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Potential theoretic properties of the gradient of a convex function on a functional space

Published online by Cambridge University Press:  22 January 2016

Nobuyuki Kenmochi  and
Yoshihiro Mizuta
Nobuyuki Kenmochi
Affiliation:
Department of Mathematics, Faculty of Education Chiba University Chiba, Japan
Yoshihiro Mizuta
Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University Hiroshima, Japan
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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.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 59 , December 1975 , pp. 199 - 215
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

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