Boundary criterion for integral operators

Integral operators of the form for the case of a finite domain Ω ⊂ R n with smooth boundary ∂Ω are considered. Conditions on the real kernel K ( x , t ) of an integral operator under which this operator satisfies a well-defined boundary condition for the corresponding differential equation are found. The application of the results is demonstrated on the example of a Sturm–Liouville equation, for which the derivation of the general form of well-posed boundary value problems is presented.