An alternative potential method for mixed steady‐state elastic oscillation problems

Abstract

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.