POINTWISE CONSTRAINED RADIALLY INCREASING MINIMIZERS IN THE QUASI-SCALAR CALCULUS OF VARIATIONS

Abstract

We prove uniform continuity of radially symmetric vector minimizers uA(x) = UA ( jxj ) to multiple integrals R BR L ( u(x); jDu(x) j ) d x on a ball BR Rd, among the Sobolev functions u( ) in A + W 1;1 0 (BR; Rm ), using a jointly convex lsc L : Rm R ! [0;1] 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 ( j u(x) j ; jDu(x) j ). Complete liberty is given for L ( S; ) to take the 1 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 ( jxj ) in this Sobolev space oscillate wildly as jxj ! 0, our minimiz- ing 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.