Variations of Deformations with Self-Contact on Lipschitz Domains

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We analyze the injective deformations on Lipschitz domains, motivated by the configurations of solid elastic materials. Our goal is to rigorously derive the first order optimality conditions when minimizing over deformations with a self-contact constraint. In a previous work, we established a variational equation for minimizers of a second-gradient hyper-elastic energy. This involved obtaining a Lagrange multiplier for the self-contact constraint as a nonnegative surface measure multiplied by the outward normal vector. For that approach, we had to assume that the boundary of the reference domain is smooth. In this work, we carry out an analogous characterization assuming only that the boundary is locally the graph of a Lipschitz function. These Lipschitz domains possess everywhere defined interior cones, which we use to characterize the displacements that produce paths of injective deformations. Our application to nonlinear elasticity results in a vector-valued surface measure that is characterized by the local nonsmooth geometry of the domain at the points of self-contact.