Boundary Value Problems for Differential Equations with a High-Order Derivative in the Boundary Condition

The paper considers boundary-value problems for the Laplace equation, the boundary conditions of which contain traces of high-order derivatives on the boundary of the domain. It is shown that there are classes of (singular) solutions in which the solution of problems exists for any (sufficiently smooth) input data, without applying conditions such as orthogonality conditions. An example of a well-posed problem, the solution of which exists and is unique, is given.