On Boundary Integral Equations for Crack Problems

A ubiquitous linear boundary-value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle. One typical example is the scattering of scalar waves by a curved crack or rigid strip (Neumann boundary condition) in two dimensions. This problem can be reduced to an integrodifferential equation, which is often regularized. We adopt a more direct approach, and prove that the problem can be reduced to a hypersingular boundary integral equation. (Similar reductions will obtain in more complicated situations.) Com­putational schemes for solving this equation are described, with special emphasis on smoothness requirements. Extensions to three-dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed.