The paper considers the properties of vector analogs of the Sobolev spaces, , which appear in the study of various models of continuous media. Korn's inequality, proved in the paper, makes it possible to reduce the problem of compactness of the imbedding operators in these spaces to the scalar case and, consequently, to apply the scalar imbedding theorem of S. L. Sobolev. Heretofore, Korn's inequality in the general form was known only for .Bibliography: 18 items.
We discuss partial regularity results concerning local minimizers u : R ⊃ Ω → R of variational integrals of the form ∫ Ω { a(| (w)|) + b(| div(w)|) } dx, where a and b are N -functions of rather general type. We prove partial regularity results under quite natural conditions between a and b. Furthermore we can extend this to the non-autonomous situation which finally leads to the study of minimizers of the functional ∫ Ω { (1 + | (w)|) p(x) 2 + (1 + | div(w)|) q(x) 2 } dx, where p and q are Lipschitz-functions. Mathematics Subject Classification (2000): 94 N 60, 74 B 20, 74 G 40, 74 G 65, 35 J 50
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.