A QUANTITATIVE SOBOLEV REGULARITY FOR ABSOLUTE MINIMIZERS INVOLVING HAMILTONIAN H(p) IN PLANE

Qianyun Miao; Fa Peng; Yuan Zhou

doi:10.1137/21M1443601

A QUANTITATIVE SOBOLEV REGULARITY FOR ABSOLUTE MINIMIZERS INVOLVING HAMILTONIAN H(p) IN PLANE

Qianyun Miao, Fa Peng^*, Yuan Zhou

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

Suppose that H ∈ C^1,1(ℝ²) satisfies that H is locally strongly convex in ℝ² and H(0) = min_p∈_ℝ2 H(p) = 0. Let Ω ⊂ ℝ² be any domain. For any u absolute minimizer for H in Ω, or equivalently, for any viscosity solution to the Aronsson equation A_H[u] = ^P2_i,j₌₁ Hp_i(Du)Hp_j (Du)ux_ix_j = 0 in Ω, the following are proven in this paper: (i) We have [H(Du)]^α ∈ W¹_loc² (Ω) whenever α > 1/2 − τ_H(0); some quantitative upper bounds are also given. Here τ_H(0) = 1/2 when H ∈ C²(ℝ²), and 0 < τ_H(0) ≤ 1/2 in general. (ii) The distributional determinant −detD²u dx is a nonnegative Radon measure in Ω and enjoys some quantitative lower/upper bounds. (iii) For all α > ₂¹ − τ_H(0), we have 〈D[H(Du)]^α, DpH(Du)〉 = 0 almost everywhere in Ω.

Original language	English
Pages (from-to)	5792-5853
Number of pages	62
Journal	SIAM Journal on Mathematical Analysis
Volume	54
Issue number	6
DOIs	https://doi.org/10.1137/21M1443601
Publication status	Published - Dec 2022

Keywords

Aronsson equation
L-variational problem
Sobolev regularity
absolute minimizer
viscosity solution

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title = "A QUANTITATIVE SOBOLEV REGULARITY FOR ABSOLUTE MINIMIZERS INVOLVING HAMILTONIAN H(p) IN PLANE",

abstract = "Suppose that H ∈ C1,1(ℝ2) satisfies that H is locally strongly convex in ℝ2 and H(0) = minp∈ℝ2 H(p) = 0. Let Ω ⊂ ℝ2 be any domain. For any u absolute minimizer for H in Ω, or equivalently, for any viscosity solution to the Aronsson equation AH[u] = P2i,j=1 Hpi(Du)Hpj (Du)uxixj = 0 in Ω, the following are proven in this paper: (i) We have [H(Du)]α ∈ W1loc2 (Ω) whenever α > 1/2 − τH(0); some quantitative upper bounds are also given. Here τH(0) = 1/2 when H ∈ C2(ℝ2), and 0 < τH(0) ≤ 1/2 in general. (ii) The distributional determinant −detD2u dx is a nonnegative Radon measure in Ω and enjoys some quantitative lower/upper bounds. (iii) For all α > 21 − τH(0), we have 〈D[H(Du)]α, DpH(Du)〉 = 0 almost everywhere in Ω.",

keywords = "Aronsson equation, L-variational problem, Sobolev regularity, absolute minimizer, viscosity solution",

author = "Qianyun Miao and Fa Peng and Yuan Zhou",

note = "Publisher Copyright: {\textcopyright} 2022 Society for Industrial and Applied Mathematics.",

year = "2022",

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AU - Peng, Fa

AU - Zhou, Yuan

PY - 2022/12

Y1 - 2022/12

N2 - Suppose that H ∈ C1,1(ℝ2) satisfies that H is locally strongly convex in ℝ2 and H(0) = minp∈ℝ2 H(p) = 0. Let Ω ⊂ ℝ2 be any domain. For any u absolute minimizer for H in Ω, or equivalently, for any viscosity solution to the Aronsson equation AH[u] = P2i,j=1 Hpi(Du)Hpj (Du)uxixj = 0 in Ω, the following are proven in this paper: (i) We have [H(Du)]α ∈ W1loc2 (Ω) whenever α > 1/2 − τH(0); some quantitative upper bounds are also given. Here τH(0) = 1/2 when H ∈ C2(ℝ2), and 0 < τH(0) ≤ 1/2 in general. (ii) The distributional determinant −detD2u dx is a nonnegative Radon measure in Ω and enjoys some quantitative lower/upper bounds. (iii) For all α > 21 − τH(0), we have 〈D[H(Du)]α, DpH(Du)〉 = 0 almost everywhere in Ω.

AB - Suppose that H ∈ C1,1(ℝ2) satisfies that H is locally strongly convex in ℝ2 and H(0) = minp∈ℝ2 H(p) = 0. Let Ω ⊂ ℝ2 be any domain. For any u absolute minimizer for H in Ω, or equivalently, for any viscosity solution to the Aronsson equation AH[u] = P2i,j=1 Hpi(Du)Hpj (Du)uxixj = 0 in Ω, the following are proven in this paper: (i) We have [H(Du)]α ∈ W1loc2 (Ω) whenever α > 1/2 − τH(0); some quantitative upper bounds are also given. Here τH(0) = 1/2 when H ∈ C2(ℝ2), and 0 < τH(0) ≤ 1/2 in general. (ii) The distributional determinant −detD2u dx is a nonnegative Radon measure in Ω and enjoys some quantitative lower/upper bounds. (iii) For all α > 21 − τH(0), we have 〈D[H(Du)]α, DpH(Du)〉 = 0 almost everywhere in Ω.

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KW - L-variational problem

KW - Sobolev regularity

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A QUANTITATIVE SOBOLEV REGULARITY FOR ABSOLUTE MINIMIZERS INVOLVING HAMILTONIAN H(p) IN PLANE

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