The Relaxation of Some Classes of Variational Integrals with Pointwise Continuous-Type Gradient Constraints

The Relaxation of Some Classes of Variational Integrals with Pointwise Continuous-Type Gradient... Relaxation problems for a functional of the type $G(u)=\int_\Omega g(x,\nabla u)\,{\rm d}x$ are analyzed, where $\Omega$ is a bounded smooth subset of $\R^N$ and $g$ is a Carath\’eodory function, when the admissible $u$ are forced to satisfy a pointwise gradient constraint of the type $\nabla u(x)\in C(x)$ for a.e. $x\in\Omega$, $C(x)$ being, for every $x\in\Omega$, a bounded convex subset of $\R^N$. The relaxed functionals $\overline{G_{PC^1(\Omega)}}$, and $\overline{G_{W^{1,\infty}(\Omega)}}$ of $G$ obtained letting $u$ vary in $PC^1(\Omega)$, the set of the piecewise $C^1$-functions in $\Omega$, and in $W^{1,\infty}(\Omega)$ respectively in the definition of $G$ are considered. Identity and integral representation results are proved under continuity-type assumptions on $C$, together with the description of the common density by means of convexification arguments. Classical relaxation results are extended to the case of the continuous variable dependence of $C$, and the non-identity features described in the measurable dependence case by De Arcangelis, Monsurr\‘o and Zappale (2004) are shown to be non-occurring. Proofs are based on the properties of certain limits of multifunctions, and on an approximation result for functions $u$ in $W^{1,\infty}(\Omega)$, with $\nabla u(x)\in C(x)$ for a.e. $x\in\Omega$, by $PC^1(\Omega)$ ones satisfying the same condition. Results in more general settings are also obtained. Applied Mathematics and Optimization Springer Journals

The Relaxation of Some Classes of Variational Integrals with Pointwise Continuous-Type Gradient Constraints

Loading next page...
Copyright © 2005 by Springer
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
Publisher site
See Article on Publisher Site

There are no references for this article.

You’re reading a free preview. Subscribe to read the entire article.

DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches


Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.



billed annually
Start Free Trial

14-day Free Trial