On C1,α-regularity for critical points of a geometric obstacle-type problem

Abstract

We consider critical points of the geometric obstacle problem on vectorial maps u:B2⊂R2→RN∫B2|∇u|2subject to u∈RN\\BN(0). Our main result is C1,α-regularity for any α<1.Technically, we split the map u=λv, where v:B2→SN−1 is the vectorial component and λ=|u| the scalar component measuring the distance to the origin. While v satisfies a weighted harmonic map equation with weight λ2, λ solves the obstacle problem for∫B2|∇λ|2+λ2|∇v|2,subject to λ≥1, where |∇v|2∈L1(B2). We then play ping-pong between the increases in the regularity of λ and v to obtain finally the C1,α-result.