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，这些方程在非线性分析中作为模型问题，在应用中也很常见。Verbitsky将结合多种分析技术，包括线性和非线性算子的离散（小波型）分解、具有迭代权重的加权范数不等式和对嵌入常数的良好控制、与格林核相关的准度量、精炼的3g不等式和非线性势理论。一个预期的新发展是任意的非负系数在低阶项和数据可以处理没有任何先验的规则性假设。这种方法的本质是引入与非线性问题内在关联的新函数空间，识别它们的前导和建立基本的嵌入定理。精确的逐点估计的解决方案和他们的梯度到边界可能是非常有价值的应用问题。提出的研究为以前广泛用于线性分析的现代谐波分析方法和势理论开辟了新的应用。非线性技术为具有奇异系数和数据的微分和积分方程提供了更精确的数学模型和数值估计。它们描述了许多具有非线性源的环境和技术现象，并广泛应用于传热、流体流动、控制理论和随机过程的研究中。

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.