Singular asymptotic solution along an elliptical edge for the Laplace equation in 3-D

• The asymptotic solution to the Laplace equation in the vicinity of an elliptical crack in a 3-D domain is provided.
• It involves the flux intensity functions that are functions along the elliptical edge and associated eigen-function of two coordinates.
• The explicit representation for the entire series is derived for the first time.
• Having this expansion, one may derive the asymptotic solution for elasticity in the vicinity of an elliptic crack.

Explicit asymptotic solutions are still unavailable for an elliptical crack or sharp V-notch in a three-dimensional elastic domain. Towards their derivation we first consider the Laplace equation. Both homogeneous Dirichlet and Neumann boundary conditions on the surfaces intersecting at the elliptical edge are considered. We derive these asymptotic solutions and demonstrate, just as for the circular edge case, that these are composed of three series with eigenfunctions and shadows depending on two coordinates.