scholarly journals $p$-Dirichlet energy minimizing maps into a complete manifold.

1996 ◽  
Vol 43 (1) ◽  
pp. 33-49
Author(s):  
Shiah-Sen Wang
Keyword(s):  
Dirichlet Energy ◽  
Complete Manifold ◽  
Minimizing Maps

scholarly journals Path Planning and Replanning for Mobile Robot Navigation on 3D Terrain: An Approach Based on Geodesic

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Kun-Lin Wu ◽  
Ting-Jui Ho ◽  
Sean A. Huang ◽  
Kuo-Hui Lin ◽  
Yueh-Chen Lin ◽  
...  
Keyword(s):  
Mobile Robot ◽  
Robot Navigation ◽  
Tangent Plane ◽  
Intersection Point ◽  
Mobile Robot Navigation ◽  
Complete Manifold ◽  
Geodesic Triangle ◽  
Planning Stage ◽  
3D Terrain ◽  
Bonnet Theorem

In this paper, mobile robot navigation on a 3D terrain with a single obstacle is addressed. The terrain is modelled as a smooth, complete manifold with well-defined tangent planes and the hazardous region is modelled as an enclosing circle with a hazard grade tuned radius representing the obstacle projected onto the terrain to allow efficient path-obstacle intersection checking. To resolve the intersections along the initial geodesic, by resorting to the geodesic ideas from differential geometry on surfaces and manifolds, we present a geodesic-based planning and replanning algorithm as a new method for obstacle avoidance on a 3D terrain without using boundary following on the obstacle surface. The replanning algorithm generates two new paths, each a composition of two geodesics, connected via critical points whose locations are found to be heavily relying on the exploration of the terrain via directional scanning on the tangent plane at the first intersection point of the initial geodesic with the circle. An advantage of this geodesic path replanning procedure is that traversability of terrain on which the detour path traverses could be explored based on the local Gauss-Bonnet Theorem of the geodesic triangle at the planning stage. A simulation demonstrates the practicality of the analytical geodesic replanning procedure for navigating a constant speed point robot on a 3D hill-like terrain.

The relaxed Dirichlet energy of mappings into a manifold

2005 ◽  
Vol 24 (2) ◽  
pp. 155-166 ◽  
Author(s):  
Mariano Giaquinta ◽  
Domenico Mucci
Keyword(s):  
Dirichlet Energy

scholarly journals Minimal Dirichlet Energy Partitions for Graphs

2014 ◽  
Vol 36 (4) ◽  
pp. A1635-A1651 ◽  
Author(s):  
Braxton Osting ◽  
Chris D. White ◽  
Édouard Oudet
Keyword(s):  
Dirichlet Energy

scholarly journals ON THE DEFINITIONS OF SOBOLEV AND BV SPACES INTO SINGULAR SPACES AND THE TRACE PROBLEM

2007 ◽  
Vol 09 (04) ◽  
pp. 473-513 ◽  
Author(s):  
DAVID CHIRON
Keyword(s):  
Sobolev Spaces ◽  
Metric Spaces ◽  
The Other ◽  
Dirichlet Energy ◽  
Singular Spaces ◽  
Other Hand ◽  
The One

The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to Korevaar and Schoen on the one hand, and Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to Ambrosio (for BV maps into metric spaces), Reshetnyak and finally to the notion of Newtonian–Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of Bourgain, Brezis and Mironescu in terms of "limit" of the space Ws,p as s → 1, 0 < s < 1, and finally following the approach proposed by Nguyen. We also establish the [Formula: see text] regularity of traces of maps in Ws,p (0 < s ≤ 1 < sp).

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