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研究有关的两个领域。分析了非光滑域上的双调和方程和多调和方程，以及二阶非光滑系数椭圆发散形式算子的边值问题。在以往的研究中发现，非光滑域上的高阶椭圆算子的可解范围取决于下域的维数。在解具有有限范数（二次元以上）的范围内仍有一个间隙。如果定义域具有四维或更大的维数，那么对于p大于2的某个值，存在具有无界p模的解。但是p的第一个值是未知的，也不知道这个差距是否随着维度的增加而减小到零。将努力对这一范围作出准确的估计。非光滑域上微分算子的研究与非光滑系数域上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的表示。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（调和测度相对于曲面测度是绝对连续的）。这并不一定是一般情况。我们要做的工作是建立一些准则，这些准则仍然可以保证通过近似过程得到积分表示。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。

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.