On Multiwell Liouville Theorems in Higher Dimensions

We consider certain subsets of the space of n × n matrices of the form K = ∪i=1SO(n)Ai, and we prove that for p > 1, q ≥ 1 and for connected Ω ′ ⊂⊂ Ω ⊂ IR, there exists positive constant a < 1 depending on n, p, q,Ω,Ω such that for ε = ‖dist(Du,K)‖ Lp(Ω) we have infR∈K ‖Du−R‖ p Lp(Ω′) ≤ Mε1/p provided u satisfies the inequality ‖D2u‖ Lq(Ω) ≤ aε1−q . Our main result holds whenever m = 2, and also for generic m ≤ n in every dimension n ≥ 3, as long as the wells SO(n)A1, . . . , SO(n)Am satisfy a certain connectivity condition. These conclusions are mostly known when n = 2, and they are new for n ≥ 3.