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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have