Yet another proof of the density in energy of Lipschitz functions

Pub Date : 2024-05-04 DOI:10.1007/s00229-024-01562-2
Danka Lučić, Enrico Pasqualetto
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Abstract

We provide a new, short proof of the density in energy of Lipschitz functions into the metric Sobolev space defined by using plans with barycenter (and thus, a fortiori, into the Newtonian–Sobolev space). Our result covers first-order Sobolev spaces of exponent \(p\in (1,\infty )\), defined over a complete separable metric space endowed with a boundedly-finite Borel measure. Our proof is based on a completely smooth analysis: first we reduce the problem to the Banach space setting, where we consider smooth functions instead of Lipschitz ones, then we rely on classical tools in convex analysis and on the superposition principle for normal 1-currents. Along the way, we obtain a new proof of the density in energy of smooth cylindrical functions in Sobolev spaces defined over a separable Banach space endowed with a finite Borel measure.

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立普次函数能量密度的另一个证明
我们提供了一个新的、简短的证明,证明了利普齐兹函数进入由带原心的计划定义的度量索博廖夫空间的能量密度(因此,更不用说进入牛顿-索博廖夫空间的能量密度)。我们的结果涵盖了指数(p\in (1,\infty )\)的一阶 Sobolev 空间,它定义在一个禀赋有界有限 Borel 度量的完全可分离度量空间上。我们的证明基于完全平滑的分析:首先,我们把问题还原到巴拿赫空间环境中,在那里我们考虑平滑函数而不是 Lipschitz 函数,然后我们依靠凸分析中的经典工具和法向 1 流的叠加原理。在此过程中,我们获得了一个新的证明,即在定义于可分离巴拿赫空间并赋有有限伯勒尔度量的索波列夫空间中,光滑圆柱函数的能量密度。
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