We prove the existence of solutions to the static contact problem with Coulomb friction, provided that the coefficient of friction is small enough. The proof employs the penalty method and a certain smoothing procedure for the friction functional. Using optimal trace estimates for the solutions of the Lamé equations, we calculate an upper bound for the admissible coefficient of friction which is greater than the corresponding bounds proposed by Necas, Jarusek and Haslinger (1980) and by Jarusek (1983).
The term electrowetting is commonly used for phenomena where shape and wetting behavior of liquid droplets are changed by the application of electric fields. We develop and analyze a model for electrowetting that combines the Navier-Stokes system for fluid flow, a phase-field model of Cahn-Hilliard type for the movement of the interface, a charge transport equation, and the potential equation of electrostatics. The model is derived with the help of a variational principle due to Onsager and conservation laws. A modification of the model with the Stokes system instead of the Navier-Stokes system is also presented. The existence of weak solutions is proved for several cases in two and three space dimensions, either with non-degenerate or with degenerate electric conductivity vanishing in the droplet exterior. Some numerical examples in two space dimensions illustrate the applicability of the model.
The existence of solutions to the dynamic contact problem with Coulomb friction for viscoelastic bodies is proved with the use of penalization and regularization methods. The contact condition, which describes the nonpenetrability of mass, is formulated in velocities. The coefficient of friction may depend on the solution but is assumed to be bounded by a certain constant.
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