Mollifier smoothing of C0-Finsler structures

Fukuoka, Ryuichi; Setti, Anderson Macedo

doi:10.1007/s10231-020-01007-z

Cited by 2 publications

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“…The term C 0 -Finsler structure given in Definition 2.3 coincides with the third definition of Finsler structure above. It was first used in [22] and it is a natural generalization of (smooth) Finsler structure.…”

Section: Preliminariesmentioning

confidence: 99%

“…In this case the theory is much less developed due to the lack of a model theory (like Riemannian geometry) and also due to the lack of tools in order to study variational problems in detail. In [22,37] the authors consider a general C 0 -Finsler structure F and create a one-parameter family of Finsler structures (F ε ) ε∈(0,1) that converges uniformly to F on compact subsets of T M . This smoothing works properly on Finsler structures F , that is, several connections of Finsler geometry and the flag curvatures of (M, F ε ) converges uniformly on compact subsets to the respective objects of (M, F ).…”

Section: Introductionmentioning

confidence: 99%

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Geodesic fields for Pontryagin type C⁰-Finsler manifolds

Rodrigues

Fukuoka

2022

ESAIM: COCV

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Let M be a differentiable manifold, T x M be its tangent space at x ε M and TM={(x,y);x ε M; y ε T x M} be its tangent bundle. A C 0 -Finsler structure is a continuous function F:TM → [0, ∞ ) such that F(x, · ):T x M → [0, ∞ ) is an asymmetric norm. In this work we introduce the Pontryagin type C 0 -Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin's maximum principle for the problem of minimizing paths. We define the extended geodesic field E on the slit cotangent bundle T * M\0 of (M,F) , which is a generalization of the geodesic spray of Finsler geometry. We study the case where E is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by E than by a similar structure on TM . Finally we show that the maximum of independent Finsler structures is a Pontryagin type C 0 -Finsler structure where E is a locally Lipschitz vector field.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%