On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

Abstract

Abstract. The mixed Dirichlet-Neumann problem for the Laplace equation in an unbounded plane domain with cuts (cracks) is studied. The Dirichlet condition is given on closed curves making up the boundary of the domain, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory and a boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density of the potentials satisfies a uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of the cuts are investigated. Introduction. The boundary of a 2-D cracked domain consists of both closed curves and open arcs (cuts).Open arcs model cracks in solids and screens or wings in fluids. Different physical processes in cracked domains can be described by boundary value problems for the Laplace equation, for example, distribution of stationary heat and electric fields in cracked solids, electric flow in cracked semiconductors, flow of an ideal fluid over several obstacles and wings, etc. Appropriate boundary conditions must be specified on the total boundary, i.e., on both closed curves and open arcs (cracks). The Neumann boundary condition reflects the nonflow (of fluid, electric current, etc.) through the boundary. The Dirichlet boundary condition corresponds to the given temperature in heat theory, fluid pressure in hydrodynamics, electric potential in electrostatics, etc.Boundary value problems with mixed boundary conditions were not treated in cracked domains by rigorous mathematical methods before. Even in the case of Laplace and Helmholtz equations the problems in domains bounded by closed curves [2], [13]- [17] and problems in the exterior of cuts (cracks) [14,16], [18]-[20] were treated separately, because different methods were used in their analysis. Previously the Neumann problem in the exterior of a cut was reduced to a hypersingular integral equation [14,16,18,19] or

Mixed boundary value problems of diffraction by a half‐plane with an obstacle perpendicular to the boundary

“…The problems are reformulated as boundary integral systems which are first analyzed in view of obtaining conditions for the existence and uniqueness of solutions within a scale of Bessel potential spaces. Several recent works could be referred as being devoted to present a mathematical analysis of wave diffraction problems governed by the Helmholtz equation and which are somehow natural physical generalizations of the original works of Sommerfeld (see ).…”