Mathematical Sciences: Partial Differential Equations and Geometric Analysis

该奖项赞助的大部分研究将集中在 数学分析中的变分问题及其偏微分方程 当达到极值时导出的微分方程。 变分问题通常反映了一些 物理原理，如作用或能量的最小化。 该提案侧重于四个领域，其中大部分代表了 继续目前正在进行的研究。 椭圆 变分问题代表一个领域。 的形式 伴随微分方程是一类非齐次微分方程， 发散型 它是制定反映一个变 涉及障碍物的问题。 方程中有强迫项 这是一个度量或分布，而不是一个函数。 一 这项工作的目标是确定的规律性的性质， 障碍物接触集的解和几何 （可能不知道）在假设的平滑度方面， 障碍的边界。 抛物变分问题也将被处理。 的 将再次考虑障碍物问题。 这类 问题比椭圆问题更难分析，因为 空间和时间成分之间缺乏对称性 的微分算子。 尽管如此， 椭圆情况预期在此适用。 另一个需要解决的变分问题是 最小化域上函数的梯度（积分） 指定函数的边界值。 一个问题 这里关注的是平滑度-有一些例子， 最小化不需要在内部平滑的函数 的领域。 工作将做的特点时，一个可以 在这种情况下，期望平滑极值函数。 第四个要处理的领域是非线性 潜力理论 这涉及到一系列问题， 微分算子，p-Laplacian，它一直是 最近很感兴趣的话题。 解 相应的微分方程享有许多性质 调和函数。 重点将放在理解 解的梯度等于零的临界集。

Much of the research sponsored by this award will focus on variational problems in mathematical analysis and the partial differential equations which derive when extrema are reached. The variational problems are normally reflections of some physical principle such as the minimization of action or energy. The proposal focuses on four areas, most of which represent continuations of research currently in progress. Elliptic variational problems represents one area. The form of the associated differential equation is one of inhomogeneous divergence type. It is formulated to reflect a variational problem involving an obstacle. The equation has a forcing term which is a measure, or distribution, rather than a function. A goal of this work is to determine the regularity properties of the solution and the geometry of the contact set of the obstacle (which may not be known) in terms of assumed smoothness of the boundary of the obstacle. Parabolic variational problems will also be treated. The obstacle problem will again be considered. This class of problems is more difficult to analyze than the elliptic because there is a lack of symmetry between the space and time components of the differential operator. Still, some results from the elliptic case are expected to be applicable here. Another variational question to be addressed is that of minimizing (integrals of) gradients of functions on domains in which the function's boundary values are specified. A question of concern here is that of smoothness - there are examples of minimizing functions which need not be smooth across the interior of the domain. Work will be done to characterize when one can expect smooth extremal functions in this context. The fourth area to be treated is that of non-linear potential theory. This concerns a number of questions about a differential operator, the p-Laplacian, which has been the subject of considerable interest recently. Solutions of the corresponding differential equations enjoy many of the properties of harmonic functions. Emphasis will be placed on understanding the critical set where the gradient of solutions equals zero.