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

  • Luís Balsa Bicho (a1) and António Ornelas (a1)


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 levelL∗∗(UA(r),0) always increases with r, a original feature of our result.



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