On elliptic and parabolic systems with x-dependent multivalued graphs

We consider elliptic and parabolic systems with multivalued x-dependent graphs. The existence of solutions for elliptic equation was established in (Ann. Inst. H. Poincare Anal. Non Linéaire 1990; 7(3):123-160; Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2004; 7 (1):23-59). We extend this result to the elliptic and parabolic systems, in particular to the systems describing a flow of non-Newtonian incompressible fluids. Contrary to these two papers we follow the spirit of the compactness method of J. L. Lions for variational-type operators, however, expanded on the framework of measure-valued solutions. The main concept consists in applying the relation between x-dependent maximal monotone graphs and 1-Lipschitz Carathéodory functions to introduce the generalized Young measures. The method was announced in the short note (C. R. Math. Acad. Sci. Paris 2005; 340(7):489-492).