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Existence and Uniqueness of Green's Functions to Nonlinear Yamabe Problems

“…x N / is uniformly bounded as t 3 1. Furthermore, as u t x u, known interior first derivative estimates [10,22], [37, theorem 1.10], [55], and interior second derivative estimates in [22] and [38, theorem 1.20] give…”

A class of Neumann problems arising in conformal geometry

“…Lemma 2.1 (see [Chen 2005;). Let be an open convex cone with vertex at the origin satisfying + n ⊂ , and let e = (1, .…”

“…where ϕ is a given smooth function on M. If τ = 1 = k and ϕ is constant, (1-1) is just the Yamabe problem, which has been solved by Yamabe, Trudinger, Aubin and Schoen (see [Lee and Parker 1987]). When τ = 1, k ≥ 2 and ϕ is constant, then (1-1) is called k-Yamabe problem, which has attracted enormous interest [Chang et al 2002; Ge and Wang 2006;Guan and Wang 2003a;2003b;Gursky and Viaclovsky 2007;Li and Li 2003;2005;Trudinger and Wang 2009;2010;Viaclovsky 2000], etc. There are many interesting works on the Yamabe problem and k-Yamabe problem on a manifold with boundary [Chen 2007;Escobar 1992b;1992a;Han and Li 1999;2000;He and Sheng 2011a;2011b;, etc.…”

Local pointwise second derivative estimates for strong solutions to the $$\sigma _k$$-Yamabe equation on Euclidean domains

“…(1.2) and their counterparts on Riemannian manifolds were first studied by Viaclovsky in [63]. Since then, these equations have been addressed by various authors -for a partial list of references, see [1][2][3][8][9][10][11][12]14,[16][17][18][19]24,27,28,[31][32][33]35,[39][40][41]43,44,46,47,53,54,56,64,65] in the positive case and [13,23,25,29,30,42,45,55] in the negative case. When k = 1, these equations reduce to the original Yamabe equation.…”

“…A priori local first and second derivative estimates play an important role in the study of the σ k -Yamabe equation, and were established in the positive case by Chen [14], Guan and Wang [27], Jin, Li and Li [39], Li and Li [40], Li [43] and Wang [65]. In the negative case, an a priori (global) C 1 estimate is proven by Gursky and Viaclovsky [30], but it is unknown whether a priori C 2 estimates hold.…”