Comparison principles for variational and hemivariational inequalities

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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.