Volume 20, Number 1, January-March 2014
|Page(s)||141 - 157|
|Published online||10 December 2013|
Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations∗
Cima-ue, Rua Romão Ramalho 59, 7000-671 Évora, Portugal
Revised: 11 March 2013
We prove uniform continuity of radially symmetric vector minimizers uA(x) = UA(|x|) to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on a ball BR ⊂ ℝd, among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using a jointly convex lsc L∗∗ : 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 optimal control problems, on constrained domains, 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, absolutely continuous and tame, in the sense that its “static level” L∗∗(UA(r),0) always increases with r, a original feature of our result.
Mathematics Subject Classification: 49J10 / 49N60
Key words: Vectorial calculus of variations / vectorial distributed-parameter optimal control / continuous radially symmetric monotone minimizers
The research leading to this paper was performed at: Cima-ue (Math Research Center of Universidade de Évora, Portugal) with financial support from “ Financiamento Plurianual do Cima-ue ” of FCT ( Fundação para a Ciência e a Tecnologia, Portugal ) in 2006 / 2012; CCM ( Math Research Center of Universidade da Madeira, Portugal), during December 2009 by A. Ornelas.
© EDP Sciences, SMAI 2013
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.