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- W3103161106 abstract "For a bounded domain $${Usubset{{mathbb{R}}^n}}$$ , consider the $${L^infty}$$ -functional involving a non-negative Hamilton function $${H:{overline{U}}times{{mathbb{R}}^n}to [0,infty)}$$ . In this paper, we will establish the uniqueness of absolute minimizers $${uin W^{1,infty}_{rm loc}(U)cap C({overline{U}})}$$ for $${H}$$ , under the Dirichlet boundary value $${gin C(partial U)}$$ , provided that: (A1) $${H}$$ is lower semicontinuous in $${overline Utimes{mathbb{R}}^n}$$ , and $${H(x,cdot)}$$ is convex for any $${xin{overline{U}}}$$ . (A2) $${ H(x,0)=min_{pin {{mathbb{R}}^n}}H(x,p)=0}$$ for any $${ xin {overline{U}}}$$ , and $${bigcup_{xin overline U}{p: H(x,p)=0}}$$ is contained in a hyperplane of $${{mathbb{R}}^n}$$ . (A3) For any $${lambda > 0}$$ , there exist $${ 0 < r_lambdaleq R_lambda < infty}$$ , with $${lim_{lambdatoinfty}r_lambda=infty}$$ , such that $$B(0,r_lambda)subset {pin{mathbb{R}}^n | H(x,p) < lambda }subset B(0,R_lambda) forall lambda > 0 {rm and} xin {overline{U}}.$$ 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 $${{mathcal{L}}_t(x,y)}$$ induced by $${H}$$ , and the identification of the absolute subminimality of $${u}$$ with convexity of the Hamilton–Jacobi flow $${tmapsto T^t u(x)}$$ ." @default.
- W3103161106 created "2020-11-23" @default.
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- W3103161106 date "2016-08-16" @default.
- W3103161106 modified "2023-09-26" @default.
- W3103161106 title "Uniqueness of Absolute Minimizers for $${L^infty}$$-Functionals Involving Hamiltonians $${H(x,p)}$$" @default.
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- W3103161106 doi "https://doi.org/10.1007/s00205-016-1033-8" @default.
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