相关于非光滑区域上椭圆算子的边值问题的函数空间实变理论及其应用

Many problems in modern mathematics could be attributed to study the boundedness of some special operators on corresponding function spaces, and the study for the boundedness of these operators closely contacts with the corresponding real variable theory of function spaces. Moreover, because of its important applications in mathematics and physics, the solvability of boundary value problems for elliptic equations on non-smooth domains has been attracted by many native and foreign scholars for a long time. In recent years, the applicant and his co-operators have studied the real variable theory of Hardy spaces of Musielak-Orlicz type associated with second order elliptic operators of divergence form operators or Schrödinger operators with Dirichlet or Neumann boundary conditions on strongly Lipschitz domains of Euclidean spaces, and the solvability of the Neumann or the Robin problem for the Laplace equation on bounded semi-convex domains. As a continuation and depth of these topics, this subject will intend to further study the real variable theory of function spaces, including Hardy type spaces and Besov spaces, associated with second order (or higher order) elliptic operators of divergence form operators or Schrödinger operators with the Robin boundary condition or the mixed boundary condition on non-smooth domains of Euclidean spaces and applications of these spaces to the regularity of the boundary value problems, and the solvability of the Robin or the mixed boundary value problem for the corresponding elliptic equations or Schrödinger equations.

现代数学中的众多问题均可归结为研究某些特殊算子在相应函数空间上的有界性, 而研究这些算子的有界性则离不开相应函数空间的实变理论. 此外, 非光滑区域上椭圆方程的边值问题因其在数学和物理中的重要应用而长期受到众多国内外学者的关注. 近年来, 申请人与合作者研究了欧氏空间中强Lipschitz区域上相关于带Dirichlet或Neumann边界条件的二阶散度型椭圆算子或薛定谔算子的Musielak-Orlicz型Hardy空间的实变理论和有界半凸区域上Laplace方程的Neuman或Robin边值问题的可解性. 作为这些课题的继续和深入, 本课题拟进一步研究相关于非光滑区域上带Robin边界或混合边界条件的二阶(或高阶)散度型椭圆算子或薛定谔算子的Hardy型空间和Besov型空间等函数空间的实变理论及其在边值问题正则性中的应用及对应椭圆方程或薛定谔方程的Robin或混合边值问题的可解性.

现代数学中的众多问题均可归结为研究某些特殊算子在相应函数空间上的有界性,而研究这些算子的有界性则离不开相应函数空间的实变理论.特别地,各种函数空间不仅为偏微分方程的适定性问题的研究提供了工作空间,而且其相应的实变特征刻画也为方程解的先验估计提供了有效的方法.在该项目中,建立了非光滑区域上相关于带Robin边界条件或混合边界条件的二阶(或高阶)散度型椭圆算子或薛定谔算子的(局部)Hardy型空间和Besov空间的包括原子特征、各种极大函数特征及Lusin面积函数特征等在内的实变特征刻画,且系统地讨论了这些与算子相关的Hardy空间或Besov空间和区域上经典的Hardy空间或Besov空间的包含关系或等价关系.获得了有界Lipschitz区域或半凸区域上二阶(或高阶)散度型椭圆方程或薛定谔方程的Robin边值问题或混合边值问题在(加权)Lebesgue空间和Hardy空间等函数空间中的全局正则性估计.所获结果改进了C.Amrouche等人[Calc.Var.Partial Differential Equations 59(2020),no.2,No.71]的主要结果.建立了有界Lipschitz区域或半凸区域上齐次的二阶散度型椭圆方程或薛定谔方程的带Lebesgue空间或Hardy空间边值的Neumann边值问题,Robin边值问题或混合边值问题的可解性.所获结果改进了Z.Shen[Indiana Univ.Math.J.43(1994),143-176]的主要结果.