Interior C^{1,1} regularity of solutions to degenerate Monge-Ampère type equations
aa r X i v : . [ m a t h . A P ] J un INTERIOR C , REGULARITY OF SOLUTIONS TO DEGENERATEMONGE-AMP`ERE TYPE EQUATIONS
FEIDA JIANG, JUHUA SHI, AND XIAO-PING YANG ∗ Abstract.
In this paper, we study the interior C , regularity of viscosity solutions for a degenerateMonge-Amp`ere type equation det[ D u − A ( x, u, Du )] = B ( x, u, Du ) when B ≥ B n − ∈ C , ( ¯Ω × R × R n ). We prove that u ∈ C , (Ω) under the A3 condition and A3w + condition respectively. In theformer case, we construct a suitable auxiliary function to obtain uniform a priori estimates directly. Inthe latter case, the main argument is to establish the Pogorelov type estimates, which are interestingindependently. Introduction
In this paper, we shall study the following degenerate Monge-Amp`ere type equation (DMATE)(1.1) det[ D u − A ( · , u, Du )] = B ( · , u, Du ) , in Ω , where Ω is a bounded domain, Du and D u denote the gradient and Hessian matrix of second orderderivatives of the unknown function u : Ω → R respectively, A : Ω × R × R n → R n × n is a symmetric n × n matrix valued function and A ∈ C , ( ¯Ω × R × R n , R n × n ), B : Ω × R × R n → R + ∪ { } is anonnegative scalar function and B n − ∈ C , ( ¯Ω × R × R n ). We shall use x, z and p to denote the pointsin Ω , R and R n , respectively.We say that A is strictly regular in Ω, if(1.2) n X i,j,k,l =1 D p k p l A ij ( x, z, p ) ξ i ξ j η k η l ≥ c | ξ | | η | , holds for all ( x, z, p ) ∈ Ω × R × R n , ξ, η ∈ R n with ξ · η = 0, and some positive constant c . If c on the right hand side in (1.2) is replaced by 0, we say that A is regular in Ω. As usual, the strictlyregular condition and regular condition are also said to be the A3 condition and the A3w condition,respectively, see [16, 17]. If (1.2) holds for c = 0 without the restriction ξ · η = 0, we call (1.2) theregular condition without orthogonality or the A3w condition without orthogonality. We introduce aparticular form of A3w condition, namely(1.3) n X i,j,k,l =1 D p k p l A ij ( x, z, p ) ξ i ξ j η k η l ≥ µ ( ξ · η ) , holds for all ( x, z, p ) ∈ Ω × R × R n , ξ, η ∈ R n , and some constant µ . We call (1.3) the A3w + condition. It is obvious that the A3w + condition implies the A3w condition. The A3w conditionwithout orthogonality implies the A3w + condition when µ ≤ Date : September 25, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Degenerate Monge-Amp`ere type equations; interior regularity; interior second derivative esti-mate; Pogorelov type estimate.This work was supported by the National Natural Science Foundation of China (No. 11771214, 11531005).*corresponding author. f solutions to Monge-Amp`ere equations. When A ≡
0, the equation (1.1) reduces to the classicalMonge-Amp`ere equation. For the case B ≥ B > B , the Pogorelov estimate forthe equation (1.1) together with the homogeneous Dirichlet boundary condition u = 0 on ∂ Ω was firstproved by Pogorelov [18]. Various versions of Pogorelov estimates for nondegenerate Monge-Amp`ereequations can be found in [4, 5, 8, 20]. For the case
B >
0, Blocki [1] proved(1.4) ( w − u ) α | D u | ≤ C, in Ω , where α = n − n ≥ α > n = 2, w ∈ C (Ω) is convex satisfying u ≤ w in Ω and lim x → ∂ Ω ( w ( x ) − u ( x )) = 0, and the constant C is independent of the lower bound of B . When A
0, the Monge-Amp`ere type equations (1.1) arise in various aspects such as optimal mass transportation problems,geometric optics and conformal geometry etc (see, for instance [9, 11, 17, 19]). The Pogorelov typeestimates of non-degenerate Monge-Amp`ere type equations were established under the assumptionsof A3w and A -boundedness conditions in [14, 15]. Without the A -boundedness condition, the interiorsecond order derivative estimates of Pogorelov type were also shown to be valid in [9] by constructing adifferent barrier function with the help of an admissible function. In the optimal mass transportationsetting, interior C regularity for non-degenerate Monge-Am`epre type equations was obtained underthe A3 condition in [17].In this paper, we investigate the interior regularity of a viscosity solution u to the degenerate Monge-Amp`ere type equation (1.1). By constructing a suitable auxiliary function to directly obtain uniform a priori estimates of second order derivatives, we first prove that u ∈ C , (Ω) under the A3 condition.Then we relax the A3 condition to the A3w + condition, by assuming some suitable additional condi-tions, we establish the Pogorelov type estimates, which are independently interesting, and further showthat the solution u has interior C , regularity.More precisely, we have the following main results. Theorem 1.1.
Let u ∈ C (Ω) ∩ C , ( ¯Ω) be a solution of the equation (1.1) in a bounded domain Ω ⊂ R n , where B is a positive function and B n − ∈ C , ( ¯Ω × R × R n ) . Assume that (1.5) D pp ˜ B ≥ − C B I, for some nonnegative constant C B , where I is the n × n identity matrix and ˜ B = log B . Assume that A ∈ C ( ¯Ω × R × R n , R n × n ) is strictly regular. Then, we have (1.6) | D u ( x ) | ≤ C, where C depends on n , dist( x, ∂ Ω) , sup Ω | Du | , k B n − k C , , k A k C and c . Before stating the next theorem, we first define the viscosity solution of the equation (1.1). Afunction u is called a viscosity subsolution (supersolution) of the equation (1.1), if for any function φ ∈ C (Ω) such that u − φ has a local maximum (minimum) at some point x ∈ Ω, there holds(1.7) det[ D φ ( x ) − A ( x , φ ( x ) , Dφ ( x ))] ≥ ( ≤ ) B ( x , φ ( x ) , Dφ ( x )) . A function u is a viscosity solution of the equation (1.1) if it is both a viscosity subsolution and aviscosity supersolution of the equation (1.1). Theorem 1.2.
Let Ω be a bounded domain in R n , and u be a viscosity solution of the equation (1.1) .Assume that A ∈ C ( ¯Ω × R × R n , R n × n ) is strictly regular, B is a nonnegative function, B n − ∈ C , ( ¯Ω × R × R n ) and B satisfies the condition (1.5) . Then, we have u ∈ C , (Ω) . Note that the constant c in Theorem 1.1 is from the strictly regular condition (1.2) of the matrix A .The second order derivative estimate (1.6) depends on c , which will blow up when c tends to 0. Inthis sense, Theorems 1.1 and 1.2 are not valid for the interior second order derivative estimate underthe A3w condition. owever, we can still obtain the interior C , regularity for the degenerate Monge-Amp`ere typeequation (1.1) under the A3w + condition with the help of suitable barrier functions. In order toconstruct the barrier functions, we can assume either the A -boundedness condition or the existence ofa strict subsolution.First, we introduce the A -boundedness condition as in [14, 19]. We say that the A -boundednesscondition holds, if there exists a function ϕ ∈ C ( ¯Ω) satisfying(1.8) [ D ij ϕ − D p k A ij ( x, z, p ) D k ϕ ( x )] ξ i ξ j ≥ | ξ | , for all ξ ∈ R n , ( x, z, p ) ∈ Ω × R × R n .Next, we introduce the definition of a strict subsolution of the equation (1.1). A function u ∈ C (Ω)is called an elliptic (a degenerate elliptic) function when its augmented Hessian matrix M [ u ] := D u − A ( x, u, Du ) > ≥ u is also a solution of the equation (1.1), we call it an elliptic (a degenerateelliptic) solution. A function u ∈ C (Ω) is said to be elliptic (degenerate elliptic) with respect to u inΩ, if M u [ u ] := D u − A ( · , u, Du ) > ≥
0) in Ω. If such a function u also satisfies(1.9) det( M u [ u ]) > B ( · , u, Du ) , at points in Ω, we call u a strict subsolution of the equation (1.1).We now formulate the Pogorelov type estimate under A3w + in the following theorem. Theorem 1.3.
Let u ∈ C (Ω) ∩ C , ( ¯Ω) be a solution of the equation (1.1) in a bounded domain Ω ⊂ R n , where B is a positive function, B n − ∈ C , ( ¯Ω × R × R n ) and B satisfies the condition (1.5) .Assume that A ∈ C ( ¯Ω × R × R n , R n × n ) satisfies the A3w + condition, and there exists a C , function w satisfying w ≥ u in Ω , w = u on ∂ Ω , which is degenerate elliptic with respect to u in Ω . Assumealso one of the following conditions: (i) A -boundedness condition (1.8) holds; (ii) there exists a strict subsolution u ∈ C (Ω) of the equation (1.1) satisfying (1.9) .Then we have the estimate (1.10) ( w − u ) τ | D u | ≤ C, in Ω , where τ = 2 if B p and τ = 1 if B p ≡ , the constant C depends on n , Ω , k B n − k C , , k A k C , sup Ω | Dw | , sup Ω | Du | . In case (ii), the constant C depends in addition on u . There is a technical reason why we restrict our attention under the A3w + condition, see Remark 4.1after the proof of Theorem 1.3. Remark . We remark that, in Theorem 1.3, if B satisfies a further condition | B p | B ≤ C for some non-negative constant C , then the estimate (1.10) can be improved to ( w − u ) | D u | ≤ C , which correspondsto the estimate (1.10) for the B p ≡ Theorem 1.4.
Under the assumptions of Theorem 1.3, assume instead that u is a viscosity solution ofthe equation (1.1) and B is a nonnegative function, and assume further that A and B are nondecreasingin z . Then we have u ∈ C , (Ω) . In order to guarantee the comparison principle, the monotonicity conditions for both A and B withrespect to z are assumed in Theorem 1.4. Remark . We emphasize that the constants C in both the estimates (1.6) in Theorem 1.1 and (1.10)in Theorem 1.3 are independent of the positive lower bound of B , so that they can be applied to obtainthe interior C , regularity for the degenerate equation (1.1). The assumption B n − ∈ C , can befound in [6, 7], which is proved to be optimal in [21] when A ≡ B is independent of z and p . hen µ ≤
0, the matrix A ≡ + condition (1.3) automatically, so that Theorem 1.3and 1.4 can apply to the standard Monge-Amp`ere equation det D u = B ( · , u, Du ).The organization of this paper is as follows. In Section 2, we introduce some properties of B when B n − ∈ C , , in Lemma 2.1 and Corollary 2.1, which are useful in deriving estimates independent ofthe lower bound of B . A fundamental barrier construction under the A3w condition is also introducedin Lemma 2.2, which will be used in Section 4 when we only assume the A3w + condition. In Section 3,we obtain interior second order derivative estimates for the Monge-Amp`ere type equation (1.1) underA3 condition, and then show the interior C , regularity for viscosity solutions of the DMATE (1.1). InSection 4, under the A3w + condition, we establish the Pogorelov type estimates for the Monge-Amp`eretype equation (1.1) by using suitable barrier functions, and apply these estimates to obtain interior C , regularity for viscosity solutions of the DMATE (1.1).2. Preliminaries
In this section, we introduce some properties of B when B n − ∈ C , ( ¯Ω × R × R n ), and a fundamentallemma of barrier construction, which will be used in later sections.In the equation (1.1), we suppose B > u ij := u ij − A ij and { ˜ u ij } := { ˜ u ij } − . Then bothmatrices { ˜ u ij } and { ˜ u ij } are positive definite. We can rewrite the equation (1.1) in the form(2.1) log det { ˜ u ij } = ˜ B, in Ω , where ˜ B := log B . By differentiating the equation (2.1) in the direction ξ ∈ R n once and twicerespectively, we have(2.2) ˜ u ij [ D ξ u ij − D ξ A ij − ( D z A ij ) D ξ u − ( D p k A ij ) D ξ u k ] = D ξ ˜ B, and ˜ u ij [ D ξξ u ij − D ξξ A ij − ( D p k p l A ij ) D ξ u k D ξ u l − ( D p k A ij ) D ξξ u k − ( D z A ij ) D ξξ u − ( D zz A ij )( D ξ u ) − D ξz A ij ) D ξ u − D zp k A ij ) D ξ uD ξ u k − D ξp k A ij ) D ξ u k ]=˜ u is ˜ u jt D ξ ˜ u ij D ξ ˜ u st + D ξξ ˜ B, (2.3)where(2.4) D ξ ˜ B = B ξ + B z D ξ u + B p k D ξ u k B , and D ξξ ˜ B = B ξξ + B zz ( D ξ u ) + B z D ξξ u + B p k D ξξ u k + B p k p l ( D ξ u l )( D ξ u k ) B + 2 B ξz D ξ u + 2 B ξp l D ξ u l + 2 B zp k ( D ξ u )( D ξ u k ) B − B ξ + B z ( D ξ u ) + B p k B p l ( D ξ u k )( D ξ u l ) B − B ξ B z u ξ + 2 B ξ B p l D ξ u l + 2 B z B p l ( D ξ u )( D ξ u l ) B . (2.5)Note that we use the standard summation convention in the context that repeated indices indicatesummation from 1 to n unless otherwise specified.We introduce the following lemma and its corollary, in order to deal with the right-hand side termof the equation (1.1). emma 2.1. Assume B n − ( x, u, Du ) ∈ C , ( ¯Ω × R × R n ) and B > , then we have (2.6) (cid:12)(cid:12)(cid:12)(cid:12) B i B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B z B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B p i B (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − q k B n − k C , (¯Ω × R × R n ) B − n − , in ¯Ω × R × R n , for i = 1 , · · · , n , and (2.7) (cid:12)(cid:12)(cid:12)(cid:12) B ij B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B iz B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B ip j B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B zz B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B zp i B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B p i p j B (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − n − k B n − k C , (¯Ω × R × R n ) B − n − , in ¯Ω × R × R n , for i, j = 1 , · · · , n .Proof. By Taylor’s formula, for any given ( x , z , p ) ∈ ¯Ω × R × R n ,0 ≤ B n − ( x, z, p ) ≤ B n − ( x , z , p ) + ∇ (cid:16) B n − (cid:17) ( x , z , p ) · ( x − x , z − z , p − p )+ 12 k B n − k C , (¯Ω × R × R n ) (cid:2) | x − x | + | z − z | + | p − p | (cid:3) , (2.8)holds for any ( x, z, p ) ∈ ¯Ω × R × R n , where ∇ := ( D x , D z , D p ). Kirszbraun’s Theorem (in Section12.10.43 in [3]) asserts that there exists an extension from ¯Ω × R × R n to R n × R × R n such that B n − ∈ C , ( R n × R × R n ) and k B n − k C , ( R n × R × R n ) = k B n − k C , (¯Ω × R × R n ) , then (2.8) holds for all( x, z, p ) ∈ R n × R × R n . Consequently, we have(2.9) (cid:16) ( B n − ) i ( x , z , p ) (cid:17) − k B n − k C , (¯Ω × R × R n ) B n − ( x , z , p ) ≤ , for i = 1 , · · · , n, (2.10) (cid:16) ( B n − ) z ( x , z , p ) (cid:17) − k B n − k C , (¯Ω × R × R n ) B n − ( x , z , p ) ≤ , and(2.11) (cid:16) ( B n − ) p i ( x , z , p ) (cid:17) − k B n − k C , (¯Ω × R × R n ) B n − ( x , z , p ) ≤ , for i = 1 , · · · , n, namely,(2.12) (cid:12)(cid:12)(cid:12) ( B n − ) i ( x , z , p ) (cid:12)(cid:12)(cid:12) ≤ q k B n − k C , (¯Ω × R × R n ) B n − ( x , z , p ) , for i = 1 , · · · , n, (2.13) (cid:12)(cid:12)(cid:12) ( B n − ) z ( x , z , p ) (cid:12)(cid:12)(cid:12) ≤ q k B n − k C , (¯Ω × R × R n ) B n − ( x , z , p ) , and(2.14) (cid:12)(cid:12)(cid:12) ( B n − ) p i ( x , z , p ) (cid:12)(cid:12)(cid:12) ≤ q k B n − k C , (¯Ω × R × R n ) B n − ( x , z , p ) , for i = 1 , · · · , n. By (2.12), (2.13) and (2.14), we have(2.15) (cid:12)(cid:12)(cid:12)(cid:12) B i B ( x , z , p ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − q k B n − k C , (¯Ω × R × R n ) B − n − ( x , z , p ) , for i = 1 , · · · , n, (2.16) (cid:12)(cid:12)(cid:12)(cid:12) B z B ( x , z , p ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − q k B n − k C , (¯Ω × R × R n ) B − n − ( x , z , p ) , and(2.17) (cid:12)(cid:12)(cid:12)(cid:12) B p i B ( x , z , p ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − q k B n − k C , (¯Ω × R × R n ) B − n − ( x , z , p ) , for i = 1 , · · · , n. Since ( x , z , p ) can be an arbitrary point in ¯Ω × R × R n , from (2.15), (2.16) and (2.17), conclusion(2.6) is proved. ext, by a direct computation, we obtain(2.18) D ij (cid:16) B n − (cid:17) = 1 n − B n − (cid:18) B ij B − n − n − B i B B j B (cid:19) , in ¯Ω × R × R n , for i, j = 1 , · · · , n . Therefore, we have from (2.18) that(2.19) (cid:12)(cid:12)(cid:12)(cid:12) B ij B (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − (cid:12)(cid:12)(cid:12) ( B n − ) ij (cid:12)(cid:12)(cid:12) B − n − + n − n − (cid:12)(cid:12)(cid:12)(cid:12) B i B (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) B j B (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − n − k B n − k C , (¯Ω × R × R n ) B − n − , in ¯Ω × R × R n , where (2.6) is used in the last inequality. Then (2.19) completes the proof of the firstinequality in (2.7). The other inequalities in (2.7) can be derived similarly to (2.19). We omit theremaining proof, in order to avoid too many repetitions. (cid:3) Remark . In fact, we can have a relaxed version of the estimate (2.6),(2.20) (cid:12)(cid:12)(cid:12)(cid:12) B i B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B z B (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) B p i B (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − k B n − k C , (¯Ω × R × R n ) B − n − , in ¯Ω × R × R n , for i = 1 , · · · , n , which can be readily verified by a direct calculation. Namely, we have(2.21) (cid:12)(cid:12)(cid:12)(cid:12) B i B (cid:12)(cid:12)(cid:12)(cid:12) = ( n − (cid:12)(cid:12)(cid:12) D i ( B n − ) (cid:12)(cid:12)(cid:12) B − n − ≤ ( n − k B n − k C , (¯Ω × R × R n ) B − n − , for i = 1 , · · · , n . The estimates for (cid:12)(cid:12) B z B (cid:12)(cid:12) and (cid:12)(cid:12)(cid:12) B p B (cid:12)(cid:12)(cid:12) can be obtained exactly in the same way.We have the following consequence of Lemma 2.1 and Remark 2.1. Corollary 2.1.
Assume B n − ( x, u, Du ) ∈ C , ( ¯Ω × R × R n ) , B > and ˜ B = log B . Then we havethe following properties:(i) (2.22) | D i ˜ B | ≤ C (cid:20) j ( | ˜ u ij | ) (cid:21) B − n − holds for i = 1 , · · · , n , where the constant C depends on n, k B n − k C , , A and sup Ω | Du | .(ii) If the condition (1.5) holds, then (2.23) D ii ˜ B ≥ − C (cid:20) j ( | ˜ u ij | ) (cid:21) B − n − − C ′ (cid:20) j ( | ˜ u ij | ) (cid:21) + n X k =1 ˜ B p k D ii u k holds for i = 1 , · · · , n , where the constant C depends on n, k B n − k C , , A and sup Ω | Du | , and the constant C ′ depends on C B and A .Proof. Choosing ξ = e i in (2.4), we have, for i = 1 , · · · , n ,(2.24) D i ˜ B = B i + B z D i u + B p k D i u k B .
It follows from (2.20) that(2.25) (cid:12)(cid:12)(cid:12)(cid:12) B i + B z D i uB (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) B i B (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) B z B (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ CB − n − , where the constant C depends on n, k B n − k C , and sup Ω | Du | . Since ˜ u ij = u ij − A ij , we obtain(2.26) (cid:12)(cid:12)(cid:12)(cid:12) B p k D i u k B (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:20) j ( | ˜ u ij | ) (cid:21) B − n − , here the constant C depends on n, k B n − k C , and A . Combining (2.24), (2.25) and (2.26), we get(2.22) and finish the proof of conclusion (i).Next, we turn to prove (ii). It follows from (2.6) and (2.7) that, for i = 1 , · · · , n , (cid:12)(cid:12)(cid:12)(cid:12) B ii + B zz ( D i u ) + B z D ii u + 2 B iz D i u + 2 B ip l D i u l + 2 B zp k D i uD i u k B (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:20) j ( | u ij | ) (cid:21) B − n − ≤ C (cid:20) j ( | ˜ u ij | ) (cid:21) B − n − , (2.27)and (cid:12)(cid:12)(cid:12)(cid:12) B i + B z u i + 2 B i B z u i + 2 B i B p l D i u l + 2 B z B p l u i D i u l B (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:20) j ( | u ij | ) (cid:21) B − n − ≤ C (cid:20) j ( | ˜ u ij | ) (cid:21) B − n − , (2.28)where the constants C depend on n, k B n − k C , , A and sup Ω | Du | . By the condition (1.5), we have B p k p l B − B p k B p l B u il u ik ≥ − C B δ kl (˜ u il + A il )(˜ u ik + A ik ) ≥ − C ′ (cid:20) j ( | ˜ u ij | ) (cid:21) , (2.29)where δ kl denotes the usual Kronecker delta, the constant C ′ depends on C B and A . Taking ξ = e in(2.5), and using (2.27), (2.28) and (2.29), we get (2.23) and finish the proof of conclusion (ii). (cid:3) Remark . We remark that ˜ B = log B satisfies the condition (1.5), if it is semi-convex in p . Theterm P nk =1 ˜ B p k D ii u k on the right hand side of (2.23) can also be dealt with in the later discussion.By the equation (1.1), we can build the relationship between B − n − and P ni =1 ˜ u ii , (˜ u ii ) = (˜ u ii ) − ,if ˜ u >
1. Therefore, a suitable barrier function is necessary to control the term C P ni =1 ˜ u ii . Weintroduce the following barrier construction lemma under the A3w condition, which is a variant ofLemma 2.1(ii) in [10] when the operator F is given by “log det”. Similar versions of such a lemma canalso be found in [9, 12]. Lemma 2.2.
Let u ∈ C ( ¯Ω) be an elliptic solution of the equation (1.1) and u ∈ C ( ¯Ω) be a strictsubsolution of the equation (1.1) satisfying (1.9) . Assume that A ∈ C ( ¯Ω × R × R n ) satisfies the A3wcondition, B ∈ C ( ¯Ω × R × R n ) is a positive function satisfying (1.5) . Then the inequality (2.30) L h e κ ( u − u ) i ≥ ε n X i =1 ˜ u ii − C, holds in Ω for sufficiently large positive constant κ and uniform positive constants ε and C , where (2.31) L = n X i,j =1 ˜ u ij D ij − n X k =1 D p k A ij ( x, u, Du ) D k ! − n X k =1 ˜ B p k D k . Proof.
Since u is a strict subsolution satisfying (1.9), by taking F = log det in Lemma 2.1(ii) in [10],following (2.17) in [10] we have(2.32) L h e κ ( u − u ) i ≥ ε n X i =1 ˜ u ii + 1 ! + C " ˜ B ( · , u, Du ) − ˜ B ( · , u, Du ) − n X k =1 ˜ B p k ( · , u, Du ) D k ( u − u ) , or large positive constant κ and uniform positive constant ε . By Taylor’s formula and the condition(1.5), we have(2.33) ˜ B ( · , u, Du ) − ˜ B ( · , u, Du ) − n X k =1 ˜ B p k ( · , u, Du ) D k ( u − u )= 12 n X k,l =1 ˜ B p k p l ( · , u, ˆ p ) D k ( u − u ) D l ( u − u ) ≥ − C B | D ( u − u ) | , where ˆ p = θDu + (1 − θ ) Du with θ ∈ (0 , (cid:3) In Lemma 2.2, if the A3w condition holds without orthogonality, the inequality barrier inequalitystill holds by replacing the barrier function e κ ( u − u ) with κ ( u − u ). Note also that if C B = 0 in condition(1.5), namely ˜ B is convex in p , then the barrier inequality (2.30) can be replaced by(2.34) L h e κ ( u − u ) i ≥ ε n X i =1 ˜ u ii + 1 ! , since the second term on the right hand side of (2.32) is nonnegative in this case.3. Interior regularity for the DMATE (1.1) under the A3 condition
In this section, by constructing an auxiliary function, we obtain interior second order derivativeestimates for the Monge-Amp`ere type equation (1.1) under the A3 condition and
B >
0. We then usethe estimates to obtain the interior regularity for the solution of the DMATE (1.1).
Proof of Theorem 1.1.
We employ the auxiliary function(3.1) G ( x, ξ ) = η ( x )˜ u ξξ , where η is a cut-off function in Ω, 0 ≤ η ≤
1, ˜ u ξξ = ˜ u ij ξ i ξ j , ˜ u ij = u ij − A ij ( x, u, Du ) and ξ ∈ R n isa unit vector. We may assume that G attains its maximum at x ∈ Ω and ξ = ξ . Without loss ofgenerality, we may assume { ˜ u ij } is diagonal at x and ξ = e . Then the function(3.2) G ( x, ξ ) = η ( x )˜ u attains its maximum at x . Denoting(3.3) ˜ G ( x ) := log G ( x, ξ ) = 2 log η + log ˜ u , then ˜ G ( x ) also attains its maximum at x . At x , we have˜ G i = 2 η i η + D i ˜ u ˜ u = 0 , ˜ G ij = 2 η ij η − η i η j η + D ij ˜ u ˜ u − D i ˜ u D j ˜ u ˜ u = 2 η ij η − η i η j η + D ij ˜ u ˜ u , (3.4) or i, j = 1 , · · · , n , and the matrix { ˜ G ij } ≤
0. From now on, we assume all the calculations are takenat x . Then it follows from { ˜ u ij } ≥
0, ˜ u ≥ ≥ ˜ u L ˜ G = ˜ u n X i,j =1 ˜ u ij D ij ˜ G = ˜ u n X i,j =1 ˜ u ij (cid:20) η ij η − η i η j η + D ij ˜ u ˜ u (cid:21) ≥ − C ˜ u η n X i =1 ˜ u ii + n X i,j =1 ˜ u ij D ij ˜ u , (3.5)where L is the linearized operator defined in (2.31). Recalling that ˜ u = u − A , we obtain n X i,j =1 ˜ u ij D ij ˜ u = n X i,j =1 ˜ u ij D ij ( u − A ) ≥ n X i,j,k,l =1 ˜ u ij [ u ij − ( D p k A ) u kij − ( D p k p l A ) u ki u lj ] − C n X i =1 ˜ u ii ! , (3.6)where C is a constant depending on A and sup Ω | Du | . By a direct computation, we have n X i,j =1 ˜ u ij u ij = n X i,j =1 ˜ u ij D u ij = n X i,j =1 ˜ u ij D (˜ u ij + A ij ) ≥ n X i,j,k,l =1 ˜ u ij [ D ˜ u ij + ( D p k A ij ) u k + ( D p k p l A ij ) u k u l ] − C n X i =1 ˜ u ii ! , (3.7)where C is a constant depending on A and sup Ω | Du | . By differentiating equation (2.1) in the direction ξ ∈ R n once and twice, we get(3.8) ˜ u ij D ξ ˜ u ij = D ξ ˜ B, and(3.9) ˜ u ij D ξξ ˜ u ij ≥ D ξξ ˜ B. Here the inequality (3.9) is obtained by using the concavity of “log det”. Inserting (3.6) and (3.7) into(3.5), we have(3.10) 0 ≥ − C ˜ u η n X i =1 ˜ u ii + D ˜ B − CD ˜ B + n X i,j,k,l =1 ˜ u ij [( D p k p l A ij ) u k u l − ( D p k p l A ) u ki u lj ] , where (3.8), (3.9) and the first equality in (3.4) are used to deal with the terms P ni,j,k =1 ˜ u ij ( D p k A ) u kij , P ni,j =1 ˜ u ij D ˜ u ij and P ni,j,k =1 ˜ u ij ( D p k A ij ) u k respectively, the terms − C (cid:0) P ni =1 ˜ u ii (cid:1) in (3.6) and(3.7) are absorbed in the first term on the right hand side of (3.10) since we can always assume ˜ u and P ni =1 ˜ u ii as large as we want. Next, we estimate the last term in (3.10). Since both { ˜ u ij } and ˜ u ij } are diagonal at x , we get n X i,j,k,l =1 ˜ u ij [( D p k p l A ij ) u k u l − ( D p k p l A ) u ki u lj ]= X i =1 n X k,l =1 ˜ u ii [( D p k p l A ii ) u k u l − ( D p k p l A ) u ki u li ]= X i =1 n X k,l =1 ˜ u ii [ D p k p l A ii (˜ u k + A k )(˜ u l + A l ) − D p k p l A (˜ u ki + A ki )(˜ u li + A li )] ≥ X i =1 ˜ u ii ( D p p A ii )˜ u − C n X i =1 ˜ u ii ˜ u . (3.11)Using (2.22) and (2.23) in Corollary 2.1, then (3.10) becomes0 ≥ X i =1 ˜ u ii ( D p p A ii )˜ u − C ˜ u η n X i =1 ˜ u ii + D ˜ B − CD ˜ B ≥ X i =1 ˜ u ii ( D p p A ii )˜ u − C ˜ u η n X i =1 ˜ u ii − C ˜ u B − n − − C ˜ u + n X k =1 ˜ B p k D u k ≥ X i =1 ˜ u ii ( D p p A ii )˜ u − C ˜ u η n X i =1 ˜ u ii − C ˜ u η B − n − − C ˜ u , (3.12)where the third order derivative term P nk =1 ˜ B p k D u k is treated by using (2.20) and the first equalityin (3.4). Note that the constant C changes from line to line in the context. Since ˜ u ≥
1, we can get n X i =1 ˜ u ii ≥ n X i =2 ˜ u ii ≥ ( n − n Y i =2 ˜ u ii ! n − = ( n − n Y i =1 ˜ u ii ! n − (˜ u ) n − ≥ ( n − B − n − . (3.13)Plugging (3.13) into (3.12), we obtain(3.14) 0 ≥ X i =1 ˜ u ii D p p A ii ˜ u − C ˜ u η n X i =1 ˜ u ii − C (˜ u ) . y the A3 condition, choosing ˜ ξ = ˜ u e and ˜ η = n P i =2 √ ˜ u ii e i , we have X i =1 ˜ u ii ( D p p A ii )˜ u = n X i,j,k,l =1 D p k p l A ij ˜ ξ i ˜ ξ j ˜ η k ˜ η l ≥ c ˜ u n X i =2 ˜ u ii ≥ n c ˜ u n X i =2 ˜ u ii + n − n c ˜ u ˜ u ≥ n c ˜ u n X i =2 ˜ u ii + n − n c ˜ u ˜ u ≥ n c ˜ u n X i =1 ˜ u ii . (3.15)Without loss of generality, we assume(3.16) 12 n c n X i =1 ˜ u ii ≥ C. Otherwise we are done. Combining (3.14), (3.15) and (3.16), we have(3.17) 0 ≥ n c ˜ u n X i =1 ˜ u ii − C ˜ u η n X i =1 ˜ u ii , which leads to(3.18) η ˜ u ≤ C. We now complete the proof of Theorem 1.1. (cid:3)
Note that the constant C in (1.6) in Theorem 1.1 is independent of the positive lower bound of B .Then the C , regularity result under the A3 condition, Theorem 1.2, follows directly from the interiorestimates in Theorem 1.1. Here we omit the proof of Theorem 1.2 since it is standard.4. Interior regularity for the DMATE (1.1) under the A3w + condition In this section, we prove the Pogorelov type estimate in Theorem 1.3 under the A3w + conditionand suitable barrier conditions, which can be applied to the interior C , regularity for solutions of theDMATE (1.1) in Theorem 1.4. Proof of Theorem 1.3.
First we note that under either (i) or (ii), we have(4.1) Lϕ ≥ ε n X i =1 ˜ u ii − C, for some positive constants ε and C . In case (i), ϕ is the function in the A -boundedness condition(1.8), and (4.1) with ε = 1 can be calculated directly from (1.8). While in case (ii), the inequality(4.1) with ϕ = e κ ( u − u ) is proved in (2.30) in Lemma 2.2.We construct the auxiliary function(4.2) h ( x, ξ ) = η α ˜ u ξξ e β | Du | + γϕ , here ϕ is the barrier function in (4.1), ˜ u ξξ = ˜ u ij ξ i ξ j , ξ = ( ξ , · · · , ξ n ) and | ξ | = 1, ˜ u ij = u ij − A ij , η = w − u and α, β, γ are positive constants to be determined.Since h ≥ h = 0 on ∂ Ω, we may assume that h attains its maximum at the point ¯ x ∈ Ωand some unit vector ¯ ξ . We may assume u (¯ x ) < w (¯ x ), namely η (¯ x ) >
0. By taking the logarithm of h ,we obtain(4.3) ¯ h ( x, ξ ) := log h ( x, ξ ) = α log η + log(˜ u ij ξ i ξ j ) + 12 β | Du | + γϕ. Thus, ¯ h also attains its maximum at the point ¯ x ∈ Ω and the vector ¯ ξ . We may assume that ¯ ξ =(1 , , · · · ,
0) and { ˜ u ij } is diagonal at ¯ x . We define v ( x ) := ¯ h ( x, ξ ) | ξ =¯ ξ = α log η + log(˜ u ) + 12 β | Du | + γϕ. (4.4)Since ¯ x is also the maximum point of v , we have(4.5) Dv (¯ x ) = 0 , and(4.6) D v (¯ x ) ≤ . It follows from (4.5), (4.6) and { ˜ u ij } ≥ Lv (¯ x ) ≤ , where L is the linearized operator defined in (2.31). By a direct computation, we have, at ¯ x ,(4.8) D i v = αD i ηη + D i ˜ u ˜ u + βD k uD ki u + γD i ϕ, and D ii v = αD ii ηη − α ( D i η ) η + D ii ˜ u ˜ u − ( D i ˜ u ) ˜ u + β n X i,k =1 (cid:0) ( D ik u ) + ( D k u ) D iik u (cid:1) + γD ii ϕ, (4.9)for i = 1 , · · · , n . Inserting (4.8) and (4.9) into (4.7), we get0 ≥ Lv (¯ x )= αη Lη − αη n X i =1 ˜ u ii ( D i η ) + 1˜ u L ˜ u − u n X i =1 ˜ u ii ( D i ˜ u ) + β n X k =1 D k uLu k + β n X i,k =1 ˜ u ii ( D ik u ) + γLϕ. (4.10)Next, we estimate each term of (4.10). From now on, all calculations are made at the maximumpoint ¯ x . We first consider the general case that B depends on p , namely B p
0. By calculations, we ave Lη = n X i =1 ˜ u ii " D ii w − ˜ u ii − A ii ( x, u, Du ) − n X k =1 ( D p k A ii ( x, u, Du )) D k η − n X k =1 ˜ B p k D k η ≥ − n + n X i =1 ˜ u ii " A ii ( x, u, Dw ) − A ii ( x, u, Du ) − n X k =1 ( D p k A ii ( x, u, Du )) D k η − n X k =1 ˜ B p k D k η ≥ − n − CB − n − + 12 n X i,k,l =1 ˜ u ii A ii,kl ( x, u, ¯ p ) D k ηD l η ≥ − n − CB − n − − µ − n X i =1 ˜ u ii ( D i η ) , (4.11)for ¯ p = (1 − θ ) Du + θDw and θ ∈ (0 , D ii w − A ii ( x, u, Dw ) ≥ + conditionis used to obtain the third inequality, µ − = − min { µ , } and µ is the constant in (1.3). Using theCauchy’s inequality, it follows from (4.11) that αη Lη ≥ − α " nη + Cη B − n − + µ − η n X i =1 ˜ u ii ( D i η ) ≥ − αnη − α Cη − CB − n − − αµ − n X i =1 ˜ u ii ( D i η ) η , (4.12)where we have assumed η (¯ x ) ∈ (0 , η (¯ x ) > u L ˜ u , we first calculate Lu . We can assume ˜ u ≥
1, otherwise we are done.By a direct computation and using (2.3) with ξ = e , we have Lu ≥ n X i =1 ˜ u ii ˜ u jj ( D ˜ u ij ) + n X i,k,l =1 ˜ u ii A ii,kl u k u l + D ˜ B − ˜ B p k D k u − C n X i,j =1 [(1 + ˜ u jj )˜ u ii ] ≥ n X i =1 ˜ u ii ˜ u jj ( D ˜ u ij ) − C n X i =1 ˜ u ii + D ˜ B − ˜ B p k D k u − C n X i,j =1 [(1 + ˜ u jj )˜ u ii ] , (4.13)where the A3w + condition is used to obtain the second inequality. With the help of (2.23) in Corollary2.1, we can further get Lu ≥ n X i,j =1 ˜ u ii ˜ u jj ( D ˜ u ij ) − C n X i =1 ˜ u ii − C n X i,j =1 [(1 + ˜ u jj )˜ u ii ] − C (1 + ˜ u ) B − n − − C (1 + ˜ u ) ≥ n X i,j =1 ˜ u ii ˜ u jj ( D ˜ u ij ) − C n X i,j =1 ˜ u jj ˜ u ii − C ˜ u B − n − − C ˜ u , (4.14)where we assume ˜ u ≥ n P i =1 ˜ u ii ≥ − ˜ B p k D k u in (4.13) is eliminated by the last term of (2.23). Next, we calculate LA . Using he definition of L , ˜ u ij = u ij − A ij and the C smoothness of A , we obtain LA ≤ C + C n X i,j =1 [(1 + ˜ u jj )˜ u ii ] + n X i,j,k,l =1 ˜ u ij D p k p l A ˜ u ki ˜ u kj + D k ˜ B ≤ C n X i,j =1 [(1 + ˜ u jj )˜ u ii + ˜ u jj ] + D k ˜ B ≤ C n X i,j =1 [(1 + ˜ u jj )˜ u ii ] + D k ˜ B ≤ C n X i,j =1 ˜ u jj ˜ u ii + C ˜ u B − n − , (4.15)where we again assume ˜ u ≥ n P i =1 ˜ u ii ≥
1. Recalling ˜ u = u − A , we get from (4.14) and(4.15) that(4.16) L ˜ u ≥ n X i,j =1 ˜ u ii ˜ u jj ( D ˜ u ij ) − C n X i,j =1 ˜ u jj ˜ u ii − C ˜ u B − n − − C ˜ u . Therefore, we have(4.17) 1˜ u L ˜ u ≥ u n X i,j =1 ˜ u ii ˜ u jj ( D ˜ u ij ) − C n X i =1 (˜ u ii + ˜ u ii ) − CB − n − . Choosing ξ = e k in (2.2), we have Lu k = n X i =1 ˜ u ii " D ii u k − n X l =1 ( D p l A ii ) D l u k − n X l =1 B p l B u lk = n X i =1 ˜ u ii D k A ii + n X i =1 ˜ u ii ( D u A ij ) u k + B k B + B z B u k , (4.18)for k = 1 , · · · , n . Hence, we have(4.19) β n X k =1 D k uLu k ≥ − βC n X i =1 ˜ u ii − βCB − n − . By a direct calculation, we have β n X i,k =1 ˜ u ii ( D ik u ) = β n X i =1 ˜ u ii (˜ u ii + A ii ) + β X k = i ˜ u ii A ik ≥ β n X i =1 ˜ u ii − βC n X i =1 ˜ u ii . (4.20)From the barrier inequality (4.1) in both cases (i) and (ii), we can also have(4.21) γLϕ ≥ ε γ n X i =1 ˜ u ii , y assuming n P i =1 ˜ u ii ≥ Cε . Now choosing α ≥ β ≥ ≥ − α Cη − βCB − n − + (cid:18) γε − βC (cid:19) n X i =1 ˜ u ii + ( β − C ) n X i =1 ˜ u ii − αC n X i =1 ˜ u ii ( D i η ) η + 1˜ u n X i,j =1 ˜ u ii ˜ u jj ( D ˜ u ij ) − u n X i =1 ˜ u ii ( D i ˜ u ) . (4.22)Splitting P ni =1 ˜ u ii ( D i η ) η into two parts, we have(4.23) n X i =1 ˜ u ii ( D i η ) η = ( D η ) η ˜ u + n X i =2 ˜ u ii ( D i η ) η . Observing that the first term on the right hand side of (4.23) can be absorbed by the first term on theright hand side of (4.22), we only need to estimate the last term in (4.23). From (4.5) and (4.8), wehave αC n X i =2 ˜ u ii ( D i η ) η = αC n X i =2 ˜ u ii ( α (cid:20) D i ˜ u ˜ u + βD k u (˜ u ki − A ki ) + γD i ϕ (cid:21) ) ≤ Cα n X i =2 ˜ u ii ((cid:18) D i ˜ u ˜ u (cid:19) + β (˜ u ii + 1) + γ ( D i ϕ ) ) ≤ u n X i =2 ˜ u ii ( D i ˜ u ) + n X i =1 (˜ u ii + ˜ u ii ) , (4.24)where we choose α = ( β + γ + 2) C . Thus, from (4.22), (4.23) and (4.24), we have0 ≥ − α Cη − βCB − n − + (cid:18) γε − βC (cid:19) n X i =1 ˜ u ii + ( β − C ) n X i =1 ˜ u ii − u n X i =2 ˜ u ii ( D i ˜ u ) + 1˜ u n X i,j =1 ˜ u ii ˜ u jj ( D ˜ u ij ) − u n X i =1 ˜ u ii ( D i ˜ u ) . (4.25)Using the Pogorelov term u P ni,j =1 ˜ u ii ˜ u jj ( D ˜ u ij ) , we have − u n X i =2 ˜ u ii ( D i ˜ u ) + 1˜ u n X i,j =1 ˜ u ii ˜ u jj ( D ˜ u ij ) − u n X i =1 ˜ u ii ( D i ˜ u ) = n X i,j =1 u ˜ u ii ˜ u jj ( D ˜ u ij ) − u ˜ u ( D ˜ u ) − u n X i =2 ˜ u ii ( D i ˜ u ) ≥ u n X i =2 ˜ u ii ( D i ˜ u ) + 2˜ u n X i =2 ˜ u ii [( D ˜ u i ) − ( D i ˜ u ) ] ≥ u n X i =2 ˜ u ii ( D i ˜ u ) + 2˜ u n X i =2 ˜ u ii ( D i A − D A i )(2 D i ˜ u + D i A − D A i ) ≥ − C ˜ u n X i =1 ˜ u ii ≥ − C n X i =1 ˜ u ii , (4.26) here Cauchy’s inequality is used in the second last inequality. Therefore, from (4.25) and (4.26), wehave(4.27) 0 ≥ − α Cη − βCB − n − + (cid:18) γε − βC (cid:19) n X i =1 ˜ u ii + ( β − C ) n X i =1 ˜ u ii . By using the key relationship (3.13) between B − n − and P ni =1 ˜ u ii , we have from (4.27) that(4.28) 0 ≥ − α Cη + (cid:18) γε − βC (cid:19) n X i =1 ˜ u ii + ( β − C ) n X i =1 ˜ u ii . By choosing β = C + 1 and γ = βCε , (4.28) becomes(4.29) 0 ≥ − α Cη + n X i =1 ˜ u ii ≥ − α Cη + ˜ u , which leads to(4.30) η ˜ u (¯ x ) ≤ α C. We then immediately get the conclusion (1.10) in the B p B p ≡ | B p | B ≤ C , (4.12) can be replaced by(4.31) αη Lη ≥ − αCη − C ′ B − n − − αµ n X i =1 ˜ u ii ( D i η ) η , (the constant C ′ = 0 when B p ≡ ≥ − αCη + ˜ u , which leads to(4.33) η ˜ u (¯ x ) ≤ αC. We then immediately get the conclusion (1.10) in the B p ≡ | B p | B ≤ C case. ThenTheorem 1.3 is proved provided η (¯ x ) ∈ (0 , η (¯ x ) >
1, (4.12) still holds. Furthermore, η in the denominators on the right hand side of(4.12) can be replaced by 1. Following the above proof, we can have(4.34) ˜ u (¯ x ) ≤ C, which also leads to the conclusion (1.10).We now complete the proof of Theorem 1.3. (cid:3) Remark . In the above proof, the A3w + condition is crucial in the critical inequality (4.11), whichis the reason why we restrict our study in the class of A satisfying A3w + . Alternative conditions toget through the inequality (4.11) can be found in (2.4), Remark 2.1 and Remark 2.2 in [15]. Note thatthe inequality (4.13), which is deduced from the A3w + condition, can also be derived by just using theA3w condition and some other conditions, see [14, 15].We are now ready to prove Theorem 1.4. The proof of Theorem 1.4.
Let Ω j be a sequence of C ∞ bounded domains such that Ω j → Ω as j → ∞ .Note that if in case (i), these domains also need to satisfy the A -boundedness condition. We can find B j ∈ C ∞ such that B j > B j tends uniformly to B in Ω and k B n − j k C , (¯Ω j × R × R n ) C for someuniform constant C , (independent of j ). From the existence result in [12], the Dirichlet problemdet( M [ u j ]) = B j in Ω j , u j = w on ∂ Ω j , has a unique classical solution u j ∈ C ( ¯Ω j ). ince A and B are nondecreasing in z , from the strong maximum principle, either u ≡ w in Ω or u < w in Ω. In the former case, since w ∈ C , (Ω), we immediately have u ∈ C , (Ω). Next, we onlyconsider the latter case when u < w in Ω. Since u j is a degenerate elliptic solution, we can have theuniform gradient estimate from [13]. By applying the Pogorelov type estimate (1.10) in the domain { u j < w − ε } for any fixed small constant ε >
0, we have(4.35) ( w − u j − ε ) τ | D u j | ≤ C, in { u j < w − ε } , where the constant C is independent of j . Thus, we have(4.36) | D u j | ≤ C, in { u j < w − ε } , where the constant C is independent of j . From the stability property of viscosity solutions [2], wehave u j → u as j → ∞ , and(4.37) u ∈ C , ( { u < w − ε } ) , for any fixed small constant ε >
0. Since the domain { u < w − ε } tends to Ω = { u < w } as ε to 0,from (4.37), we finally get u ∈ C , (Ω). (cid:3) References
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E-mail address : [email protected] Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China
E-mail address : [email protected]@nju.edu.cn