几类具非标准增长的拟线性椭圆和抛物型方程的研究

This project is mainly devoted to the study of some class of quasilinear elliptic and parabolic equations with non-standard growth conditions. This kind of equations has rich physical significance and broad applications. Many practical problems, such as elastic mechanics, electro-rheological fluid dynamics and image processing, are all can be come down to the quasilinear elliptic and parabolic equations with non-standard growth conditions to solve. Although a large number of important achievements have been obtained for such equations, lots of physical phenomena cannot be given an accurate and reasonable explanation by the known mathematical theory of these equations. There are still many profound mathematical problems which are worth further investigation. We will study the wellposeness of solutions of these equations in the framework of weak solutions, renormalized solutions and entropy solutions and the inner link between them, prove the global gradient estimates of weak solutions for the elliptic and parabolic p(x)-Laplace equations and prove the asymptotic behavior of solutions when the exponent p(x) goes to 1 and infinity.

本项目主要致力于几类具有非标准增长性条件的拟线性椭圆和抛物型偏微分方程的研究.这类方程有着丰富的物理意义和广泛的应用背景,如弹性力学,电流变流体动力学和图像处理等实际问题都可以归结为具非标准增长性条件的拟线性椭圆与抛物型方程来描述.尽管对于这类方程的研究已取得许多重要成果,但许多物理现象尚不能从这些方程已有的数学理论中得到准确和合理的解释,仍然有许多深刻的数学问题值得进一步探讨.我们将在弱解、重整化解和熵解等框架下研究这些方程解的适定性及各种解之间的内在联系,证明椭圆与抛物型p(x)-Laplace方程弱解的全局梯度估计以及指数p(x)趋于1和趋于无穷时解的渐近行为.

本项目主要致力于几类来源于弹性力学、电流变流体动力学和图像处理的非标准拟线性椭圆和抛物型偏微分方程的研究。主要内容包括：(1) 在重整化解和熵解等框架下研究这些方程解的适定性及各种解之间的内在联系；（2）在Lebesgue可积数据下建立这些偏微分方程初边值问题弱解的全局梯度估计以及在加权的Lorentz空间和Lorent-Morrey空间的Calderón-Zygmund 估计；(3) 证明分数阶偏微分方程正解的存在性。 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。