Mathematical Sciences: Higher Order Operators in Non-Smooth Domains

数学科学：非光滑域中的高阶算子

• 批准号: 9001574
• 负责人: Jill Pipher
• 金额: $ 4.78万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-15 至 1992-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9001574&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Higher; Order; Operators

Work on this project focuses on two areas related to the study of solutions of elliptic partial differential equations. These are the analysis of the biharmonic and polyharmonic equations on non-smooth domains and boundary value problems for second order elliptic divergence form operators with non-smooth coefficients. In past work it was discovered that the range of solvability of higher order elliptic operators on non-smooth domains depended on the dimension of the underlying domain. There remains a gap in the range in which the solution will have a finite norm (above the quadratic). If the domain has dimension four or greater then solutions exist with unbounded p-norms for some value of p greater than two. But the first value of p is not known, nor is it known whether this gap reduces to zero as the dimension increases. Work will be done in an effort to obtain sharp estimates on this range. There is a close connection between the study of differential operators on non-smooth domains and those with non-smooth coefficients. Research will concentrate on establishing representations of solutions through integrals of boundary values. This is a particularly delicate matter. In the case of smooth coefficient operators, the integrals are Lebesgue integrals (the harmonic measure is absolutely continuous with respect to surface measure). This is not necessarily the case in general. Work will be done in establishing criteria which will still guarantee the integral representation through approximation procedures. Once these are established, one can then begin to study the behavior of solutions as the variable is made to approach the domain boundary.

该项目的工作重点是与椭圆偏微分方程的解研究相关的两个领域。 这些是非光滑域上的双调和和多调和方程以及具有非光滑系数的二阶椭圆散度形式算子的边值问题的分析。 在过去的工作中发现，非光滑域上高阶椭圆算子的可解范围取决于基础域的维数。 在解具有有限范数（二次以上）的范围内仍然存在差距。 如果域的维度为四或更大，则对于某个大于二的 p 值，存在具有无界 p 范数的解。 但 p 的第一个值是未知的，也不知道这个差距是否会随着维度的增加而减小到零。 我们将努力获得对此范围的准确估计。 非光滑域上的微分算子的研究与非光滑系数域上的微分算子的研究有着密切的联系。 研究将集中于通过边界值的积分建立解决方案的表示。 这是一个特别微妙的问题。 在平滑系数算子的情况下，积分是勒贝格积分（调和测量相对于表面测量绝对连续）。 一般来说，情况不一定如此。 我们将制定标准，以保证通过近似程序的积分表示。 一旦这些建立起来，人们就可以开始研究当变量接近域边界时解的行为。