Uniqueness of Absolute Minimizers for L∞ -Functionals Involving Hamiltonians H(x, p)

Abstract

For a bounded domain U⊂ Rn, consider the L∞-functional involving a non-negative Hamilton function H: U¯ × Rn→ [0 , ∞). In this paper, we will establish the uniqueness of absolute minimizers u∈Wloc1,∞(U)∩C(U¯) for H, under the Dirichlet boundary value g∈ C(∂U), provided that: (A1) H is lower semicontinuous in U¯ × Rn, and H(x, ·) is convex for any x∈ U¯. (A2) H(x,0)=minp∈RnH(x,p)=0 for any x∈ U¯ , and ⋃ x∈U¯{ p: H(x, p) = 0 } is contained in a hyperplane of Rn. (A3) For any λ > 0 , there exist 0 < rλ≦ Rλ< ∞, with lim λ→∞rλ= ∞, such that (Formula presented.). This generalizes the uniqueness theorem by Jensen (Arch Ration Mech Anal 123:51–74, 1993), Jensen et al. (Arch Ration Mech Anal 190:347–370, 2008), Armstrong et al. (Arch Ration Mech Anal 200:405–443, 2011) and Koskela et al. (Arch Ration Mech Anal 214:99–142, 2014) to a large class of Hamiltonian functions H(x, p) with x-dependence. As a corollary, we confirm an open question on the uniqueness of absolute minimizers posed by Jensen et al. (Arch Ration Mech Anal 190:347–370, 2008). The proofs rely on the geometric structure of the action function Lt(x, y) induced by H, and the identification of the absolute subminimality of u with convexity of the Hamilton–Jacobi flow t↦ Ttu(x).