Potential theoretic properties of the gradient of a convex function on a functional space
Nobuyuki Kenmochi, Yoshihiro Mizuta · Nagoya Mathematical Journal · 1975
In the previous paper [11], introducing the notions of potentials and of capacity associated with a convex function Φ given on a regular functional space we discussed potential theoretic properties of the gradient ∇Φ which were originally introduced and studied by Calvert [5] for a class of nonlinear monotone operators in Sobolev spaces. For example: (i) The modulus contraction operates. (ii) The principle of lower envelope holds. (iii) The domination principle holds. (iv) The contraction Tk onto the real interval [0, k] (k > 0) operates. (v) The strong principle of lower envelope holds. (vi) The complete maximum principle holds.