Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T00:18:03.047Z Has data issue: false hasContentIssue false
  • English
  • Français

Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations∗

Published online by Cambridge University Press:  10 December 2013

Luís Balsa Bicho  and
António Ornelas
Luís Balsa Bicho
Affiliation:
Cima-ue, Rua Romão Ramalho 59, 7000-671 Évora, Portugal. [email protected]; [email protected]
António Ornelas
Affiliation:
Cima-ue, Rua Romão Ramalho 59, 7000-671 Évora, Portugal. [email protected]; [email protected]
Get access

Abstract

We prove uniform continuity ofradially symmetric vector minimizersuA(x) = UA(|x|)to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on aball BR ⊂ ℝd,among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using ajointly convex lsc L∗∗ : ℝm×ℝ → [0,∞]withL∗∗(S,·) evenand superlinear. Besides such basic hypotheses,L∗∗(·,·) is assumed to satisfy alsoa geometrical constraint, which we callquasi − scalar; the simplest example being thebiradial caseL∗∗(|u(x)|,|Du(x)|).Complete liberty is given forL∗∗(S,λ)to take the ∞ value, so that our minimization problem implicitly also representse.g. distributed-parameteroptimal control problems, onconstrained domains, under PDEs or inclusions inexplicit or implicit form. While generic radial functionsu(x) = U(|x|) inthis Sobolev space oscillate wildly as |x| → 0, our minimizingprofile-curve UA(·) is, incontrast, absolutely continuous andtame, in the sense that its“static level” L∗∗(UA(r),0)always increases with r, a original feature of our result.

Keywords

Vectorial calculus of variationsvectorial distributed-parameter optimal controlcontinuous radially symmetric monotone minimizers
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 141 - 157
Copyright
© EDP Sciences, SMAI 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bicho, L.B. and Ornelas, A., Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients. Nonlinear Anal. 74 (2011) 7061–7070. Google Scholar
C. Carlota, S. Chá and A. Ornelas, Existence of radially increasing minimizers for nonconvex vectorial multiple integrals in the calculus of variations or optimal control, preprint.
Cellina, A. and Perrotta, S., On minima of radially symmetric fuctionals of the gradient. Nonlinear Anal. 23 (1994) 239–249. Google Scholar
Cellina, A. and Vornicescu, M., On gradient flows. J. Differ. Eqs. 145 (1998) 489–501. Google Scholar
Crasta, G., Existence, uniqueness and qualitative properties of minima to radially-symmetric noncoercive nonconvex variational problems. Math. Z. 235 (2000) 569–589. Google Scholar
Crasta, G., On the minimum problem for a class of noncoercive nonconvex functionals. SIAM J. Control Optim. 38 (1999) 237–253. Google Scholar
Crasta, G. and Malusa, A., Euler-Lagrange inclusions and existence of minimizers for a class of non-coercive variational problems. J. Convex Anal. 7 (2000) 167–181. Google Scholar
I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland, Amsterdam (1976).
Krömer, S., Existence and symmetry of minimizers for nonconvex radially symmetric variational problems. Calc. Var. PDEs 32 (2008) 219–236. Google Scholar
Krömer, S. and Kielhöfer, H., Radially symmetric critical points of non-convex functionals. Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 1261–1280. Google Scholar
P. Pedregal and A. Ornelas (editors), Mathematical methods in materials science and enginneering. CIM 1997 summerschool with courses by N. Kikuchi, D. Kinderlehrer, P. Pedregal. CIM www.cim.pt (1998).
J. Yeh, Lectures on Real Analysis. World Scientific, Singapore (2006).