Mathematical Sciences: Presidential Young Investigator Award

这位总统青年研究员的工作将集中在与椭圆型偏微分方程解的研究相关的两个领域。分析了非光滑区域上的双调和多调和方程以及具有非光滑系数的二阶椭圆型散度型算子的边值问题。在过去的工作中，人们发现高阶椭圆算子在非光滑区域上的可解性范围依赖于基础区域的维度。在解将具有有限范数(在二次以上)的范围内仍有一个间隙。如果区域具有四维或更大的维度，则对于p大于2的某个值，存在具有无界p-范数的解。但p的第一个值是未知的，也不知道该间隙是否随着维度的增加而减小到零。将努力在这一范围内获得准确的估计。非光滑区域上的微分算子的研究与具有非光滑系数的微分算子的研究有着密切的联系。研究将集中在通过边值积分建立解的表示法上。这是一个特别微妙的问题。在光滑系数算子的情况下，积分是勒贝格积分(调和测度关于曲面测度是绝对连续的)。大体上不一定是这样。将开展工作，以确定仍将通过近似程序保证积分表示的标准。一旦建立了这些，就可以开始研究当变量接近区域边界时解的行为。

Work by this Presidential Young Investigator will focus 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 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.