Model oblique derivative problem for the heat equation with a discontinuous boundary function

Abstract

The oblique derivative problem for the heat equation is considered in a model formulation with a boundary function that can be discontinuous and with the boundary condition understood as the limit in the normal direction almost everywhere on the lateral boundary of the domain. An example is given showing that the solution is not unique in this formulation. A solution is sought in the parabolic Zygmund space H₁, which is an analogue of the parabolic Hölder space for an integer smoothness exponent. A subspace of H₁ is introduced in which the existence and uniqueness of the solution is proved under suitable assumptions about the data of the problem.