# Download Cartesian Currents in the Calculus of Variations II: by Mariano Giaquinta PDF

R U {+oo}, we have (i) rcf Ii BU {+oo} with non-empty effective domain E )f and with an affine minorant. By Theorem 3 C f < oo on co (D f) and C f = +oo on V \ co (Df), by Proposition 3 rCf = r(Cf). Being Cf also continuous on the interior of co (D f), we then get Proposition 4.

Envelop. Since the pointwise supremum of convex functions is clearly convex, Cf is convex. Moreover Theorem 2 reads as Theorem 4. Let f : V -+ IR U {+oo} and n = dim V. Then we have n+1 n+1 Cf (x) = inf { \if (xi) i=1 I l Ai > 0, E Ai = 1, xi E V } i=1 . c.. In this case the epigraph of f epif :_ {(x,y)EVxRjy? f(x)} is convex and closed. Therefore Hahn-Banach's theorem yields Proposition 2. c.. Then f is never -oo if and only if there exists an affine map P : V -+ 1R such that \$(x) < f (x) V x E V.

Regular Variational Integrals 8 Proof. Denote by LT (x) Lebesgue's extension Lr(x) := inf{l f f(xk) I {xk} C Y, Xk - x} . , that the set Ct :_ {xEXILT(x)