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

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

Received: 6 November 2012
Revised: 11 March 2013

Abstract

We prove uniform continuity of radially symmetric vector minimizers u_A(x) = U_A(|x|) to multiple integrals ∫_BRL^**(u(x), |Du(x)|) dx on a ball B_R ⊂ ℝ^d, among the Sobolev functions u(·) in A+W₀^1,1 (B_R, ℝ^m), using a jointly convex lsc L^∗∗ : ℝ^m×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 U_A(·) is, in contrast, absolutely continuous and tame, in the sense that its “static level” L^∗∗(U_A(r),0) always increases with r, a original feature of our result.