Advances in the general theory of boundary value problems for partial differential equations

The issues of the general theory of boundary value problems for partial differential equations are discussed, the report contains the author's results. Three new directions of the theory development will be discussed. The first is the structure of the domain of definition of the maximal extension of the general differential operator, questions about traces on the boundary of the domain of the $L_2$-solution of the general equation and the properties of these traces. The second is new methods for studying general boundary value problems and the results obtained. The third is new applications of ill-posed boundary value problems in problems of geometry, algebra and analysis.