Abstract

We consider a Cauchy problem of an abstract evolution inclusion on a real Hilbert space associated with subdifferentials of time-dependent proper lower semicontinuous convex functions. The abstract evolution inclusion, which is treated in this paper, contains not only convex functions but also inner products of the Hilbert space depending on unknown functions. Especially, we call such structures for convex functions and inner products "quasi-variational structures for convex functions and inner products". The main purposes of this paper are to show the existence of global-in-time solutions to the Cauchy problem, which has a quasi-variational structure, and to apply this existence result to a phase field model with a quasi-variational boundary condition.

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Keywords

  • Evolution inclusion
  • quasi-variational structures
  • inner products
  • initial-value problems
  • global-in-time solutions

Mathematics Subject Classification

34G25, 47J35, 49J40, 58E35

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Suggested citation

A. Ito. “Evolution Inclusion on a Real Hilbert Space with Quasi-Variational Structure for Inner Products.” Journal of Convex Analysis 26 (2019), No. 4, 1187–1254. https://doi.org/10.68381/jca26065

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.