Gradient estimates and Harnack inequalities for Yamabe-type parabolic equations on Riemannian manifolds

Gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces and applications

“…Applying the Bochner formula to ℎ and using the inequality |∇ 2 ℎ| ≥ 1 (Δℎ) 2 (obtained by Cauchy-Schwarz inequality) yields…”

“…which is known not to admit any bounded solution different from constant on nonnegative Bakry-Émery Ricci tensor [1]. Suppose one chooses specific values for the functions ( ) and ( ) and constant , (1.1) becomes Yamabe problem for being a constant [2] (see next section and [3] for further discussion). Another special case of (1.1) when ( ) ≡ 0 and restricted to 1 ≤ < ( + 2)∕( − 2) was studied by Gidas and Spruck [4].…”

“…Since the appearance of these works by Li and Yau [45], Hamilton [39] and Souplet and Zhang [61] several results on generalizing, extending and improving classical gradient estimates with several applications have appeared in literature., including but not limited to those ones cited above. See also [4,7,18,19,22,23,28,29,30,33,37,40,44,51,55,56,61,67,69,70]. The gradient estimates discussed in this paper are broadly categorized into two, namely, elliptic type and parabolic type.…”