The logarithmic Sobolev capacity

Abstract

This paper presents various functional-geometric aspects of the logarithmic Sobolev capacity Caplog⁡,γ,p - a capacity generated by the logarithmic Sobolev space Wplog⁡,γ, where (γ,p)∈(0,∞)×[1,∞). This new nonlinear-inhomogeneous-nontrivial capacity is closely related to the logarithmic Hausdorff capacity. Not only continuously sharp embedding property of Wplog⁡,γ 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 Caplog⁡,γ,p; find the first variation of the logarithmic perimeter via the corresponding logarithmic mean curvature; characterize the hypersurface of a constant logarithmic mean curvature.