assumptions. Essential to this approach is an introduction of new function spaces intrinsically associated with nonlinear problems, identification of their preduals and establishing basic imbedding theorems. Exact pointwise estimates of solutions and their gradients up to the boundary could be extremely valuable in applied problems. The proposed research develops new applications for modern methods of harmonic analysis and potential theory which previously were used extensively in linear analysis. Nonlinear techniques lead to more accurate mathematical models and numerical estimates for differential and integral equations with singular coefficients and data. They describe many environmental and technological phenomena with nonlinear sources, and are widely used in the studies of heat transfer, fluid flow, control theory, and stochastic processes.