Comparison principles for variational and hemivariational inequalities

次上解方法已被证明是证明各种非线性椭圆型和抛物型边值问题的存在性和比较结果的有力工具。这种方法不仅提供了定性的结果，例如解的存在性和界，而且还用于这类问题的数值分析。我们的主要目的是将这种方法的思想推广到椭圆型和抛物型变分半变分不等式，并研究这类问题的解集的结构。就我们所知，似乎没有任何出版物系统地提到变分-半变分不等式的下上解方法。在这一推广中面临的主要困难是建立一个与变分方程特殊情况下通常的上、下解概念相容的适当的上、下解概念。

The sub-supersolution method has been proved to be a powerful tool for proving existence- and comparison results for a variety of nonlinear elliptic and parabolic boundary value problems. This method not only provides qualitative results, such as, e.g., existence and bounds of solutions, but is also used in the numerical analysis of such problems. Our main goal is to extend the idea of this method to elliptic and parabolic variational-hemivariational inequalities, and to study the structure of the solution set of such kind of problems. To the best of our knowledge it seems that there has but been any publication that adresses the sub-supersolution method for variational-hemivariational inequalities in a systematic way. The main difficulty one is faced in this extention is creating an appropirate notion of sub- and supersolution that is compatible with the usual notion of sub-supersolution in the special case of variational equations.