The main objective of the research was study of elliptic partial differential equations. Partial differential equations are used to mathematically describe behaviour of many real life phenomena and arise practically everywhere. Equations of this type can be encountered in physics, material science, geometry, probability and many other disciplines.
In many real life applications, the equations that arise have certain singularities. For example the domain of equation can have corners, cusps or the coefficients of equation itself might be discontinuous. 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. For these reasons it is very important to consider these situations mathematically.
Our research have made substrantial progress in studying these phenomena. We have established solvability of the Dirichlet boundary value problem under assumption that the coefficients satisfy certain Carleson condition. We have established this result for both divergence and non-divergence form elliptic equations. We have also studied the Dirichlet problem with BMO data and have established equivalence between solvability of this problem and so called A_\infty condition.