Solvability of elliptic partial differential equations with rough coefficients; the boundary value problems

本研究旨在研究椭圆型偏微分方程。偏微分方程用于数学描述许多现实生活现象的行为，并且几乎无处不在，例如物理，材料科学，几何，概率论和许多其他学科。在许多现实生活模型中，例如在材料科学或物理学中，系数可能是不连续的（模拟材料中的杂质或裂纹），因此研究具有低正则系数的方程是有意义的。这里，系数的不连续是许多材料含有杂质（异物）这一事实的数学表达式，这些杂质会在一定程度上改变所研究对象的性质。因此，从数学上考虑这些情况并了解求解这些方程的方法是非常重要的。这就是我们在研究中要做的。正如这个项目的名字所暗示的，我们将重点关注三种基本类型的边值问题。我们对不同边值问题的认识是参差不齐的，特别是一类边值问题比另外两类边值问题探索得更好。我们的目标是纠正这种情况，使我们对这些边值问题的认识达到大致相同的水平。

The proposed research aims to study elliptic partial differential equations. Partial differential equations are used to mathematically describe behaviour of many real life phenomena and arise practically everywhere, such as in physics, material science, geometry, probability and many other disciplines. In many real life models such as in material science or physics the coefficients can be discontinuous (modelling impurities in the material or cracks) and so it makes sense to study equations withlow regularity coefficients. Here the discontinuity of coefficients is the mathematical expression of the fact that many materials contain impurities (foreign objects) that somewhat change the properties of studied objects. It is therefore very important to consider these situations mathematically and understand way of solving such equations. This is what we are going to do in our research. As the name of this project suggests we are going to focus on three basic types of boundary value problems. Our knowledge about different boundary value problems is quite uneven, in particular one type of boundary value problem is much better explored than the other two. We aim to remedy this situation and bring our knowledge about these boundary value problems to approximately same level.