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Two approaches to minimax formula of the additive eigenvalue for quasiconvex Hamiltonians

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Atsushi Nakayasu

Analysis of PDEs · Submitted 21 Dec 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton-Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily convex.

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arXiv ID1412.6735 (version 1)
Pith IDXN62C5ZW, Pith's permanent record of this paper and its review
CitationAtsushi Nakayasu. "Two approaches to minimax formula of the additive eigenvalue for quasiconvex Hamiltonians." arXiv:1412.6735 (2014). https://pith.science/paper/XN62C5ZW
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arXiv listing abstract v1. Results and the main claim are underlined; .

Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton-Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily convex. The first proof, which is similar to known proofs for convex Hamiltonians, invokes a Jensen-like inequality for quasiconvex functions instead of the standard Jensen's inequality. The second proof is completely different with elementary calculations. It is based on convergence of derivatives of mollified Lipschitz continuous functions whose proof is also given. These methods also relate to an approximation problem of viscosity solutions.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton-Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily convex. contribution 0.99 0.94 0.81 asserted 1.00
s1 The first proof, which is similar to known proofs for convex Hamiltonians, invokes a Jensen-like inequality for quasiconvex functions instead of the standard Jensen's inequality. method 0.94 0.81 0.09 asserted 1.00
s2 The second proof is completely different with elementary calculations. method 0.82 0.89 0.07 asserted 1.00
s3 It is based on convergence of derivatives of mollified Lipschitz continuous functions whose proof is also given. method 0.99 0.89 0.07 asserted 1.00
s4 These methods also relate to an approximation problem of viscosity solutions. interpretation 0.90 0.68 0.08 asserted 0.96

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Original abstract

Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton-Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily convex. The first proof, which is similar to known proofs for convex Hamiltonians, invokes a Jensen-like inequality for quasiconvex functions instead of the standard Jensen's inequality. The second proof is completely different with elementary calculations. It is based on convergence of derivatives of mollified Lipschitz continuous functions whose proof is also given. These methods also relate to an approximation problem of viscosity solutions.

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