基于调和分析方法的Lipschitz区域上椭圆型方程组的L^p预解式估计及相关问题的研究

L^p resolvent estimates for elliptic systems on Lipschitz domains take an important role in spectral theory and analytic semigroup theory, and they are closely related to the solvability of elliptic boundary value problems. Based on our previous work that has initially established L^p resolvent estimates in Lipschitz domains for strongly elliptic systems of second order with constant coefficients and homogeneous Neumann boundary conditions, this project is to investigate the span of the index p in the previous L^p resolvent estimates, starting from the structures of elliptic systems with a spectral parameter by using several harmonic analysis techniques such as C-Z decomposition, interpolation theory, nontangential-maximal-function estimates and Good-λ inequalities. Furthermore, we extend these results to the case of strongly elliptic systems of second order with VMO coefficients and inhomogeneous Neumann boundary conditions on Lipschitz domains, by using the duality argument, superposition principle and perturbation theory. Meanwhile we consider the problem whether this kind of variable coefficient elliptic partial differential operators generates a bounded analytic semigroup in L^p. This project is concerned with L^p resolvent estimates by the techniques that are intimately related to harmonic analysis on Euclidean spaces and the structures of elliptic systems on non-smooth domains, which is valuable and important for us to combine harmonic analysis, PDE and functional analysis.

Lipschitz区域上椭圆型方程组的L^p预解式估计在谱论和解析半群理论的研究中占有重要地位，并与椭圆边值问题的可解性密切相关。本项目在我们前期工作已部分解决Lipschitz区域上带有齐次Neumann边值的二阶常系数强椭圆型方程组的L^p预解式估计问题的基础上，拟从含谱参数方程组的结构出发，通过采用C-Z分解、算子内插、非切向极大函数估计和Good-λ不等式等调和分析方法，进一步研究上述L^p预解式估计中指标p的最大取值区间问题。进而通过对偶论证、叠加原理及系数扰动等实变方法，将所得结果推广到Lipschitz区域上带有非齐次Neumann边值和VMO系数的二阶强椭圆型方程组上，同时考虑此类变系数椭圆型算子生成L^p有界解析半群的问题。本课题是将调和分析与非光滑区域上椭圆型方程组的结构特征相结合来研究谱论中的预解算子族有界性问题，这对调和分析、PDE与泛函分析的相互交叉大有裨益。

二阶椭圆型方程组在Lipschitz区域上的L^p预解式估计问题对于谱理论和解析半群理论的研究具有重要意义，并与椭圆边值问题的可解性密切相关。.本项目在我们前期工作已部分解决Lipschitz区域上带有齐次Neumann边值的二阶常系数强椭圆型方程组的L^p预解式估计问题的基础上，从含谱参数方程组的结构出发，通过采用C-Z分解、反向Hölder估计、John-Nirenberg不等式及系数扰动等调和分析方法，进一步证明了上述L^p预解式估计中指标p的取值范围同样适用于Lipschitz区域上带有VMO系数的二阶强椭圆型方程组的情形。进而通过对偶论证和叠加原理等实变方法，将所得结果推广到了Lipschitz区域上满足非齐次Neumann边值条件的带有VMO系数的二阶强椭圆型方程组上，并建立起上述变系数椭圆算子的预解式在指标p<2时的全局L^p梯度估计的结果。最终，证明了一般Lipschitz区域上带有非齐次Neumann边值和VMO系数的二阶强椭圆型算子具有生成L^p有界解析半群的性质。.本课题是将调和分析与非光滑区域上椭圆型方程组的结构特征相结合来研究谱理论中的预解算子有界性问题，这对调和分析、PDE与泛函分析的相互交叉大有裨益。.课题在研期间，项目组已发表或录用论文5篇，待发表论文4篇；培养研究生2名，其中一名已取得博士学位，另一名已取得硕士学位。