A comparative study of generalised solution concepts for elliptic partial differential equations using nonsmooth analysis techniques

椭圆型偏微分方程解是科学和工程的核心问题。有许多解的概念，如弱解和粘性解的概念，但这些概念之间的关系还没有完全理解。我们将利用最优化理论和非光滑分析的最新进展来研究这个主要问题。我们的方法是使用各种近似及其相关的二阶次微分，每一种都隐含着一个广义解概念和相关的抽象凸性。特别注意，包括计算细节，将给予方程具有与另一种类型的解非常不同的解。

The solution of ellpitic partial differential equations is central to science and engineering. There are a number of solution concepts, such as those of weak solutions and viscosity solutions, but the relations between these are incompletely understood. We shall investigate this major question using recent advances in optimisation theory and nonsmooth analysis. Our approach is to use various approximations and their associated second-order subdifferentials, each of which implies a generalised solution concept and associated abstract convexity. Particular attention, including computational details, will be given to equations which have very different solutions of one type from those of another.