Elliptic partial differential equations: theory and applications

在许多试图描述现实世界中各种物理现象的数学模型中，偏微分方程组(PDE‘s)处于核心地位。正因为如此，偏微分方程式才是最重要的。从数学上讲，偏微分方程式是一个非常广泛的领域，通常集中在某些类别的偏微分方程上。我的专业领域是椭圆型偏微分方程领域，特别是半线性和拟线性椭圆型PDE领域。 我建议研究各种在结构上不是变分的椭圆型方程。在变分偏微分方程组的情况下，人们可以附加一个临界点与偏微分方程组的解相对应的能量泛函。这通常允许人们使用非常发达的变分演算机制，例如极小-极大技巧，来获得存在性和多重性结果。当方程不再是变分方程时，人们没有可用的变分机制，必须使用其他方法，例如可以尝试拓扑学方法。此外，我还建议研究一些椭圆问题，这些问题本质上是变分的，但却是超临界的，这通常不允许采用变分方法。但这些问题的难度从相当直接到相当具有挑战性。但其中一些问题非常不适合硕士研究生，不需要太多背景知识。

Partial differential equations (pde's) lie at the heart of many mathematical models which attempt to describe various physical phenomena in the real world.    Because of this understanding pde's is of the utmost importance.   Mathematically pde's is a very broad area and generally one concentrates on certain classes of pde's.   My area of expertise is in the area of elliptic partial differential equations and in particular in the area of semilinear and quasilinear elliptic pde's. I propose to examine various elliptic  equations which are not variational in structure.  In the case of variational pde's one can attach an energy functional whose critical points correspond to solutions of the pde.  Often this allows one to use the very well developed machinery of variational calculus, for instance min-max techniques, to obtain existence and multiplicity results.  When the equation is no longer variational one does not have this variational machinery available and one must utilize other methods, for instance one can try topological methods.   In addition I propose to study some elliptic problems which are variational in nature but are supercritical, which in general, does not allow for a variational approach.    These problems range in difficulty from fairly straightforward to quite challenging.    Some of these problems are very  suitable for a masters student and do not require a lot of background.