Asymptotic expansion at infinity of solutions of Monge-Ampère type equations
AAsymptotic expansion at infinity of solutions ofMonge-Amp`ere type equations
Zixiao Liu, Jiguang Bao * February 18, 2021
Abstract
We obtain a quantitative expansion at infinity of solutions for a kind of Monge-Amp`eretype equations that origin from mean curvature equations of Lagrangian graph ( x, Du ( x )) andrefine the previous study on zero mean curvature equations and the Monge-Amp`ere equations. Keywords:
Monge-Amp`ere equation, Mean curvature eqution, Asymptotic expansion.
MSC 2020:
In 2018, Wang-Huang-Bao [31] studied the second boundary value problem of Lagrangian meancurvature equation of gradient graph ( x, Du ( x )) in ( R n × R n , g τ ) , where Du denotes the gradientof scalar function u and g τ = sin τ δ + cos τ g , τ ∈ (cid:104) , π (cid:105) is the linearly combined metric of standard Euclidean metric δ = n (cid:88) i =1 dx i ⊗ dx i + n (cid:88) j =1 dy j ⊗ dy j , with the pseudo-Euclidean metric g = n (cid:88) i =1 dx i ⊗ dy i + n (cid:88) j =1 dy j ⊗ dx j . They proved that for domain Ω ⊂ R n , if u ∈ C (Ω) is a solution of F τ (cid:0) λ (cid:0) D u (cid:1)(cid:1) = f ( x ) , x ∈ Ω , (1.1)then Df ( x ) is the mean curvature of gradient graph ( x, Du ( x )) in ( R n × R n , g τ ) . Previously,Warren [32] proved that when f ( x ) ≡ C for some constants C , the mean curvature of ( x, Du ( x )) * Supported in part by Natural Science Foundation of China (11871102 and 11631002). a r X i v : . [ m a t h . A P ] F e b s zero. In (1.1), f ( x ) is a scalar function with sufficient regularity, λ (cid:0) D u (cid:1) = ( λ , λ , · · · , λ n ) are n eigenvalues of Hessian matrix D u and F τ ( λ ) := n n (cid:88) i =1 ln λ i , τ = 0 , √ a + 12 b n (cid:88) i =1 ln λ i + a − bλ i + a + b , < τ < π , −√ n (cid:88) i =1
11 + λ i , τ = π , √ a + 1 b n (cid:88) i =1 arctan λ i + a − bλ i + a + b , π < τ < π , n (cid:88) i =1 arctan λ i , τ = π ,a = cot τ, b = (cid:112) | cot τ − | .If τ = 0 , then (1.1) becomes the Monge-Amp`ere type equation det D u = e nf ( x ) in R n . (1.2)For f ( x ) being a constant C , there are Bernstein-type results by J¨orgens [21], Calabi [8] andPogorelov [29], which state that any convex classical solution of (1.2) must be a quadratic poly-nomial. See Cheng-Yau [9], Caffarelli [3], Jost-Xin [22] and Li-Xu-Simon-Jia [24] for differentproofs and extensions. For f ( x ) − C having compact support, there are exterior Bernstein-typeresults by Ferrer-Mart´ınez-Mil´an [13] for n = 2 and Caffarelli-Li [6], which state that any convexsolution must be asymptotic to quadratic polynomials at infinity (for n = 2 we need additional ln -term). For f ( x ) − C vanishing at infinity, there are similar asymptotic results by Bao-Li-Zhang [2]. For f ( x ) − C being a periodic function or asymptotically periodic function, there areclassification results by Caffarelli-Li [7], Teixeira-Zhang [30] etc.If τ = π , then (1.1) becomes the Lagrangian mean curvature equation n (cid:88) i =1 arctan λ i (cid:0) D u (cid:1) = f ( x ) in R n . (1.3)For f ( x ) being a constant C , there are Bernstein-type results by Yuan [33, 34], which state thatany classical solution of (1.3) and D u ≥ (cid:40) − KI, n ≤ , − ( √ + (cid:15) ( n )) I, n ≥ , or C > n − π, (1.4)must be a quadratic polynomial, where I denote the unit n × n matrix, K is a constant and (cid:15) ( n ) is asmall dimensional constant. For f ( x ) − C having compact support, there is an exterior Bernstein-type result by Li-Li-Yuan [25], which states that any classical solution of (1.3) with (1.4) must beasymptotic to quadratic polynomials at infinity (for n = 2 we need additional ln -term).2or general τ ∈ [0 , π ] , for f ( x ) being a constant C , there are Bernstein-type results undersuitable semi-convex conditions by Warren [32], which is based on the results of J¨orgens [21]-Calabi [8]-Pogorelov [29], Flanders [14] and Yuan [33, 34]. For f ( x ) − C having compactsupport, there are exterior Bernstein-type results when n ≥ in our earlier work [27], which statethat any classical solution of (1.1) with suitable semi-convex conditions must be asymptotic toquadratic polynomial at infinity. There are also higher order expansions at infinity, which give theprecise gap between exterior maximal/minimal gradient graph and the entire case. Such higherorder expansions problem was considered for the Yamabe equation and σ k -Yamabe equation byHan-Li-Li [17], which refines the study by Caffarelli-Gidas-Spruck [5], Korevaar-Mazzeo-Pacard-Schoen [23], Han-Li-Teixeira [18] etc.In this paper, we obtain asymptotic expansion at infinity of classical solutions of F τ ( λ ( D u )) = f ( x ) in R n , (1.5)where n ≥ , τ ∈ [0 , π ] and f ( x ) is a perturbation of f ( ∞ ) := lim x →∞ f ( x ) at infinity. Thispartially refines previous study [2, 6, 19, 25, 27] etc.Our first result considers asymptotic behavior and higher order expansions of general classicalsolution of (1.5). Hereinafter, we let ϕ = O m ( | x | − k (ln | x | ) k ) with m ∈ N , k , k ≥ denote | D k ϕ | = O ( | x | − k − k (ln | x | ) k ) as | x | → + ∞ for all ≤ k ≤ m . Let x T denote the transpose of vector x ∈ R n , Sym ( n ) denote the setof symmetric n × n matrix, H nk denote the k -order spherical harmonic function space in R n , DF τ ( λ ( A )) denote the matrix with elements being value of partial derivative of F τ ( λ ( M )) w.r.t M ij variable at matrix A and [ k ] denote the largest natural number no larger than k . Theorem 1.1.
Let u ∈ C ( R n ) be a classical solution of (1.5) , where f ∈ C ( R n ) is C m outsidea compact subset of R n and satisfies lim sup | x |→∞ | x | ζ + k | D k ( f ( x ) − f ( ∞ )) | < ∞ , ∀ k = 0 , , , · · · , m (1.6) for some ζ > and m ≥ . Suppose either of the following holds(1) D u > for τ = 0 ;(2) u ( x ) ≤ C (1 + | x | ) and D u > ( − a + b ) I, ∀ x ∈ R n (1.7) for some constant C , for τ ∈ (0 , π ) ;(3) u ( x ) ≤ C (1 + | x | ) and D u > − I, ∀ x ∈ R n (1.8) for some constant C , for τ = π .Then there exist c ∈ R , b ∈ R n and A ∈ Sym ( n ) with F τ ( λ ( A )) = f ( ∞ ) such that u ( x ) − (cid:18) x T Ax + bx + c (cid:19) = (cid:26) O m +1 ( | x | − min { n,ζ } ) , ζ (cid:54) = n,O m +1 ( | x | − n (ln | x | )) , ζ = n, (1.9) as | x | → + ∞ . emark 1.2. The matrix A in Theorem 1.1 also satisfies A > in case (1) , A > ( − a + b ) I incase (2) and A > − I in case (3) respectively. Remark 1.3.
Notice that in condition (1.6) , we only require m ≥ , which is an improvementto the results for m ≥ by Bao-Li-Zhang [2]. It would be an interesting to determin sharplower bounds for m in Theorem 1.1. There has been an example in [2] that shows the decay rateassumption ζ > in (1.6) is optimal. We also have the following higher order expansions for ζ > n , which gives a finer character-istic of the error term in (1.9).
Theorem 1.4.
Under conditions of Theorem 1.1, there exist c ∈ R , c k ( θ ) ∈ H nk with k =1 , , · · · , n − [2 n − ζ ] − such that u ( x ) − (cid:18) x T Ax + bx + c (cid:19) − c ( x T ( DF τ ( λ ( A ))) − x ) − n − n − [2 n − ζ ] − (cid:88) k =1 c k ( θ ) (cid:0) x T ( DF τ ( λ ( A ))) − x (cid:1) − n − k = (cid:26) O m ( | x | − min { n,ζ } ) , min { n, ζ } − n (cid:54)∈ N ,O m ( | x | − min { n,ζ } (ln | x | )) , min { n, ζ } − n ∈ N , (1.10) as | x | → + ∞ , where θ = ( DF τ ( λ ( A ))) − x ( x T ( DF τ ( λ ( A ))) − x ) . Remark 1.5.
By computing F τ ( λ ( D u )) of radially symmetric u of form C | x | + C | x | − k , wefind expansions (1.9) and (1.10) are optimal for all ζ > in the sense that the series of k doesn’texists or cannot be taken up to n − [2 n − ζ ] when < ζ ≤ n or ζ > n respectively since c n − [2 n − ζ ] doesn’t belong to space H n − [2 n − ζ ] n in general. The paper is organized as follows. In section 2 we prove that the Hessian matrix D u convergeto some constant matrix A ∈ Sym ( n ) at infinity, in order to make preparation for proving Theorem1.1. In the next two sections we give the proofs of Theorems 1.1 and 1.4 respectively based on thedetailed analysis of the solutions of non-homogeneous linearized equations.Hereinafter, we let B r ( x ) denote a ball centered at x ∈ R n with radius r . Especially for x = 0 ,we let B r := B r (0) . For any open subset Ω ⊂ R n , we let Ω denote the closure of Ω and Ω c denotethe complement of Ω in R n . In this section, we study the asymptotic behavior at infinity of Hessian matrix of classical solutionsof (1.5). We prove a weaker convergence than (1.9) in Theorem 1.1 and D u has bounded C α norm for some < α < under a weaker assumption on f . By interior regularity as Lemma17.16 of [16] and extension theorem as Theorem 6.10 of [12], we may change the value of u, f ona compact subset of R n and prove only for u ∈ C ,α ( R n ) and f ∈ C α ( R n ) .4 heorem 2.1. Let u be as in Theorem 1.1, f ∈ C α ( R n ) for some < α < and satisfy lim sup | x |→∞ (cid:32) | x | ζ | f ( x ) − f ( ∞ ) | + | x | α + ζ (cid:48) [ f ] C α ( B | x | ( x )) (cid:33) < ∞ (2.1) (1) with some ζ > , ζ (cid:48) > for τ = 0 ;(2) with some ζ > , ζ (cid:48) > for τ ∈ (0 , π ) ;(3) with some ζ > , ζ (cid:48) > for τ = π .Then there exist (cid:15) > , A ∈ Sym ( n ) with F τ ( λ ( A )) = f ( ∞ ) and C > such that || D u || C α ( R n ) ≤ C, and (cid:12)(cid:12) D u ( x ) − A (cid:12)(cid:12) ≤ C | x | (cid:15) , ∀ | x | ≥ . The proof is separated into three subsections according to three different range of τ . τ = 0 case In τ = 0 case, (1.5) becomes the Monge-Amp`ere equation (1.2). Theorem 2.2.
Let u ∈ C ( R n ) be a convex viscosity solution of det D u = ψ ( x ) in R n (2.2) with u (0) = min R n u = 0 , where < ψ ∈ C ( R n ) and ψ n − ∈ L n ( R n ) . Then there exists a linear transform T satisfying det T = 1 such that v := u ◦ T satisfies (cid:12)(cid:12)(cid:12)(cid:12) v − | x | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x | − ε , ∀ | x | ≥ . for some C > and ε > . Theorem 2.2 can be found in the proof of Theorem 1.2 in [2], which is based on the level setmethod by Caffarelli-Li [6].
Corollary 2.3.
Let u ∈ C ( R n ) be a convex viscosity solution of (1.5) with f ∈ C ( R n ) satisfies lim sup | x |→∞ | x | ζ | f ( x ) − f ( ∞ ) | < ∞ for some ζ > . Then there exists a linear transform T satisfying det T = 1 such that v := u ◦ T satisfies (cid:12)(cid:12)(cid:12)(cid:12) v − exp( f ( ∞ ))2 | x | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x | − ε , ∀ | x | ≥ (2.3) for some C > and ε > . roof. By a direct computation, (cid:101) u ( x ) := 1exp( f ( ∞ )) ( u ( x ) − Du (0) x − u (0)) is a convex viscosity solution of det D (cid:101) u = e n ( f ( x ) − f ( ∞ )) =: (cid:101) f ( x ) in R n . By a direct computation, | (cid:101) f ( x ) − | ≤ C | x | − ζ for some C > and (cid:90) R n \ B (cid:12)(cid:12)(cid:12) ( (cid:101) f ( x )) n − (cid:12)(cid:12)(cid:12) n dx ≤ C (cid:90) R n \ B (cid:12)(cid:12)(cid:12) (cid:101) f ( x ) − (cid:12)(cid:12)(cid:12) n dx ≤ C (cid:90) R n \ B | x | − ζn dx < ∞ . The result follows immediately by applying Theorem 2.2 to (cid:101) u .As a consequence, we have the following convergence of Hessian matrix for solutions of (2.2).The proof is similar to the one in Bao-Li-Zhang [2] and in Caffarelli-Li [6]. Since there are somedifferences from their proof, we provide the details here for reading simplicity. Theorem 2.4.
Let u ∈ C ( R n ) be a convex viscosity solution of (1.5) , f ∈ C α ( R n ) satisfy (2.1) for some < α < , ζ > and ζ (cid:48) > . Then u ∈ C ,α ( R n ) , || D u || C α ( R n ) ≤ C, (2.4) and u − (cid:18) x T Ax + bx + c (cid:19) = O ( | x | − (cid:15) ) (2.5) as | x | → ∞ , where (cid:15) := min { ε, ζ, ζ (cid:48) } , ε is the positive constant from Theorem 2.2, A ∈ Sym ( n ) with det A = exp( nf ( ∞ )) , b ∈ R n , c ∈ R and C > .Proof. By Corollary 2.3, there exist a linear transform T , ε > and C > such that v := u ◦ T satisfies (2.3). Step 1: prove C α boundedness of Hessian (2.4). Let v R ( y ) = (cid:18) R (cid:19) v (cid:18) x + R y (cid:19) , | y | ≤ for | x | = R > . By (2.3), (cid:107) v R (cid:107) C ( B ) ≤ C for some C > for all R ≥ . Then v R satisfies det (cid:0) D v R ( y ) (cid:1) = exp (cid:18) nf (cid:18) x + R y (cid:19)(cid:19) =: f R ( y ) in B . (2.6)By a direct computation, there exists C > uniform to x such that || f R − exp( nf ( ∞ )) || C ( B ) ≤ CR − ζ y , y ∈ B , | f R ( y ) − f R ( y ) || y − y | α = | f ( z ) − f ( z ) || z − z | α · ( R α ≤ CR − ζ (cid:48) , where z i := x + R y i ∈ B | x | ( x ) . Applying the interior estimate by Caffarelli [3], Jian-Wang [20]on B , we have (cid:13)(cid:13) D v R (cid:13)(cid:13) C α ( B ) ≤ C (2.7)and hence C I ≤ D v R ≤ CI in B (2.8)for some C independent of R . For any | x | = R ≥ , we have | D v ( x ) | = | D v R (0) | ≤ || D v R || C ( B ) ≤ C. (2.9)For any x , x ∈ B c with < | x − x | ≤ | x | , let R := | x | > , by (2.7), (cid:12)(cid:12) D v ( x ) − D v ( x ) (cid:12)(cid:12) | x − x | α = (cid:12)(cid:12)(cid:12) D v R (0) − D v R (cid:16) x − x ) | x | (cid:17)(cid:12)(cid:12)(cid:12) | x − x | α ≤ [ D v R ] C α ( B ) · (cid:16) | x | (cid:17) α ≤ CR − α . For any x , x ∈ B c with | x − x | ≥ | x | , by (2.9), | D v ( x ) − D v ( x ) || x − x | α ≤ α · || D v || C ( R n ) ≤ C. Since the linear transform T from Theorem 2.2 is invertible, (2.4) follows immediately. Step 2: prove convergence speed at infinity (2.5). Let w ( x ) := v ( x ) − exp( f ( ∞ ))2 | x | and w R ( y ) := (cid:18) R (cid:19) w (cid:18) x + R y (cid:19) , | y | ≤ for | x | = R ≥ . By (2.3) in Theorem 2.2, (cid:107) w R (cid:107) C ( B ) ≤ CR − ε . Applying Newton-Leibnitz formula between (2.6) and det(exp( f ( ∞ )) I ) = exp( nf ( ∞ )) , (cid:102) a ij ( y ) D ij w R = f R ( y ) − exp( nf ( ∞ )) in B , where (cid:102) a ij ( y ) = (cid:82) D M ij (det (cid:0) I + tD w R ( y ) (cid:1) ) dt .By (2.7) and (2.8), there exists constant C independent of | x | = R > such that IC ≤ (cid:102) a ij ≤ CI in B , (cid:107) (cid:102) a ij (cid:107) C α ( B ) ≤ C.
7y interior Schauder estimates, see for instance Theorem 6.2 of [16], (cid:107) w R (cid:107) C ,α (cid:18) B (cid:19) ≤ C (cid:16) (cid:107) w R (cid:107) C ( B ) + (cid:107) f R − exp( nf ( ∞ )) (cid:107) C α ( B ) (cid:17) ≤ CR − min { ε,ζ,ζ (cid:48) } . (2.10)The result (2.5) follows immediately by scaling back. Remark 2.5.
In the proof of Theorem 2.4, the interior Schauder estimates used in (2.10) can bereplaced by the W , ∞ type estimates (see for instance Remark 1.3 of [11]), || w R || W , ∞ ( B ) ≤ C (cid:16) (cid:107) w R (cid:107) C ( B ) + (cid:107) f R − exp( nf ( ∞ )) (cid:107) C α ( B ) (cid:17) ≤ CR − min { ε,ζ,ζ (cid:48) } . Remark 2.6.
The condition (2.1) in Theorem 2.4 holds if for some
C > , | x | ζ | f ( x ) − f ( ∞ ) | + | x | ζ (cid:48) | Df ( x ) | ≤ C, ∀ | x | > . (2.11) Even if f ( x ) is C , condition (2.1) is weaker than (2.11). For example, we consider f ( x ) := e −| x | sin( e | x | ) . On the one hand, Df ( x ) doesn’t admit a limit at infinity, hence f doesn’t satisfycondition (2.11). On the other hand, for any | x | = R > and z , z ∈ B | x | ( x ) , | f ( z ) − f ( z ) || z − z | α ≤ e −| z | (cid:12)(cid:12) sin( e | z | ) − sin( e | z | ) (cid:12)(cid:12) | z − z | α + sin( e | z | ) (cid:12)(cid:12) e −| z | − e −| z | (cid:12)(cid:12) | z − z | α ≤ Ce − R · | z − z || z − z | α ≤ Ce − R · R − α for constant C independent of R . Hence f satisfies condition (2.1) for all α ∈ (0 , and any ζ, ζ (cid:48) > . This finishes the proof of Theorem 2.1 for τ = 0 case. τ ∈ (0 , π ) case In this subsection, we deal with τ ∈ (0 , π ) case by Legendre transform and the results in previoussubsection.Let f ∈ C α ( R n ) satisfy (2.1) for some < α < , ζ > , ζ (cid:48) > and u ∈ C ,α ( R n ) be aclassical solution of (1.5) satisfying (2). Let u ( x ) := u ( x ) + a + b | x | , then D u = D u + ( a + b ) I > bI in R n . (2.12)Let ( (cid:101) x, v ) be the Legendre transform of ( x, u ) , i.e., (cid:26) (cid:101) x := Du ( x ) ,Dv ( (cid:101) x ) := x, (2.13)8nd we have D v ( (cid:101) x ) = (cid:0) D u ( x ) (cid:1) − = ( D u ( x ) + ( a + b ) I ) − < b I. Let ¯ v ( (cid:101) x ) := 12 | (cid:101) x | − bv ( (cid:101) x ) . (2.14)By a direct computation, D ¯ u ( R n ) = R n and (cid:101) λ i (cid:0) D ¯ v (cid:1) = 1 − b · λ i + a + b = λ i + a − bλ i + a + b ∈ (0 , . (2.15)Thus ¯ v ( (cid:101) x ) satisfies the following Monge-Amp`ere type equation det D ¯ v = exp (cid:26) b √ a + 1 f (cid:18) b ( (cid:101) x − D ¯ v ( (cid:101) x )) (cid:19)(cid:27) =: g ( (cid:101) x ) in R n . (2.16) Step 1:
There exists C > such that C | x | ≤ | (cid:101) x | ≤ C | x | , ∀ | x | > . (2.17)We prove the two inequalities in (2.17) separately.By the definition of (cid:101) x = D ¯ u ( x ) and (2.12), | (cid:101) x − (cid:101) | = | D ¯ u ( x ) − D ¯ u (0) | > b | x | . Hence by triangle inequality, | (cid:101) x | ≥ −| (cid:101) | + | (cid:101) x − (cid:101) | > −| (cid:101) | + 2 b | x | , (2.18)and the first inequality of (2.17) follows immediately.By the quadratic growth condition in (1.7), we prove the linear growth result of Du ( x ) . Infact, for any | x | ≥ , let e := Du ( x ) | Du ( x ) | ∈ ∂B . By Newton-Leibnitz formula and (1.7) u ( x + | x | e ) = u ( x ) + (cid:90) | x | e · Du ( x + se ) d s = u ( x ) + (cid:90) | x | (cid:90) s e · D u ( x + te ) · e d t d s + (cid:90) | x | e · Du ( x ) d s ≥ u ( x ) + ( − a + b )2 | x | + | Du ( x ) | · | x | . (2.19)Furthermore by (1.7), there exists C > independent of | x | ≥ such that | Du ( x ) | ≤ | x | (cid:18) C (1 + | ( x + | x | e ) | ) + C (1 + | x | ) + a − b | x | (cid:19) ≤ C (1 + | x | ) . Hence there exists
C > such that | Du ( x ) | ≤ C (1 + | x | ) , ∀ x ∈ R n . (2.20)9y (2.20), there exists C > such that | (cid:101) x | = | Du ( x ) + ( a + b ) x | ≤ | Du ( x ) | + ( a + b ) | x | ≤ C ( | x | + 1) . The second inequality of (2.17) follows immediately.Now we study equation (2.16) by applying Theorem 2.4 and Remark 2.6, which require aknowledge on the asymptotic behavior of g ( (cid:101) x ) . Step 2: g ( (cid:101) x ) satisfies condition (2.1). By the equivalence (2.17), lim (cid:101) x →∞ g ( (cid:101) x ) = exp (cid:26) b √ a + 1 f ( ∞ ) (cid:27) =: g ( ∞ ) ∈ (0 , . By a direct computation, | (cid:101) x | ζ | g ( (cid:101) x ) − g ( ∞ ) | = e b √ a f ( ∞ ) | (cid:101) x | ζ (cid:12)(cid:12)(cid:12) (cid:101) x − D ¯ v ( (cid:101) x )2 b (cid:12)(cid:12)(cid:12) ζ · (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) x − D ¯ v ( (cid:101) x )2 b (cid:12)(cid:12)(cid:12)(cid:12) ζ · (cid:12)(cid:12)(cid:12)(cid:12) e b √ a ( f ( (cid:101) x − D ¯ v ( (cid:101) x )2 b ) − f ( ∞ )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x | ζ (cid:12)(cid:12)(cid:12)(cid:12) e b √ a ( f ( x ) − f ( ∞ )) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x | ζ | f ( x ) − f ( ∞ ) | < C. For any (cid:101) y, (cid:101) z ∈ B | (cid:101) x | · b ( (cid:101) x ) , (cid:101) y (cid:54) = (cid:101) z with | (cid:101) x | > C , by (2.12) we have y, z ∈ B | x | ( x ) , | (cid:101) y − (cid:101) z | ≥ b | y − z | > and y (cid:54) = z. Thus by condition (2.1), | g ( (cid:101) y ) − g ( (cid:101) z ) || (cid:101) y − (cid:101) z | α ≤ (2 b ) − α exp { b √ a +1 f ( y ) } − exp { b √ a +1 f ( z ) }| y − z | α ≤ C [ f ] C α ( B | x | ( x )) . (2.21)Thus g ( (cid:101) x ) satisfies (2.1) for < α < , ζ > and ζ (cid:48) > as given.By Theorem 2.4, we have || D ¯ v || C α ( R n ) ≤ C and ¯ v − (cid:18) (cid:101) x T (cid:101) A (cid:101) x + (cid:101) b · (cid:101) x + (cid:101) c (cid:19) = O ( | (cid:101) x | − (cid:15) ) (2.22)for some < (cid:101) A ∈ Sym ( n ) satisfying det (cid:101) A = g ( ∞ ) , (cid:101) b ∈ R n , (cid:101) c ∈ R and C, (cid:15) > . Step 3: we finish the proof of Theorem 2.1 (2). By strip argument as in [25, 27] etc, weprove that I − (cid:101) A is invertible. In fact, by (2.15), (cid:101) A ≤ I and it remains to prove λ i ( (cid:101) A ) < for all i = 1 , , · · · , n . Arguing by contradiction and rotating the (cid:101) x -space to make (cid:101) A diagonal, we mayassume that (cid:101) A = 1 . By (2.22) with the definition of Legendre transform (2.14) and (2.18), thereexists (cid:101) b such that x = D v ( (cid:101) x ) = (cid:101) b + O ( | (cid:101) x | − (cid:15) ) as | (cid:101) x | → ∞ . (2.23)This becomes a contradiction to (2.17). 10et A := 2 b (cid:16) I − (cid:101) A (cid:17) − − ( a + b ) I. By a direct computation, F τ ( λ ( A )) = f ( ∞ ) and (cid:12)(cid:12) D u ( x ) − A (cid:12)(cid:12) = 2 b (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) I − D ¯ v ( (cid:101) x ) (cid:1) − − (cid:16) I − (cid:101) A (cid:17) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | D ¯ v ( (cid:101) x ) − (cid:101) A |≤ C | (cid:101) x | (cid:15) ∀ | x | ≥ . By the equivalence (2.17), we have (cid:12)(cid:12) D u ( x ) − A (cid:12)(cid:12) ≤ C | x | (cid:15) , ∀ | x | ≥ . (2.24)Furthermore, by (2.14), for any x, y ∈ R n , (cid:12)(cid:12) D u ( x ) − D u ( y ) (cid:12)(cid:12) = 2 b (cid:12)(cid:12)(cid:12)(cid:0) I − D ¯ v ( (cid:101) x ) (cid:1) − − (cid:0) I − D ¯ v ( (cid:101) y ) (cid:1) − (cid:12)(cid:12)(cid:12) . By (2.24), D ¯ v ( (cid:101) x ) is bounded away from and I , it follows that ∃ C > such that (cid:12)(cid:12) D u ( x ) − D u ( y ) (cid:12)(cid:12) ≤ bC (cid:12)(cid:12) D ¯ v ( (cid:101) x ) − D ¯ v ( (cid:101) y ) (cid:12)(cid:12) (2.25)Combining (2.25) and the equivalence (2.17), D u has bounded C α norm.So far, we finished the proof of Theorem 2.1 for τ ∈ (0 , π ) case. τ = π case In this subsection, we deal with τ = π case by Legendre transform and analysis on the Poissonequations.Let f ∈ C α ( R n ) satisfy (2.1) for some < α < , ζ, ζ (cid:48) > and u ∈ C ,α ( R n ) be a classicalsolution of (1.5) satisfying (3). Let u ( x ) := u ( x ) + 12 | x | , then D u > in R n . By equation (1.5), for all i = 1 , , · · · , n , − λ i ( D ¯ u ) ≥ − n (cid:88) j =1 λ j ( D ¯ u ) ≥ √
22 inf R n f. Thus there exists δ > such that D u ( x ) > δI, ∀ x ∈ R n . Let ( (cid:101) x, v ) be the Legendre transform of ( x, u ) as in (2.13) and we have < D v ( (cid:101) x ) = ( D u ( x )) − < δ I.
11y a direct computation, D ¯ u ( R n ) = R n and v ( (cid:101) x ) satisfies the following Poisson equation ∆ v = − √ f ( Dv ( (cid:101) x )) =: g ( (cid:101) x ) in R n . (2.26) Step 1:
There exists C > such that (2.17) holds. The proof is separated into two parts similarly.By the definition of Legendre transform in (2.13), | (cid:101) x − (cid:101) | = | D ¯ u ( x ) − D ¯ u (0) | > δ | x | . Hence by triangle inequality, | (cid:101) x | ≥ −| (cid:101) | + | (cid:101) x − (cid:101) | > −| (cid:101) | + δ | x | and the first inequality of (2.17) follows immediately. The second inequality of (2.17) followssimilarly by (1.8) and (2.19). Step 2:
Asymptotic behavior of g ( (cid:101) x ) at infinity. By the equivalence (2.17), g ( (cid:101) x ) = − √ f ( x ) → − √ f ( ∞ ) =: g ( ∞ ) as | (cid:101) x | → + ∞ . Similar to the proof of (2.21), we have lim sup | (cid:101) x |→ + ∞ (cid:32) | (cid:101) x | ζ | g ( (cid:101) x ) − g ( ∞ ) | + | (cid:101) x | α + ζ (cid:48) [ g ] C α ( B | (cid:101) x | ( (cid:101) x )) (cid:33) < ∞ for the give < α < , ζ, ζ (cid:48) > . Step 3:
Asymptotic behavior of v ( (cid:101) x ) at infinity.Since (2.1) remains when ζ > becomes smaller, we only need to prove for < ζ < casefor reading simplicity. By a direct computation, ∆ | x | − ζ = c n,ζ | x | − ζ in B c . Thus there existsubsolution v and supersolution v of Poisson equation ∆ (cid:101) v = g ( (cid:101) x ) − g ( ∞ ) in R n (2.27)with v, v = O ( | (cid:101) x | − ζ ) as | x | → ∞ . By Perron’s method (see for instance [2, 10, 26]) and interiorregularity, we have a classical solution (cid:101) v ∈ C ,α ( R n ) of (2.27) with (cid:101) v = O ( | (cid:101) x | − ζ ) as | (cid:101) x | → ∞ .For any | (cid:101) x | = R ≥ , let (cid:101) v R ( y ) := (cid:18) R (cid:19) (cid:101) v ( (cid:101) x + R y ) , y ∈ B . Then (cid:101) v R satisfies ∆ (cid:101) v R = g ( (cid:101) x + R y ) − g ( ∞ ) =: g R ( y ) in B . By a direct computation, || g R || C α ( B ) ≤ CR − min { ζ,ζ (cid:48) } and || (cid:101) v R || C ( B ) ≤ CR − ζ .
12y interior Schauder estimates, we have || (cid:101) v R || C ,α ( B / ) ≤ CR − min { ζ,ζ (cid:48) } and then (cid:101) v ( (cid:101) x ) = O ( | (cid:101) x | − min { ζ,ζ (cid:48) } ) as | (cid:101) x | → ∞ . Then ∆( v − (cid:101) v ) = g ( ∞ ) in R n and D ( v − (cid:101) v ) is bounded. By Liouville type theorem, v − (cid:101) v is a quadratic function and hence v − (cid:18) (cid:101) x T (cid:101) A (cid:101) x + (cid:101) b (cid:101) x + (cid:101) c (cid:19) = O ( | (cid:101) x | − min { ζ,ζ (cid:48) } ) for some (cid:101) A ∈ Sym ( n ) with trace (cid:101) A = g ( ∞ ) , (cid:101) b ∈ R n and (cid:101) c ∈ R . Similarly we have (2.23) and (cid:101) A is invertible. Taking A := (cid:101) A − − I and the result follows similar to τ ∈ (0 , π ) case. (1.5) In this section, we prove Theorem 1.1. As an integral part of the preparation, we analyze thelinearized equation of (1.5) and obtain the asymptotic behavior at infinity. The major difficulty isthat the linearized equation is not homogeneous.
Consider the linear elliptic equation Lu := a ij ( x ) D ij u ( x ) = f ( x ) in R n , (3.1)where the coefficients are uniformly elliptic, satisfying || a ij || C α ( R n ) < ∞ , (3.2)for some < α < and | a ij ( x ) − a ij ( ∞ ) | ≤ C | x | − ε , (3.3)for some < ( a ij ( ∞ )) ∈ Sym ( n ) and ε, C > . Theorem 3.1.
Let v be a classical solution of (3.1) that bounded from at least one side, thecoefficients satisfy (3.2) and (3.3) and f ∈ C ( R n ) satisfy lim sup | x |→ + ∞ | x | ζ | f ( x ) | < ∞ (3.4) for some ζ > . Then there exists a constant v ∞ such that v ( x ) = v ∞ + (cid:26) O (cid:0) | x | − min { n,ζ } (cid:1) , ζ (cid:54) = n,O (cid:0) | x | − n (ln | x | ) (cid:1) , ζ = n, (3.5) as | x | → ∞ . L is equivalent to the Green’s functionof Laplacian under conditions (3.2) and (3.3). More precisely, let G L ( x, y ) be the Green’s functioncentered at y , there exists constant C such that C − | x − y | − n ≤ G L ( x, y ) ≤ C | x − y | − n , ∀ x (cid:54) = y, | D x i G L ( x, y ) | ≤ C | x − y | − n , i = 1 , · · · , n, ∀ x (cid:54) = y, (cid:12)(cid:12) D x i D x j G L ( x, y ) (cid:12)(cid:12) ≤ C | x − y | − n , i, j = 1 , · · · , n, ∀ x (cid:54) = y. (3.6)By an elementary estimate as in Bao-Li-Zhang [2], we construct a solution that vanishes at infinity.More rigorously, we introduce the following result. Lemma 3.2.
There exists a bounded strong solution u ∈ W ,ploc ( R n ) with p > n of (3.1) satisfying u ( x ) = (cid:26) O ( | x | − min { n,ζ } ) , ζ (cid:54) = n,O ( | x | − n (ln | x | )) , ζ = n, as | x | → ∞ .Proof. By (3.6) and Calder´on-Zygmund inequality, w ( x ) := (cid:90) R n G L ( x, y ) f ( y ) d y belongs to W ,ploc ( R n ) for p > n and is a strong solution of (3.1) (see for instance [1, 35]). Itremains to compute the vanishing speed at infinity. Let E := { y ∈ R n , | y | ≤ | x | / } ,E := { y ∈ R n , | y − x | ≤ | x | / } ,E := R n \ ( E ∪ E ) . By a direct computation, (cid:90) E | x − y | n − f ( y ) d y ≤ C (cid:90) B | x | f ( y ) d y · | x | − n ≤ (cid:26) C | x | − min { n,ζ } , ζ (cid:54) = n,C | x | − n (ln | x | ) , ζ = n. Similarly, we have | x | ≤ | y | in E and hence (cid:90) E | x − y | n − f ( y ) d y ≤ C (cid:90) | x − y |≤ | x | | x − y | n − d y · | x | ζ ≤ C | x | − ζ . Now we separate E into two parts E +3 := { y ∈ E : | x − y | ≥ | y |} , E − := E \ E +3 . Then (cid:90) E +3 | x − y | n − · | y | ζ d y ≤ (cid:90) | y |≥ | x | | y | n + ζ − d y ≤ C | x | − ζ (cid:90) E − | x − y | n − · | y | ζ d y ≤ (cid:90) | y − x |≥ | x | | y − x | n + ζ − d y ≤ C | x | − ζ . Hence there exists
C > such that | w ( x ) | ≤ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) E ∪ E ∪ E | x − y | n − f ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:26) C | x | − min { n,ζ } , ζ (cid:54) = n,C | x | − n (ln | x | ) , ζ = n. Proof of Theorem 3.1.
We may assume without loss of generality that v is bounded from below,otherwise consider − v instead. Let w ( x ) be the bounded strong solution of (3.1) from Lemma3.2, then (cid:101) v := v − w − inf R n ( v − w ) ≥ is a strong solution of (3.1) with f ≡ . By interior regularity, (cid:101) v is a positive classical solution.By Theorem 2.2 in [25], (cid:101) v ( x ) = (cid:101) v ∞ + O (cid:0) | x | − n (cid:1) as | x | → ∞ , for some constant (cid:101) v ∞ . Then the result follows immediately from Lemma 3.2. Remark 3.3. If v is a classical solution of (3.1) with | Dv ( x ) | = O ( | x | − ) as | x | → ∞ and f ∈ C ( R n ) satisfy (3.4), then v is bounded from at least one side. The proof is similar to f ≡ case, which can be found in Corollary 2.1 of [25]. Let u ∈ C ( R n ) be a classical solution of (1.5), where f satisfies (1.6) for some ζ > , m ≥ and either of cases (1)-(3) holds. By extension and interior estimates, we may assume that u ∈ W ,ploc ( R n ) for some p > n . By Theorem 2.1, Hessian matrix D u have finite C α norm on R n andconverge to some A ∈ Sym ( n ) at a H¨older speed as in (2.24).Let v := u ( x ) − x T Ax . Applying Newton-Leibnitz formula between F τ (cid:0) λ (cid:0) D v + A (cid:1)(cid:1) = f ( x ) and F τ ( λ ( A )) = f ( ∞ ) , we have a ij ( x ) D ij v := (cid:90) D M ij F τ (cid:0) λ ( tD v + A ) (cid:1) d t · D ij v = f ( x ) − f ( ∞ ) =: f ( x ) (3.7)For any e ∈ ∂B , by the concavity of operator F , the partial derivatives v e := D e v and v ee := D e v are strong solutions of (cid:99) a ij ( x ) D ij v e := D M ij F τ (cid:0) λ ( D v + A ) (cid:1) D ij v e = f e ( x ) , (3.8)and (cid:99) a ij ( x ) D ij v ee ≥ f ee ( x ) . (3.9)15y Theorem 2.1, there exist (cid:15) > and C > such that (cid:12)(cid:12) a ij ( x ) − D M ij F τ ( λ ( A )) (cid:12)(cid:12) + (cid:12)(cid:12) (cid:99) a ij ( x ) − D M ij F τ ( λ ( A )) (cid:12)(cid:12) ≤ C | x | (cid:15) . By condition (1.6) and constructing barrier functions for (3.9), there exists
C > such that for all x ∈ R n , v ee ( x ) ≤ (cid:26) C | x | − min { n,ζ +2 } , ζ (cid:54) = n − ,C | x | − n (ln | x | ) , ζ = n − . By the arbitrariness of e , λ max (cid:0) D v (cid:1) ( x ) ≤ (cid:26) C | x | − min { n,ζ +2 } , ζ (cid:54) = n − ,C | x | − n (ln | x | ) , ζ = n − . By (1.6) and the ellipticity of equation (3.7), λ min (cid:0) D v (cid:1) ( x ) ≥ − Cλ max (cid:0) D v (cid:1) − C | f ( x ) | ≥ (cid:26) − C | x | − min { n,ζ +2 } , ζ (cid:54) = n − , − C | x | − n (ln | x | ) , ζ = n − . Hence (cid:12)(cid:12) D v ( x ) (cid:12)(cid:12) ≤ (cid:26) C | x | − min { n,ζ +2 } , ζ (cid:54) = n − ,C | x | − n (ln | x | ) , ζ = n − . By Theorem 2.1, the coefficients a ij , (cid:99) a ij has bounded C α norm on exterior domain. Since ζ > , applying Remark 3.3 to equation (3.8), for any e ∈ ∂B , v e ( x ) is bounded from one sideand there exists b e ∈ R such that v e ( x ) = b e + (cid:26) O (cid:0) | x | − min { n,ζ +1 } (cid:1) , ζ (cid:54) = n − ,O (cid:0) | x | − n (ln | x | ) (cid:1) , ζ = n − , as | x | → ∞ . (3.10)Picking e as n unit coordinate vectors of R n , we found b ∈ R n from (3.10) and let v ( x ) := v ( x ) − bx = u ( x ) − (cid:18) x T Ax + bx (cid:19) . By (3.10), | Dv ( x ) | = | ( ∂ x v − b , · · · , ∂ x n v − b n ) | = (cid:26) O (cid:0) | x | − min { n,ζ +1 } (cid:1) , ζ (cid:54) = n − ,O (cid:0) | x | − n (ln | x | ) (cid:1) , ζ = n − , as | x | → ∞ . By (3.7), a ij ( x ) D ij v = a ij ( x ) D ij v = f ( x ) . By the arguments above again, there exists c ∈ R such that v ( x ) = c + (cid:26) O ( | x | − min { n,ζ } ) , ζ (cid:54) = n,O (cid:0) | x | − n (ln | x | ) (cid:1) , ζ = n, as | x | → ∞ . Notice that here we used ζ > for | D ¯ v | = O ( | x | − ) and f = O ( | x | − ζ ) at infinity. Let Q ( x ) := x T Ax + bx + c. Then | u − Q | = | v − c | = (cid:26) O ( | x | − min { n,ζ } ) , ζ (cid:54) = n,O ( | x | − n (ln | x | )) , ζ = n, as | x | → ∞ . u . For | x | ≥ , let E ( y ) = (cid:18) | x | (cid:19) ( u − Q ) (cid:18) x + | x | y (cid:19) . Then by Newton-Leibnitz formula, a ij ( y ) D ij E ( y ) = F τ (cid:0) λ ( A + D E ( y )) (cid:1) − F τ ( λ ( A )) = f ( x + | x | y ) − f ( ∞ ) =: f ( y ) in B , where a ij ( y ) = (cid:90) D M ij F τ (cid:0) λ ( A + tD E ( y )) (cid:1) d t. By the Evans-Krylov estimate and interior Schauder estimate (see for instance Chap.8 of [4] andChap.6 of [16]), for all < α < , we have || E || C ,α ( B ) ≤ C ( || E || C ( B ) + || f || C α ( B ) ) ≤ C ( || E || C ( B ) + || f || C ( B ) )= (cid:26) O ( | x | − min { n,ζ } ) , ζ (cid:54) = n,O ( | x | − n (ln | x | )) , ζ = n, as | x | → ∞ . By taking further derivatives and iterate, we have for all k ≤ m + 1 , (cid:16) | x | (cid:17) k − (cid:12)(cid:12) D k ( u − Q )( x ) (cid:12)(cid:12) = | D k E (0) |≤ C k ( || E || C ( B ) + || f || C k − ,α ( B ) ) ≤ C k ( || E || C ( B ) + || f || C k − ( B ) )= (cid:26) O ( | x | − min { n,ζ } ) , ζ (cid:54) = n,O ( | x | − n (ln | x | )) , ζ = n, as | x | → ∞ . This finishes the proof of Theorem 1.1.
In this section, we consider asymptotic expansion at infinity for classical solutions of (1.5). As-sume that u, f are as in Theorem 1.1. Let a ij , f and v be as in (3.7) and subsection 3.2 respectively.In the following, we only need to focus on ζ > n case as explained in Remark 1.5. It followsfrom (1.9) in Theorem 1.1, (cid:12)(cid:12) a ij ( x ) − D M ij F τ ( λ ( A )) (cid:12)(cid:12) ≤ C (cid:12)(cid:12) D v ( x ) (cid:12)(cid:12) = O m − (cid:0) | x | − n (cid:1) and hence D M ij F τ ( λ ( A )) D ij v = f − ( a ij ( x ) − D M ij F τ ( λ ( A ))) D ij v =: g ( x )= O m ( | x | − ζ ) + O m − (cid:0) | x | − n (cid:1) = O m − ( | x | − min { n,ζ } ) by (1.6) as | x | → ∞ . 17et Q := [ D M ij F τ ( λ ( A ))] and (cid:101) v ( x ) := v ( Qx ) . Then ∆ (cid:101) v ( x ) = g ( Qx ) =: (cid:101) g ( x ) in R n . (4.1)By a direct computation, (cid:101) v = O m +1 ( | x | − n ) and (cid:101) g = O m − (cid:16) | x | − min { n,ζ } (cid:17) . Let ∆ S n − be the Laplace-Beltrami operator on unit sphere S n − ⊂ R n and Λ = 0 , Λ = n − , Λ = 2 n, · · · , Λ k = k ( k + n − , · · · , be the sequence of eigenvalues of − ∆ S n − with eigenfunctions Y (0)1 = 1 , Y (1)1 ( θ ) , Y (1)2 ( θ ) , · · · , Y (1) n ( θ ) , · · · , Y ( k )1 ( θ ) , · · · , Y ( k ) m k ( θ ) , · · · i.e., − ∆ S n − Y ( k ) m ( θ ) = Λ k Y ( k ) m ( θ ) , ∀ m = 1 , , · · · , m k . By Lemmas 3.1 and 3.2 of [27], there exists a solution (cid:101) v (cid:101) g of ∆ (cid:101) v (cid:101) g = (cid:101) g in R n \ B with (cid:101) v (cid:101) g = (cid:26) O m (cid:0) | x | − min { n,ζ } (cid:1) , min { n, ζ } − n / ∈ N ,O m (cid:0) | x | − min { n,ζ } (ln | x | ) (cid:1) , min { n, ζ } − n ∈ N . Thus v ( x ) := (cid:101) v − (cid:101) v (cid:101) g is harmonic on R n \ B with v = O ( | x | − n ) as | x | → ∞ . By sphericalharmonic expansions, there exist constants C (1) k,m , C (2) k,m such that v = ∞ (cid:88) k =0 m k (cid:88) m =1 C (1) k,m Y ( k ) m ( θ ) | x | k + ∞ (cid:88) k =0 m k (cid:88) m =1 C (2) k,m Y ( k ) m ( θ ) | x | − n − k . By the vanishing speed of v , we have C (1) k,m = 0 for all k, m . Thus similar to the proof of Lemma3.3 in [27], there exist constants c k,m with k ∈ N , m = 1 , · · · , m k such that (cid:101) v = [ ζ ] − n (cid:88) k =0 m k (cid:88) m =1 c k,m Y ( k ) m ( θ ) | x | − n − k + O m (cid:16) | x | − ζ (cid:17) , n < ζ < n, ζ (cid:54)∈ N , ζ − n − (cid:88) k =0 m k (cid:88) m =1 c k,m Y ( k ) m ( θ ) | x | − n − k + O m (cid:16) | x | − ζ (ln | x | ) (cid:17) , n < ζ < n, ζ ∈ N , n − (cid:88) k =0 m k (cid:88) m =1 c k,m Y ( k ) m ( θ ) | x | − n − k + O m (cid:0) | x | − n (ln | x | ) (cid:1) , n ≤ ζ. By rotating backwards by Q − , the results in Theorem 1.4 follow immediately.18 eferences [1] Robert A. Adams and John J. F. Fournier. Sobolev spaces , volume 140 of
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Graduate Texts in Mathematics .Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation.Z.Liu & J. BaoSchool of Mathematical Sciences, Beijing Normal UniversityLaboratory of Mathematics and Complex Systems, Ministry of EducationBeijing 100875, ChinaEmail: [email protected]
Email: [email protected]@bnu.edu.cn