ON THE SOLVABILITY OF SOME NONLOCAL PROBLEMS FOR THE POISSON EQUATION WITH INVOLUTION

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Abstract

 In this paper, certain linear operators with argument transformations are defined within the class of smooth functions. These operators are introduced using matrices of involution-type mappings. Subsequently, the specified operators are used to define a non-local analogue of the Laplace operator and corresponding boundary operators. For the obtained non-local analogue of the Poisson equation, solvability questions for certain boundary value problems are investigated. The boundary conditions of the considered problems are specified as relations between the values of the unknown function at different points and, thus, belong to Bitsadze–Samarskii type problems. Theorems on the existence and uniqueness of the solution to the investigated problems are proved. It is shown that the well-posedness of the considered problems depends essentially on the coefficients of the introduced linear transformation operators. Using the Green's function for the classical Dirichlet and Neumann problems, the explicit form of the Green's function for the considered problems is constructed. Moreover, integral representations of the solutions to these problems are also obtained using the constructed Green's function. In addition, the study examines the structure of the transformation operators and analyzes their properties that influence the stability of the solution. The obtained results are compared with classical local models, which makes it possible to identify the advantages of the nonlocal approach. It is noted that the proposed methods can also be applied to other types of elliptic equations involving argument transformations.