ABSTRACT

We consider a mixed variational problem governed by a nonlinear operator and a set of constraints. Existence, uniqueness and convergence results for this problem have already been obtained in the literature. In this current paper we complete these results by proving the well-posedness of the problem, in the sense of Tykhonov. To this end we introduce a family of approximating problems for which we state and prove various equivalence and convergence results. We illustrate these abstract results in the study of a frictionless contact model with elastic materials. The process is assumed to be static and the contact is with unilateral constraints. We derive a weak formulation of the model which is in the form of a mixed variational problem with unknowns being the displacement field and the Lagrange multiplier. Then, we prove various results on the corresponding mixed problem, including its well-posedness in the sense of Tykhonov, under various assumptions on the data. Finally, we provide mechanical interpretation of our results.

摘要

我们考虑由非线性算子和一组约束控制的混合变分问题。该问题的存在性、唯一性和收敛性结果已经在文献中得到。在本文中，我们通过在 Tykhonov 的意义上证明问题的适定性来完成这些结果。为此，我们引入了一系列近似问题，我们陈述并证明了各种等价和收敛结果。我们在弹性材料无摩擦接触模型的研究中说明了这些抽象结果。假定该过程是静态的，并且接触具有单边约束。我们推导出模型的弱公式，它是混合变分问题的形式，未知数是位移场和拉格朗日乘子。然后，我们在数据的各种假设下证明了相应混合问题的各种结果，包括其在 Tykhonov 意义上的适定性。最后，我们对我们的结果进行机械解释。