Abstract
We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.
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Dal Maso, G., Orlando, G. & Toader, R. Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length. Nonlinear Differ. Equ. Appl. 22, 449–476 (2015). https://doi.org/10.1007/s00030-014-0291-0
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DOI: https://doi.org/10.1007/s00030-014-0291-0