Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.
Gustafsson, B., Sebbar, A. Hadamard’s variational formula in terms of stress and strain tensors. Anal.Math.Phys. 12, 29 (2022). https://doi.org/10.1007/s13324-021-00632-y
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