On the reduction of the domain of correction for Laplace equation

• Authors: Igor Neygebauer
• Venue: F-and-B'11 Proceedings of the 4th WSEAS international conference on Finite differences - finite elements - finite volumes - boundary elements
• Year: 2011

The method of additional conditions gives a well characteristic of the finite stresses near the tip of a crack of the Griffith's problem in fracture mechanics, where is supposed of a finite elastic potential which entails the zero value of the J-integral. In this paper we apply this method to the probably most frequently occurring partial differential equation governing the behavior of certain physical quantities. The Laplace equation is considered in the wedge domain. For the given Dirichlet problem the derivative of the solution has a singularity at the origin for an obtuse angle. The solution is unique. The singularity can be considered as impossible from the physical point of view. Therefore the Laplace equation must be considered as an approximate equation for the physical problems for which sometimes the more general equations are unknown. There is suggested an additional condition which follows from the classical Green's formula and from the supposed to be bounded solution and its derivatives, and that can characterize the nonsingular solution at the origin. The second approximation is considered. The maximum principle is investigated. The possibility to reduce the region of correction is considered.