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