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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. lmbb@uevora.pt; antonioornelas@icloud.com
António Ornelas
Affiliation:
Cima-ue, Rua Romão Ramalho 59, 7000-671 Évora, Portugal. lmbb@uevora.pt; antonioornelas@icloud.com
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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

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