Some strengthening of the property of Hölder continuity of solutions of a second-order elliptic equation

It is well known (De Giorgi, Nash), that generalized solutions of a second-order elliptic equation with measurable bounded coefficients are Hцlder continuous in the interior of the domain. The talk describes the properties which are intermediate between the integral property of a solution to belong to the Sobolev space and the pointwise property of its interior Hölder continuity. All these properties are united by using the special function space. Any solution belongs to the introduced space. This inclusion gives some new properties which do not follow from Hölder continuity and belonging to the Sobolev space. In near terms analogous global properties for solutions of the Dirichlet problem with integrable square boundary function were studied.