Advances in Mathematics ( IF 1.5 ) Pub Date : 2021-09-16 , DOI: 10.1016/j.aim.2021.107993 Liguang Liu 1 , Suqing Wu 2, 3 , Jie Xiao 4 , Wen Yuan 2
This paper presents various functional-geometric aspects of the logarithmic Sobolev capacity - a capacity generated by the logarithmic Sobolev space , where . This new nonlinear-inhomogeneous-nontrivial capacity is closely related to the logarithmic Hausdorff capacity. Not only continuously sharp embedding property of into certain logarithmic weighted Lebesgue space and the tracing inequalities are established, but also the logarithmic perimeter and the Lebesgue measure are utilized to: reformulate ; find the first variation of the logarithmic perimeter via the corresponding logarithmic mean curvature; characterize the hypersurface of a constant logarithmic mean curvature.
中文翻译:
对数 Sobolev 容量
本文介绍了对数 Sobolev 容量的各种功能几何方面 - 由对数 Sobolev 空间生成的容量 , 在哪里 . 这种新的非线性非齐次非平凡容量与对数 Hausdorff 容量密切相关。不仅具有连续尖锐的嵌入特性 进入某些对数加权 Lebesgue 空间并建立追踪不等式,但也利用对数周长和 Lebesgue 测度来: ; 通过相应的对数平均曲率找到对数周长的第一个变化;表征恒定对数平均曲率的超曲面。