Mathematical Sciences: Presidential Young Investigator Award

这位总统青年调查员的工作将侧重于 与椭圆型偏微分方程解的研究有关的两个领域 微分方程 这些是双调和的分析 非光滑区域和边界上的多重调和方程 二阶椭圆散度型的值问题 非光滑系数的算子。 在过去的工作中，发现可解性的范围 非光滑区域上高阶椭圆算子的 在底层域的维度上。 仍然存在一个缺口 在解将具有有限范数的范围内（上述 quadratic）。 如果域具有四维或更大的维度，则 对于某些p值，存在具有无界p-范数的解 大于两个。 但是p的第一个值是未知的， 它知道这个间隙是否会随着尺寸的减小而减小到零， 增大 将努力工作， 估计这个范围。 对这一问题的研究与 非光滑域上的微分算子和非光滑域上的微分算子 平滑系数 研究将集中于建立 通过边界积分表示解 价值观 这是一个特别微妙的问题。 的情况下 光滑系数算子，积分是勒贝格 积分（调和测度绝对连续， 表面测量）。 这不一定是在 将军 将努力制定标准， 仍然通过近似保证积分表示 程序. 一旦这些都建立起来，然后就可以开始 研究解的行为作为变量， 接近域边界。

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.