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The problem of properly defining Sobolev spaces on a subset of the boundary of a domain is briefly addressed in the appendix of H. Berninger's dissertation (248pp): page.mi.fu-berlin.de/haiko/dissertation_Berninger.pdf You might especially find the references valuable.
I meant that positivity of $f$ does not help in determining whether or not $log \circ f$ is concave. My answer should've been a reply to wmmiao's answer of course but I lack the permission to reply.
