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

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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.