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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 uniformcontinuity of radiallysymmetric vector minimizers uA(x) = UA(|x|) to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on a ballBR ⊂ ℝd, among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using a jointlyconvexlscL∗∗ : ℝm×ℝ → [0,∞] with L∗∗(S,·) even and superlinear. Besides such basic hypotheses, L∗∗(·,·) is assumed to satisfy also a geometrical constraint, which we call quasi − scalar; the simplest example being the biradial case L∗∗(|u(x)|,|Du(x)|). Complete liberty is given for L∗∗(S,λ) to take the ∞ value, so that our minimization problem implicitly also represents e.g. distributed-parameter optimalcontrol problems, on constraineddomains, under PDEs or inclusions in explicit or implicit form. While generic radial functions u(x) = U(|x|) in this Sobolev space oscillate wildly as |x| → 0, our minimizing profile-curve UA(·) is, in contrast, absolutelycontinuous and tame, in the sense that its “staticlevel” 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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