On any compact Riemannian manifold with no boundary, the integral of the squared Hessian of sqrt(u) is bounded by a universal constant times the integral of u times the squared Hessian of log u.
Winkler , Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm
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A functional inequality between Hessians in spaces with non-zero curvature
On any compact Riemannian manifold with no boundary, the integral of the squared Hessian of sqrt(u) is bounded by a universal constant times the integral of u times the squared Hessian of log u.