On elliptic and parabolic systems with <i>x</i>‐dependent multivalued graphs

Abstract

Abstract 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). Copyright © 2006 John Wiley &amp; Sons, Ltd.