The symmetric Galerkin boundary element method is used to solve boundary value problems by keeping the symmetric nature of the matrix obtained after discretization. The matrix elements are obtained from a double integral involving the double derivative of Green's operator, which is highly singular. The paper presents a regularization of the hypersingular integrals which depend only on the properties of Green's tensor. The method is presented in the case of Laplace's operator, with an example of application. The case of elasticity is finally addressed theoretically. showing an easy extension to any case of anisotropy.