非标准拟线性椭圆和抛物型方程的Calderón-Zygmund估计

Regularity theory is one of the hottest and very challenging issues in the field of modern partial differential equations, and it is also the foundation for the study of existence, uniqueness and asymptotic behavior of solutions. This project is mainly devoted to the study of regularity for some kind of non-standard quasilinear elliptic and parabolic partial differential equations, which come from elastic mechanics, materials science and electro-rheological fluid dynamics. The main contents are the following: (1) establish the global Calderón-Zygmund estimates in quasiconvex domains and composite Reifenberg flat domains for the non-uniformly elliptic and parabolic (p,q)-Laplacian type equations; (2) study the Calderón-Zygmund estimates and nonlinear potential estimates of renormalized solutions of the quasilinear elliptic and parabolic equations with Borel measure datum right-hand side in fractional Sobolev spaces; (3) discuss the optimal conditions imposed on the discontinuous coefficients and the boundary of non-smooth domains and prove the Calderón-Zygmund estimates for the irregular obstacle problems of these elliptic and parabolic equations. The project not only can provide some new ideas for the study of regularity theory of the non-standard quasilinear elliptic and parabolic equations, but also can provide some necessary theoretical basis for the applications of partial differential equations in other disciplines.

正则性理论是近代偏微分方程领域中的热点和极具挑战的问题之一，亦是研究解的存在唯一性和渐近行为等的基础. 本项目主要研究几类非标准拟线性椭圆和抛物型偏微分方程的正则性. 这些方程来源于弹性力学、材料科学和电流变流体动力学. 主要内容包括：（1）建立非一致椭圆和抛物型(p,q)-Laplacian方程在拟凸区域和复合的Reifenberg平坦区域的全局Calderón-Zygmund估计；（2）研究右端项为Borel测度数据的拟线性椭圆和抛物型方程的重整化解在分数阶Sobolev空间的Calderón-Zygmund估计和非线性位势估计；（3）讨论不连续系数与非光滑区域边界的最优条件，证明这些非标准椭圆和抛物型方程非正则障碍问题的Calderón-Zygmund估计. 本项目将为非标准拟线性椭圆和抛物型方程的正则性研究提供新的思路，亦可为偏微分方程在其他学科中的应用提供必要的理论依据.

本项目主要研究几类来源于材料科学、弹性力学和流体动力学的非标准拟线性椭圆和抛物型偏微分方程的适定性与正则性。主要内容包括：(1) 在重整化解、熵解和弱解的框架下证明这些方程解的适定性及各种解之间的内在联系；（2）建立非一致椭圆和抛物型方程在加权的Lebesgue空间和Morrey空间等不同函数空间的Calderón-Zygmund估计；（3）在Borel测度数据下研究这些椭圆和抛物型方程的Calderón-Zygmund估计和非线性位势估计。上述研究内容不仅可以丰富非标准拟线性椭圆和抛物型方程的正则性理论，亦可为偏微分方程在其他学科中的应用提供必要的理论支持。