Calculus of Variations and Geometric Measure Theory

G. Dal Maso - G. Orlando - R. Toader

Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length

created by dalmaso on 11 Mar 2014
modified by orlando on 25 May 2015


Published Paper

Inserted: 11 mar 2014
Last Updated: 25 may 2015

Journal: NoDEA Nonlinear Differential Equations Appl.
Volume: 22
Number: 3
Pages: 449--476
Year: 2015
Doi: 10.1007/s00030-014-0291-0


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.