Pogorelov type estimates for a class of Hessian quotient equations

Pogorelov type estimates for a class of Hessian quotient equations in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$

“…The last inequality in (3.4) holds because of the estimate (3.3). If one goes back to the proof of [7, Theorem 1.1], he or she would find that for calculations done at a point, the above inequality has nearly same form with (3.5) of [7] except the last term in the RHS -the last term in the RHS of (3.4) is AT 11 (n − 1) coth(c + ) while the one in (3.5) of [7] is A n i=1 T ii . However, the main role of A in the evaluation of each term in (3.5) of [7] is to absorb those negative terms appearing.…”

“…If one goes back to the proof of [7, Theorem 1.1], he or she would find that for calculations done at a point, the above inequality has nearly same form with (3.5) of [7] except the last term in the RHS -the last term in the RHS of (3.4) is AT 11 (n − 1) coth(c + ) while the one in (3.5) of [7] is A n i=1 T ii . However, the main role of A in the evaluation of each term in (3.5) of [7] is to absorb those negative terms appearing. Since, in our setting here, T ii (x 0 ) can be controlled at x 0 and (n − 1) coth(c + ) is a constant (which will not break the role of A), with the help of properties of σ k -operator and structure equations listed in Section 2, our Pogorelov type estimates in Theorem 1.1 follows by using a similar argument (with necessary modifications 6 ) to the rest part of the proof of [7,Theorem 1.1].…”

“…In this section, we will use an idea (which is similar to that in [7,9]) to give the proof of Theorem 1.1.…”

“…(6) In fact, one can consider the problem (1.6) defined over more general bounded domains (with smooth boundary) in Lorentz-Minkowski space R n+1 1 , and similar conclusion can be obtainedsee Remark 3.1 for details. (7) Since in our previous works [15,16], the PCPs therein were considered for spacelike graphic functions defined over bounded domains (with boundary) in the hyperbolic plane…”

“…Proof. See [7,Proposition 2.3] for the proof of (2.10). The proof of (2.11) is almost the same with that of [2, Lemma 2.2] and we prefer to omit here.…”