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.