Integral representations of solutions of coercive solvable problems for the Laplace equation
Kal’menov T.
December 2026Springer Science and Business Media Deutschland GmbH
Boundary Value Problems
2026#2026Issue 1
Modeling of many stationary physical processes is reduced to the study of boundary value problems for elliptic equations. Starting with the works of A.V. Bitsadze and A.A. Samarskiy, well-posed, but not necessarily boundary value problems for the Laplace equation were investigated, which later became widely developed also for hyperbolic and parabolic equations. At the same time, the study of regular boundary value problems is closely related to the theory of correct restrictions and extensions of differential operators. The description of general regular boundary value problems and the corresponding correct extensions in the case of general linear elliptic equations is given by M.I. Vishik, and the correct but not necessarily boundary value problems and correct restrictions are given by M. Otelbaev. The main method of these works is the method of self-adjoint extensions of symmetric operators, constructed by J. von Neumann. The main aim of this work is to find the domain of definition of the coercive correct restriction for the inhomogeneous Laplace equation in integral form. Using the criterion of the boundary value of the integral operator, the boundary conditions of the coercive correct extension of the minimal Laplace operator are described, that is, the description of regular boundary value problems for the inhomogeneous Laplace equation is given. Using the representation of the domain of definition of the coercive correct extension and the trace property of the potential of a simple layer, the Green’s function of the Dirichlet problem for the Laplace equation is constructed. In contrast to the works of M.I. Vishik and M. Otelbaev, when proving the statements, we use the methods of mathematical physics, namely the method of the Newtonian potential and the potential of a simple layer. In this case, the potential of a simple layer is reduced to a special volumetric potential.
Coercive solvable problems , Green’s function , Integral representation of solutions , Laplace equation , Theory of restrictions and extensions
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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Institute of Mathematics and Mathematical Modeling
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