高阶散度型椭圆和抛物方程解的正则性研究

The analysis of regularity for generalized solutions of partial differential equations has gained much attention in recent years. This study will provide an important basis for the "difference" between the generalized solutions and the classical solutions. Applying a refinement of the technique in harmonic analysis, the Large-M-inequality principle and perturbation theory in PDE, the project will derive some decay estimates involving the upper-level set, and then study the Dirichlet problems of higher order elliptic equations, the Cauchy-Dirichlet problems of higher order parabolic equations and relevant parabolic obstacle problems as follows: 1) the global Lorentz regularity will be investigated for higher order elliptic and parabolic equations of p-Laplacian type with discontinuous coefficients; 2) the global regularity in weighted Lorentz spaces will be obtained for p-Laplacian parabolic obstacle problems with weak regular datum; 3) a global Lorentz estimate for the variable power of the optimal gradient of weak solutions will be established for higher order linear parabolic equations with partially regularity coefficients over Reifenberg domains. The study of the project will not only enrich and improve the regularity theory of elliptic and parabolic equations in divergence form, but also provide important theoretical support for the research of PDE and other related fields.

偏微分方程广义解的正则性研究一直受到高度关注。此研究将为广义解与古典解的“差异”问题提供重要依据。本项目运用精细的调和分析技巧，大M不等式原理以及偏微分方程的扰动理论建立相关上水平集的测度衰减估计，进而对高阶椭圆方程的Dirichlet问题、高阶抛物方程的Cauchy-Dirichlet问题以及相关的抛物障碍问题开展以下研究：1) 研究具不连续系数的高阶p-Laplacian型椭圆和抛物方程的整体Lorentz正则性；2) 考虑在弱正则数据下高阶p-Laplacian型抛物障碍问题的整体加权Lorentz正则性；3) 建立在部分正则系数和Reifenberg区域假设下高阶线性抛物方程解高阶导数在变指数幂下的整体Lorentz估计。本项目研究不仅丰富与完善散度型椭圆和抛物方程解的正则性理论，而且为偏微分方程等相关领域的研究提供重要的理论支撑。

高阶偏微分方程广义解的正则性问题一直是偏微分方程理论研究的难点。 本项目主要研究了具不连续系数的高阶线性椭圆和抛物方程弱解的变指Lebesgue正则性，高阶p-Laplacian 型椭圆和抛物方程广义解的Lorentz正则性，以及定义在非光滑区域上的高阶非线性抛物障碍问题弱解的加权Lorenz正则性。在项目的资助下，项目负责人积极组织和参加学术会议，在相关领域获得一些研究成果，为后续研究工作奠定基础。