Comparison between the non-crossing and the non-crossing on lines properties. (English) Zbl 1471.46031
Summary: In the recent paper [G. De Philippis and A. Pratelli, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 37, No. 1, 181–224 (2020; Zbl 1446.46018)], it was proved that the closure of the planar diffeomorphisms in the Sobolev norm consists of the functions which are non-crossing (NC), i.e., the functions which can be uniformly approximated by continuous one-to-one functions on grids. A deep simplification of this property is to consider curves instead of grids, so considering functions which are non-crossing on lines (NCL). Since the NCL property is much easier to verify, it would be extremely positive if they actually coincide, while it is only obvious that NC implies NCL. We show that in general NCL does not imply NC, but the implication becomes true with the additional assumption that \(\det(D u) > 0\) a.e., which is a very common assumption in nonlinear elasticity.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 26B10 | Implicit function theorems, Jacobians, transformations with several variables |
| 58C25 | Differentiable maps on manifolds |
Citations:
Zbl 1446.46018References:
| [1] | De Philippis, G.; Pratelli, A., The closure of planar diffeomorphisms in Sobolev spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 37, 1, 181-224 (2020) · Zbl 1446.46018 |
| [2] | Müller, S.; Spector, S. J., An existence theory for nonlinear elasticity that allows for cavitation, Arch. Ration. Mech. Anal., 131, 1-66 (1995) · Zbl 0836.73025 |
| [3] | Šverák, V., Regularity properties of deformations with finite energy, Arch. Ration. Mech. Anal., 100, 105-127 (1988) · Zbl 0659.73038 |
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