| Abstract: |
| We consider a mathematical model that describes the equilibrium of a nonlinear elastic membrane which can arrive in contact with a rigid obstacle, the so-called foundation. The membrane is fixed on its boundary, is acted upon by a vertical force, the contact is frictionless and is modelled with the well-known Signorini boundary condition. The novelty is that we model the material`s behaviour with a Hencky-type elastic constitutive law and, therefore, the problem is governed by a $p$-Laplacian operator. Using such a constitutive assumption makes the model nonstandard from a mechanical point of view and challenging from a mathematical point of view. We use a Weierstrass-type minimization argument to prove the unique weak solvability of the model. Then, we turn to the well-posedness analysis of the problem, which represents the main novelty of this manuscript. Thus, we introduce three well-posedness concepts, compare them and state and prove strong and weak well-posedness results. We also discuss the problem of finding an optimal well-posedness result. Finally, we apply these results to prove that the solution depends continuously with respect to the density of applied forces and the initial gap. We end our paper with an Appendix in which we present some preliminary results used in this manuscript. |
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