Boundary problems for the second order elliptic equations with rough coefficients

The main focus of the meeting was on boundary value problems for general differential operators L = −divA∇. Here A is an elliptic matrix with variable coefficients, given by complex-valued bounded and measurable functions. Such operators arise naturally in many problems of pure mathematics as well as in numerous applications. In particular, they describe a wide array of physical phenomena in rough, anisotropic media. Thus, one of the central questions is: what kind of medium yields solvable boundary problems, or, mathematically, what are the sharp conditions on the matrix A responsible for the solvability of problem −divA∇u = 0 in a given domainΩ ⊂ R, u|∂Ω = f , with boundary data, for instance, in L(∂Ω). Despite tremendous advances in the elliptic theory over the past half a century, this question remains largely open.