一类奇异椭圆方程解的边界估计及其存在性

Researches of singular elliptic equations are in the stage of rapid development have become an important area of modern researches to partial differential equations and many scholars P.H.Rabinowitz，H.Brezis，Yiming Long, Fanghua Lin,Juncheng Wei pay attentions to it. However, in recent years,urgent need to research another singular elliptic equations along with the indispensable application MEMS（micro electron mechanical system）for the developments of defense precision instruments and microelectronic components.The project intends to study the boundary estimate and existence of solutions for a class of singular elliptic equations having a very important theoretical significance and application value. Our main concerns are the studying the regulation estimate of weak solutions and H-loc solutions using measure estimated singular set method;developing the general theory of prior estimate;studying the relation of coefficient and regulation using characteristic function comparison;studying the existence of weak solutions with high singular using new sub-super method.the results of the initial establishment of the abstract and the development of related methods. A definite breakthrough in theory and methods may be made for from regulation point of view in-depth reveal the essential relations on singular elliptic equations and classic elliptic equations and contribute to the further development of the theory of elliptic equations.

奇异椭圆方程的研究正处在迅速发展的阶段，已成为现代微分方程研究的重要领域，引起许多学者如P.H.Rabinowitz、H.Brezis、龙以明、林芳华、魏军成等的极大关注。尤其，近年来随着MEMS（micro electron mechanical system）等理论在国防精密仪器研制和当今微电子元件研发中不可或缺的应用，迫切需要考虑一类奇异椭圆方程。本项目拟研究此类奇异椭圆方程解的边界估计及其存在性，具有重要的理论意义和应用价值。主要内容包括:采用奇异集测度估计方法研究弱解、H-loc解的近边正则性估计，发展奇异椭圆方程先验估计的一般理论方法；采用特征函数比较方法研究奇异项系数性态与解的正则性关系，得到解的最佳正则估计；采用新上下解方法研究高奇性椭圆方程弱解的存在性。项目的完成将为从正则性角度深入揭示奇异椭圆方程与经典椭圆方程的本质联系提供新的思路和方法，进一步丰富椭圆方程理论。

奇异椭圆方程的研究正处在迅速发展的阶段，已成为现代微分方程研究的重要领域。本项目研究的奇异椭圆方程解的边界估计及其存在性，具有重要的理论意义和应用价值。主要内容包括:奇异椭圆方程弱解的正则性估计，通过构造无奇异性的逼近方程，利用经典椭圆方程的上下方法，得到逼近方程的弱解序列；通过集值分析方法，得到了奇异椭圆方程弱解的最大模和梯度模的先验。采用奇异集测度估计方法研究弱解、H-loc解的近边正则性估计，借助于Moser迭代技巧给出奇异椭圆方程先验估计的一般理论方法得到高奇性椭圆方程弱解的存在性。首次运用了闭锥理论以及算子方法给出了分数阶奇异微分方程弱解的存在性及其充分条件。是对奇异微分方程研究领域的进一步探索和发展，也为本项目的后续工作打下了基础。