On the solvability of some boundary value problems with involution

This article is devoted to the study of the solvability of some boundary value problems with involution. In the space $ R ^ {n} $, the map $ Sx = -x $ is introduced. Using this mapping, a nonlocal analogue of the Laplace operator is introduced, as well as a boundary operator with an inclined derivative. Boundary-value problems are studied that generalize the well-known problem with an inclined derivative. Theorems on the existence and uniqueness of the solution of the problems under study are proved. In the Helder class, the smoothness of the solution is also studied. Using well-known statements about solutions of a boundary value problem with an inclined derivative for the classical Poisson equation, exact orders of smoothness of a solution to the problem under study are found.