Regularity of absolute minimizers for continuous convex Hamiltonians

2019/01/08 by Fa Peng, Fa, Peng, Changyou Wang +3 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1901.02379

Abstract

For any n≥ 2, Ω⊂\rn, and any given convex and coercive Hamiltonian function H∈ C⁰(\rn), we find an optimal sufficient condition on H, that is, for any c∈\mathbb R, the level set H^-1(c) does not contains any line segment, such then any absolute minimizer u∈ AM_H(Ω) enjoys the linear approximation property. As consequences, we show that when n=2, if u∈ AM_H(Ω) then u∈ C¹; and if u∈ AM_H(\rr²) satisfies a linear growth at the infinity, then u is a linear function on \rr². In particular, if H is a strictly convex Banach norm ‖⋅‖ on \mathbb R², e.g. the l_α-norm for 1

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