Convergence of an iterative scheme for the Monge-Ampère eigenvalue problem with general initial data
aa r X i v : . [ m a t h . A P ] J un CONVERGENCE OF AN ITERATIVE SCHEME FOR THE MONGE-AMP`EREEIGENVALUE PROBLEM WITH GENERAL INITIAL DATA
NAM Q. LE
Abstract.
In this note, we revisit an iterative scheme, due to Abedin and Kitagawa (InverseIteration for the Monge-Amp`ere Eigenvalue Problem, arXiv:2001.01291v2, to appear in
Proc. Amer.Math. Soc. ), to solve the Monge-Amp`ere eigenvalue problem on a general bounded convex domain.Using a nonlinear integration by parts, we show that the scheme converges for all convex initial datahaving finite and nonzero Rayleigh quotient to a nonzero Monge-Amp`ere eigenfunction. Introduction and statement of the main result
In this note, we revisit an iterative scheme, due to Abedin and Kitagawa in their recent paper [1],to solve the Monge-Amp`ere eigenvalue problem on a bounded open convex domain Ω in R n ( n ≥ ( det D w = λ | w | n in Ω ,w = 0 on ∂ Ω . Before recalling relevant results, it is convenient to introduce some notation. Let K = { w ∈ C (Ω) : w is convex, nonzero in Ω , w = 0 on ∂ Ω } . When u is merely a convex function on Ω, by an abuse of notation, we use det D u dx to denotethe Monge-Amp`ere measure associated with u ; see Section 2.For a convex function u on Ω, we define its Rayleigh quotient by(1.2) R ( u ) = R Ω | u | det D u dx R Ω | u | n +1 dx . Implicit in the definition (1.2) is the requirement that k u k L n +1 (Ω) < ∞ . For general bounded convex domains Ω ⊂ R n , the existence, uniqueness and variational charac-terization of the Monge-Amp`ere eigenvalue, and uniqueness of convex Monge-Amp`ere eigenfunctionswere obtained in [11]. They are the singular counterparts of those established by Lions [13] and Tso[15] in the smooth, uniformly convex setting. For the purpose of this note, we recall here part of[11, Theorem 1.1]. Theorem 1.1. ( [11] ) Let Ω be a bounded open convex domain in R n . Define λ = λ [Ω] by (1.3) λ [Ω] = inf w ∈ K R ( w ) . Then, the following facts hold:(i) (Existence) The infimum in (1.3) is achieved by a nonzero convex solution w ∈ C ,β (Ω) ∩ C ∞ (Ω) for all β ∈ (0 , to the eigenvalue problem (1.1). The constant λ [Ω] is called theMonge-Amp`ere eigenvalue of Ω and w is called a Monge-Amp`ere eigenfunction of Ω .(ii) (Uniqueness) If the pair (Λ , ˜ w ) satisfies det D ˜ w = Λ | ˜ w | n in Ω where Λ > is a positiveconstant and ˜ w ∈ K , then Λ = λ [Ω] and ˜ w = mw for some positive constant m . Mathematics Subject Classification.
Key words and phrases.
Eigenvalue problem, Monge-Amp`ere equation, Iterative scheme.The research of the author was supported in part by the National Science Foundation under grant DMS-1764248.
In [1], Abedin and Kitagawa introduce an iterative scheme(1.4) ( det D u k +1 = R ( u k ) | u k | n in Ω ,u k +1 = 0 on ∂ Ωto solve the Monge-Amp`ere eigenvalue problem (1.1). Here u k +1 is a convex Aleksandrov solutionof (1.4). We refer to Theorem 2.3 for the existence of u k +1 and to Definition 2.2 for the notion ofAleksandrov solutions to the Monge-Amp`ere equation.An interesting feature of the iterative scheme (1.4) is that the sequence { u k } ∞ k =0 is obtained byrepeatedly inverting the Monge-Amp`ere operator with Dirichlet boundary condition. One notesthat similar inverse iteration methods have been considered for the p -Laplace equation [2, 3, 10].Abedin and Kitagawa establish the first inverse iteration result for the eigenvalue problem of a fullynonlinear degenerate elliptic equation for a large class of initial data. Their main convergence resultstates as follows. Theorem 1.2. ( [1, Theorem 1.4] ) Let Ω ⊂ R n be a bounded and convex domain. Let u ∈ C (Ω) bea function satisfying:(i) u is convex and u ≤ on ∂ Ω ; (ii) R ( u ) < ∞ ; (iii) det D u ≥ in Ω .For k ≥ , define the sequence u k ∈ K to be the solutions of the Dirichlet problem (1.4). Then { u k } converges uniformly on Ω to a non-zero Monge-Amp`ere eigenfunction u ∞ of Ω . Furthermore, lim k →∞ R ( u k ) = λ [Ω] . The conditions (i) and (iii) in Theorem 1.2 were used in [1] to show that, in the iterative scheme(1.4), the sequence { u k } satisfies u k ≤ ˆ w for all k ≥ w is a Monge-Amp`ere eigenfunctionof Ω with k ˆ w k L ∞ (Ω) = ( λ [Ω]) − /n . This implies the lower bound k u k k L ∞ (Ω) ≥ ( λ [Ω]) − /n whichguarantees the nontriviality of the limit u ∞ of u k . Here we call a function w trivial if w ≡ Remark 1.3.
Clearly (i) and (iii) in Theorem 1.2 imply that R ( u ) > . Without (i) and (iii) inTheorem 1.2, R ( u ) can be and thus the scheme (1.4) gives u k ≡ for all k ≥ . For example,if u is a nonzero affine function, then R ( u ) = 0 . Thus, to get the nontriviality of the limit u ∞ of u k , if it exists, we need to require that R ( u ) be nonzero. In this note, we remove the restrictions (i) and (iii) in Theorem 1.2. We show that the iterativescheme (1.4) converges for all convex initial data having finite and nonzero Rayleigh quotient to anonzero Monge-Amp`ere eigenfunction of Ω. Thus our result covers all possible convex functions onΩ as initial data for the scheme (1.4).
Theorem 1.4.
Let Ω ⊂ R n ( n ≥ ) be a bounded and convex domain. Let u ∈ C (Ω) be a nonzeroconvex function on Ω with < R ( u ) < ∞ . For k ≥ , define the sequence u k ∈ K to be thesolutions of the Dirichlet problem (1.4). Then { u k } converges uniformly on Ω to a non-zero Monge-Amp`ere eigenfunction u ∞ of Ω . Furthermore, for all k ≥ and any fixed nonzero Monge-Amp`ereeigenfunction w , we have [ R ( u k )] /n − ( λ [Ω]) /n ≤ ( λ [Ω]) /n R Ω ( | u k +1 | − | u k | ) | w | n dx R Ω | u || w | n dx ≤ C ( u , Ω , n ) Z Ω | u ∞ − u k | dx. The nontriviality of our limit under the nonzero finiteness of the Rayleigh quotient R ( u ) is dueto an eventual regularity of the scheme (Proposition 3.1) and an important monotonicity formuladuring the scheme (Lemma 3.2). The proof of this monotonicity formula is based on a nonlinearintegration by parts , established in [11], which was designed to prove uniqueness results for theMonge-Amp`ere equations and systems of Monge-Amp`ere equations [11, 12]. Remark 1.5.
Theorem 1.4 also bounds the convergence rate of R ( u k ) to the Monge-Amp`ere eigen-value λ [Ω] in terms of the convergence rate of u k to the nonzero Monge-Amp`ere eigenfunction u ∞ . TERATIVE SCHEME FOR THE MONGE-AMPERE EIGENVALUE PROBLEM 3
Compared to the inverse iteration methods for the p -Laplace equation in [2, 3, 10] , this type ofestimate seems to be new. The rest of this note is organized as follows. In Section 2, we recall basic facts on the Monge-Amp`ere equation and prove a reverse Aleksandrov estimate in Proposition 2.5. In Section 3, weshow the eventual smoothness and a new monotonicity formula for the iterative scheme (1.4). Theproof of Theorem 1.4 will be given in Section 4. In Section 5, we make some remarks on the energycharacterization of the Monge-Amp`ere eigenfunctions.2.
The Monge-Amp`ere equation and a reverse Aleksandrov estimate
Here, we recall some basic facts on the Monge-Amp`ere equation on open convex domain Ω of R n ; see the books by Figalli [7] and Guti´errez [8] for more details. We will establish a reverseAleksandrov estimate in Proposition 2.5 that could be of independent interest.For a convex function u : Ω → R , we define the subdifferential of u at x ∈ Ω by ∂u ( x ) := { p ∈ R n : u ( y ) ≥ u ( x ) + p · ( y − x ) ∀ y ∈ Ω } . Below is a precise definition of the Monge-Amp`ere measure of a convex function u : Ω → R ; see also[7, Definition 2.1] and [8, Theorem 1.1.13]. Definition 2.1 (Monge-Amp`ere measure) . Let u : Ω → R be a convex function. The Monge-Amp`ere measure, M u , associated with the convex function u is defined by M u ( E ) = | ∂u ( E ) | where ∂u ( E ) = [ x ∈ E ∂u ( x ) , for each Borel set E ⊂ Ω . If u ∈ C (Ω), then M u = det D u ( x ) dx in Ω. Definition 2.2 (Aleksandrov solutions) . Given an open convex set Ω and a Borel measure µ on Ω,we call a convex function u : Ω → R an Aleksandrov solution to the Monge-Amp`ere equationdet D u = µ, if µ = M u as Borel measures. When µ = f dx we will say for simplicity that u solvesdet D u = f. In this note, we use det D u to denote the Monge-Amp`ere measure M u for a general convexfunction u . Thus, for all Borel set E ⊂ Ω, Z E | u | det D u dx = Z E | u | dM u. Now, we recall the basic existence and uniqueness result for solutions to the Dirichlet problemwith zero boundary data for the Monge-Amp`ere equation; see [7, Theorem 2.13], [8, Theorem 1.6.2],and [9, Theorem 1].
Theorem 2.3 (The Dirichlet problem) . Let Ω be a bounded open convex domain in R n , and let µ be a nonnegative Borel measure in Ω . Then there exists a unique convex function u ∈ C (Ω) that isan Aleksandrov solution of ( det D u = µ in Ω ,u = 0 on ∂ Ω . For later reference, we state the celebrated Aleksandrov’s maximum principle for the Monge-Amp`ere equation; see [7, Theorem 2.8] and [8, Theorem 1.4.2].
NAM Q. LE
Theorem 2.4 (Aleksandrov’s maximum principle) . Let Ω ⊂ R n be an open, bounded and convexdomain. Let u ∈ C (Ω) be a convex function. If u = 0 on ∂ Ω , then | u ( x ) | n ≤ C ( n )(diamΩ) n − dist( x, ∂ Ω) Z Ω det D u dx for all x ∈ Ω . We have the following proposition which will play a crucial role in the proof of Theorem 1.4.
Proposition 2.5 (Reverse Aleksandrov estimate) . Let Ω be a bounded open convex domain in R n .Let λ [Ω] be the Monge-Amp`ere eigenvalue of Ω and let w be a nonzero Monge-Amp`ere eigenfunctionof Ω . Assume that u ∈ C (Ω) ∩ C (Ω) is a strictly convex function in Ω with u = 0 on Ω and satisfies Z Ω (det D u ) /n | w | n − dx < ∞ . Then (2.1) Z Ω ( λ [Ω]) /n | u || w | n dx ≥ Z Ω (det D u ) /n | w | n dx. Remark 2.6.
Compared to Theorem 2.4, the function u appears on the dominating side in (2.1)in Proposition 2.5. For this reason, (2.1) can be viewed as a sort of reverse Aleksandrov estimate.Moreover, this estimate is sharp. When u is a Monge-Amp`ere eigenfunction of Ω , (2.1) is anequality. To prove Proposition 2.5, we recall the following nonlinear integration by parts established in [11,Proposition 1.7].
Proposition 2.7 (Nonlinear integration by parts) . Let Ω be a bounded open convex domain in R n .Let u, v ∈ C (Ω) ∩ C (Ω) be strictly convex functions in Ω with u = v = 0 on ∂ Ω . If Z Ω (det D u ) n (det D v ) n − n dx < ∞ , and Z Ω det D v dx < ∞ , then Z Ω | u | det D v dx ≥ Z Ω | v | (det D u ) n (det D v ) n − n dx. Proof of Proposition 2.5.
We apply Proposition 2.7 to u and v = w . Then, using det D w = λ [Ω] | w | n , we get Z Ω | u | λ [Ω] | w | n = Z Ω | u | det D w dx ≥ Z Ω | w | (det D u ) /n (det D w ) n − n dx = Z Ω ( λ [Ω]) n − n (det D u ) /n | w | n dx. Dividing the first and last expressions in the above estimates by ( λ [Ω]) n − n , we obtain (2.1). (cid:3) Eventual smoothness and a new monotonicity formula for the iterative scheme
In this section, we show the eventual smoothness and a new monotonicity formula for the iterativescheme (1.4). They are stated in Proposition 3.1 and Lemma 3.2.We have the following eventual smoothness of solutions to the iterative scheme (1.4). The proofis a modification of the proof of Proposition 2.8 in [11].
Proposition 3.1.
Let Ω be an open, bounded convex set in R n with nonempty interior. Let u ∈ C (Ω) be a nonzero convex function on Ω with < R ( u ) < ∞ . For k ≥ , define the sequence u k ∈ K to be the solutions of the Dirichlet problem (1.4). Then, u ∈ C , n (Ω) , and u k +1 is strictlyconvex in Ω and u k +1 ∈ C k, n (Ω) for all k ≥ . TERATIVE SCHEME FOR THE MONGE-AMPERE EIGENVALUE PROBLEM 5
We recall that a convex function u on an open bounded convex domain Ω is said to be strictlyconvex in Ω, if for any x ∈ Ω and p ∈ ∂u ( x ), u ( z ) > u ( x ) + p · ( z − x ) for all z ∈ Ω \{ x } . Proof of Proposition 3.1.
First, using the Aleksandrov’s maximum principle in Theorem 2.4, we notethat each u k +1 is uniformly bounded, that is M k +1 = k u k +1 k L ∞ (Ω) < ∞ . The regularity C , n (Ω) of u is a consequence of the Aleksandrov maximum principle. Since u
0, and R ( u ) >
0, we have u
0. The convexity of u shows that u < Claim. u k +1 is strictly convex in Ω and u k +1 ∈ C k, n (Ω) for all k ≥ k = 1. For each ε ∈ (0 , M ), let Ω ′ := Ω( ε ) = { x ∈ Ω : u ( x ) ≤ − ε } .Since u ∈ C (Ω) is convex, the set Ω( ε ) is convex with nonempty interior. Note that, since | u | > u ∈ C , n (Ω), by continuity, | u | ≥ m ( n, ε ) > ′ . Since λ [Ω] m n ( n, ε ) ≤ det D u = R ( u ) | u | n ≤ R ( u ) M n in Ω ′ and u = − ε on ∂ Ω ′ , the function u is strictly convex in Ω ′ by the localization theorem of Caffarelli [4] (see also [7,Theorem 4.10] and [8, Corollary 5.2.2]). Moreover, u ∈ C , n (Ω ′ ). Now, using Caffarelli’s C ,α estimates [5], we have u ∈ C , n loc (Ω ′ ). Since ε ∈ (0 , M ) is arbitrary, we conclude u ∈ C , n (Ω) and u is strictly convex in Ω.Suppose the claim holds up to k − k ≥
2. We show it also holds for k . For each ε k ∈ (0 , M k +1 ), let Ω( ε k ) = { x ∈ Ω : u k +1 ( x ) ≤ − ε k } . Since u k +1 ∈ C (Ω) is convex, the set Ω( ε k )is convex with nonempty interior. Let us denote Ω ′ k = Ω( ε k ) for brevity. Note that, by continuity, | u k | ≥ m ( n, k, ε ) > ′ k .Since λ [Ω] m n ( n, k, ε ) ≤ det D u k +1 = R ( u k ) | u k | n ≤ R ( u k ) M nk in Ω ′ k and u k +1 = − ε k on ∂ Ω ′ k , thefunction u k +1 is strictly convex in Ω ′ k by the localization theorem of Caffarelli. By the inductionhypothesis, u k ∈ C k − , n (Ω ′ k ). In the interior of Ω ′ k , the equation det D u k +1 = R ( u k ) | u k | n nowbecomes uniformly elliptic with C k − , n right hand side. Therefore we have u k +1 ∈ C k, n loc (Ω ′ k ).Since ε k ∈ (0 , M k +1 ) is arbitrary, we conclude u k +1 ∈ C k, n (Ω) and u k +1 is strictly convex in Ω. (cid:3) Our key observation is the following monotonicity result for the iterative scheme (1.4). Note that R ( u k ) ≥ λ [Ω] for all k ≥ . Lemma 3.2 (Monotonicity formula for the iterative scheme) . Let Ω be a bounded open convexdomain in R n . Let u ∈ C (Ω) be a nonzero convex function on Ω with < R ( u ) < ∞ . Let w be anonzero Monge-Amp`ere eigenfunction of Ω . Consider the iterative scheme (1.4). If k ≥ , then Z Ω | u k +1 || w | n dx ≥ Z Ω | u k || w | n dx + [ R ( u k )] /n − ( λ [Ω]) /n ( λ [Ω]) /n Z Ω | u k || w | n dx. Proof of Lemma 3.2.
By Proposition 3.1, we have u k +1 ∈ C , n (Ω) for all k ≥
3. We apply Propo-sition 2.5 to u k +1 and recall det D u k +1 = R ( u k ) | u k | n , NAM Q. LE to get Z Ω | u k +1 || w | n dx ≥ λ [Ω]) /n Z Ω (det D u k +1 ) /n | w | n dx = [ R ( u k )] n ( λ [Ω]) /n Z Ω | u k || w | n dx = Z Ω | u k || w | n dx + [ R ( u k )] /n − ( λ [Ω]) /n ( λ [Ω]) /n Z Ω | u k || w | n dx. The monotonicity property is thus proved. (cid:3)
We recall the following monotonicity property in [1, Lemma 3.1].
Lemma 3.3. ( [1, Lemma 3.1] ) Let Ω be an open, bounded convex set in R n with nonempty interior.Let u ∈ C (Ω) be a nonzero convex function on Ω with < R ( u ) < ∞ . Consider the iterativescheme (1.4). Then for all k ≥ , we have R ( u k +1 ) k u k +1 k nL n +1 (Ω) ≤ R ( u k ) k u k k nL n +1 (Ω) . Lemma 3.3 was stated and proved in [1] for u satisfying (i), (ii) and (iii) in Theorem 1.2. However,the proof in [1] only used the assumptions 0 < R ( u ) < ∞ and u is convex. We include here theshort proof of Lemma 3.3 for reader’s convenience. Proof of Lemma 3.3.
The proof follows by multiplying both sides of the first equation of (1.4) by | u k +1 | , integrating over Ω and then using the H¨older inequality: R ( u k +1 ) k u k +1 k n +1 L n +1 (Ω) = Z Ω | u k +1 | det D u k +1 dx = R ( u k ) Z Ω | u k | n | u k +1 | ≤ R ( u k ) k u k k nL n +1 (Ω) k u k +1 k L n +1 (Ω) . Using u k +1 k ≥
0, we obtain the claimed monotonicity property. (cid:3) Convergence of the iterative scheme
In this section, we prove Theorem 1.4.Some of our arguments in
Step 2 of the proof of Theorem 1.4 are similar to those in the proof ofTheorem 1.2 in [1]. However, since we can obtain the convergence of R ( u k ) to λ [Ω] from Lemma3.2, we can avoid using the continuity property of the energy R Ω | u k | det D u k dx for a convergingsequence of convex functions u k with an upper bound on the density of the Monge-Amp`ere measuredet D u k (see [1, Lemma 2.9]). Moreover, the monotone property in Lemma 3.2 also allows us toquickly conclude that the whole sequence u k converges to the same limit. Proof of Theorem 1.4.
We fix a nonzero Monge-Amp`ere eigenfunction w . The assumptions on u imply that u k k ≥
0. The proof is split into several steps.
Step 1: The whole sequence R ( u k ) converges to λ [Ω] . Using the monotonicity property established in Lemma 3.2, we find that if k ≥
3, then(4.1) k u k k L ∞ (Ω) ≥ R Ω | u k || w | n dx R Ω | w | n dx ≥ R Ω | u || w | n dx R Ω | w | n dx ≥ c ( n, Ω , u ) > . For each k ≥
1, using R ( u k ) ≥ λ [Ω], we obtain from Lemma 3.3 that(4.2) k u k k nL n +1 (Ω) ≤ R ( u ) k u k nL n +1 (Ω) R ( u k ) ≤ R ( u ) k u k nL n +1 (Ω) λ [Ω] < ∞ . TERATIVE SCHEME FOR THE MONGE-AMPERE EIGENVALUE PROBLEM 7
This implies that the increasing sequence R Ω | u k || w | n dx is bounded from above and thus convergesto a limit L (4.3) lim k →∞ Z Ω | u k || w | n = L ∈ (0 , ∞ )where we used (4.1) to get L > k ≥
3, taking into account the full monotonicity property in Lemma 3.2, we get[ R ( u k )] /n − ( λ [Ω]) /n ≤ ( λ [Ω]) /n R Ω ( | u k +1 | − | u k | ) | w | n dx R Ω | u k || w | n dx ≤ ( λ [Ω]) /n R Ω ( | u k +1 | − | u k | ) | w | n dx R Ω | u || w | n dx . (4.4)Letting k → ∞ in (4.4) and recalling (4.3), we conclude that the whole sequence R ( u k ) convergesto λ [Ω]:(4.5) lim k →∞ R ( u k ) = λ [Ω] . Step 2: Convergence of u k to a nontrivial Monge-Amp`ere eigenfunction u ∞ of Ω . Next, applying the Aleksandrov estimate in Theorem 2.4 to u k +1 where k ≥
0, and then usingthe H¨older inequality together with (4.2), we find k u k +1 k nL ∞ (Ω) ≤ C ( n, Ω) Z Ω det D u k +1 dx = C ( n, Ω) R ( u k ) Z Ω | u k | n dx ≤ C ( n, Ω) R ( u k ) k u k k nL n +1 (Ω) | Ω | n +1 ≤ C ( n, Ω , u ) . Hence, we obtain the uniform L ∞ bound k u k k L ∞ (Ω) ≤ C ( n, Ω , u ) < ∞ . From the Aleksandrov estimate, we have the uniform C , n (Ω) bound for u k when k ≥ k u k k C , n (Ω) ≤ C ( n, Ω , u ) . Therefore, up to extracting a subsequence, we have the following uniform convergence u k j → u ∞ u ∞ ∈ C (Ω) with u ∞ = 0 on ∂ Ω while we also have the uniform convergence u k j +1 → w ∞ w ∞ ∈ C (Ω) with w ∞ = 0 on ∂ Ω.Thus, letting j → ∞ in det D u k j +1 = R ( u k j ) | u k j | n , using (4.5) and the weak convergence of the Monge-Amp`ere measure (see [7, Corollary 2.12] and [8,Lemma 5.3.1]), we get(4.6) det D w ∞ = λ [Ω] | u ∞ | n . Letting j → ∞ in the following monotonicity property (see Lemma 3.3) R ( u k j +1 ) k u k j +1 k nL n +1 (Ω) ≤ R ( u k j +1 ) k u k j +1 k nL n +1 (Ω) ≤ R ( u k j ) k u k j k nL n +1 (Ω) , and recalling (4.5), we find that k w ∞ k L n +1 (Ω) = k u ∞ k L n +1 (Ω) . NAM Q. LE
However, from (4.6), we have R ( w ∞ ) k w ∞ k n +1 L n +1 (Ω) = Z Ω | w ∞ | det D w ∞ dx = λ [Ω] Z Ω | u ∞ | n | w ∞ | dx ≤ λ [Ω] k u ∞ k nL n +1 (Ω) k w ∞ k L n +1 (Ω) = λ [Ω] k w ∞ k n +1 L n +1 (Ω) . Since R ( w ∞ ) ≥ λ [Ω], we must have R ( w ∞ ) = λ [Ω], and the inequality above must be an equality,but this gives u ∞ = cw ∞ for some constant c >
0. Thus, from (4.6), we havedet D w ∞ = c n λ [Ω] | w ∞ | n . It follows from the uniqueness part of Theorem 1.1 that c = 1 and w ∞ = u ∞ is a Monge-Amp`ereeigenfunction of Ω. Passing to the limit in Lemma 3.2, we have Z Ω | u ∞ || w | n = lim k →∞ Z Ω | u k || w | n = L. With this property and the uniqueness up to positive multiplicative constants of the Monge-Amp`ereeigenfunctions of Ω, we conclude that the limit u ∞ does not depend on the subsequence u k j . Thisshows that the whole sequence u k converges to a nonzero Monge-Amp`ere eigenfunction u ∞ of Ω. Step 3: Convergence estimate for [ R ( u k )] /n − ( λ [Ω]) /n . Let k ≥
3. By (4.4), and the fact that R Ω | u k || w | n dx increases to L = R Ω | u ∞ || w | n dx , we havethe estimates [ R ( u k )] /n − ( λ [Ω]) /n ≤ ( λ [Ω]) /n R Ω ( | u k +1 | − | u k | ) | w | n dx R Ω | u || w | n dx ≤ ( λ [Ω]) /n R Ω ( | u ∞ | − | u k | ) | w | n dx R Ω | u || w | n dx ≤ C ( n, Ω , u ) Z Ω | u ∞ − u k | dx. The last statement of the theorem follows. (cid:3) Energy characterization of Monge-Amp`ere eigenfunctions
In this section, we make some remarks on the energy characterization of the Monge-Amp`ereeigenfunctions motivated from the proof of Theorem 1.4 in Section 4.Observe that the Monge-Amp`ere measure of each u k j +1 has density R ( u k j ) | u k j | n which is uni-formly bounded from above by a positive constant independent of k . Thus, using the continuityproperty of the energies R Ω | u k j +1 | det D u k j +1 dx (see [1, Lemma 2.9]), we getlim j →∞ Z Ω | u k j +1 | det D u k j +1 dx = Z Ω | w ∞ | det D w ∞ dx so, by (4.5)(5.1) R ( w ∞ ) = lim j →∞ R ( u k j +1 ) = λ [Ω] . We would like to show that w ∞ is a Monge-Amp`ere eigenfunction of Ω. In the proof of Theorem1.4, we used the monotonicity property of the scheme (1.4) given by Lemma 3.3. Finding a directproof from (5.1) leads us to the following question: Question 5.1.
Assume that u ∈ K satisfies R ( u ) = λ [Ω] . Is u a Monge-Amp`ere eigenfunction? TERATIVE SCHEME FOR THE MONGE-AMPERE EIGENVALUE PROBLEM 9
It is well known that for a connected, open and bounded domain in Ω ⊂ R n , if v ∈ W , (Ω) \{ } satisfies Z Ω | Dv | dx = λ Z Ω | v | dx where λ is the first eigenvalue of the Laplace operator with zero boundary condition in Ω then v is in fact a first eigenfunction of the Laplace operator on Ω.An affirmative answer to Question 5.1 will provide a nonlinear analogue of the above result. Wepartially answer Question 5.1 in the affirmative under the hypotheses of interior C regularity andstrict convexity for u . In view of the regularity for the Monge-Amp`ere eigenfunction in Theorem1.1, these conditions are natural though quite restrictive. Lemma 5.2.
Let Ω be a bounded open convex domain in R n . Let λ [Ω] be the Monge-Amp`ereeigenvalue of Ω . Assume that u ∈ K ∩ C (Ω) with D u > in Ω and R ( u ) = λ [Ω] . Then u is aMonge-Amp`ere eigenfunction of Ω . Using Proposition 3.1 and (4.6), we find that the function w ∞ in the proof of Theorem 1.4 satisfiesthe hypotheses of Lemma 5.2 so we can also use this lemma to conclude that w ∞ is a Monge-Amp`ereeigenfunction of Ω, thus bypassing the arguments after (4.6) in Step 2 there.
Proof of Lemma 5.2.
Suppose that u ∈ K ∩ C (Ω), with D u > R ( u ) = λ [Ω]. Then u isuniform convex in each compact subset of Ω, that is, if E ⊂ Ω is compact then D u ≥ c ( E, u ) > E .By multiplying a positive constant to u , we can assume that k u k L n +1 (Ω) = 1. Let v ∈ C ∞ c (Ω).Then, using the uniform convexity of u in each compact subset of Ω, we have u + tv ∈ K for | t | small. Let f ( t ) = R ( u + tv ) = R Ω ( − u − tv ) det D ( u + tv ) dx R Ω ( − u − tv ) n +1 dx . Then, using k u k L n +1 (Ω) = 1 and R ( u ) = λ [Ω], we can compute f ′ (0) = Z Ω ( − v ) det D u dx + Z Ω ( − u ) ddt | t =0 det D ( u + tv ) dx − ( n + 1) λ [Ω] Z Ω | u | n ( − v ) dx. Let ( U ij ) be the cofactor matrix of the Hessian D u = ( u ij ) ≡ (cid:16) ∂ u∂x i ∂x j (cid:17) . Then ddt | t =0 det D ( u + tv ) = U ij v ij . Integrating by parts twice, using u ∈ C (Ω), and the fact that each row and each column of ( U ij )is divergence free, we get Z Ω ( − u ) ddt | t =0 det D ( u + tv ) dx = Z Ω ( − u ) U ij v ij dx = Z Ω ( − v ) U ij u ij dx = n Z Ω ( − v ) det D u dx. Therefore, f ′ (0) = ( n + 1) (cid:20)Z Ω ( − v ) det D u − λ [Ω] Z Ω | u | n ( − v ) dx (cid:21) . Since f has a minimum value at 0, we have f ′ (0) = 0. As a consequence, we find that Z Ω ( − v ) det D u − λ [Ω] Z Ω | u | n ( − v ) dx = 0for all v ∈ C ∞ c (Ω). It follows that det D u = λ [Ω] | u | n in Ω and u is a Monge-Amp`ere eigenfunctionof Ω. (cid:3) It would be interesting to remove the assumption of strict convexity in Lemma 5.2. The difficultyhere is that we could not use the variations u + tv for all v ∈ C ∞ c (Ω) and all t small. Without thestrict convexity of u , we can only use the variations of the form u + tv where v ∈ K and t ≥ Lemma 5.3.
Assume that Ω is a bounded open, and uniformly convex domain in R n with ∂ Ω ∈ C . Let λ [Ω] be the Monge-Amp`ere eigenvalue of Ω . Suppose that u ∈ K ∩ C (Ω) ∩ C , (Ω) and R ( u ) = λ [Ω] . Then, u is a Monge-Amp`ere eigenfunction of Ω . The proof of Lemma 5.3 relies on an interesting result of Lions [14] on the characterization of thedual cone of the cone of convex functions using second derivatives of positive symmetric matrices.In fact, Question 5.1 fits into the framework of Calculus of variations with a convexity constraint.The assumption u ∈ C , (Ω) in Lemma 5.3 is motivated by the classical result of Lions [13] whichsays that if Ω is a bounded open, and uniformly convex domain in R n with ∂ Ω ∈ C , then theMonge-Amp`ere eigenfunctions of Ω are C , (Ω). Proof of Lemma 5.3.
Suppose that u ∈ K ∩ C (Ω) ∩ C , (Ω), and R ( u ) = λ [Ω]. By multiplying apositive constant to u , we can assume that k u k L n +1 (Ω) = 1. Let v ∈ K ∩ C (Ω). Then, u + tv ∈ K for all t ≥
0. Let f ( t ) = R Ω ( − u − tv ) det D ( u + tv ) dx R Ω ( − u − tv ) n +1 dx . As in the proof of Lemma 5.2, we can compute f ′ (0) = Z Ω ( − v ) det D u dx + Z Ω ( − u ) U ij v ij dx − ( n + 1) λ [Ω] Z Ω | u | n ( − v ) dx where ( U ij ) is the cofactor matrix of the Hessian D u = ( u ij ) ≡ (cid:16) ∂ u∂x i ∂x j (cid:17) . Integrating by partstwice, using u ∈ C (Ω) ∩ C , (Ω), we get Z Ω ( − u ) U ij v ij dx = Z Ω ( − v ) U ij u ij dx = n Z Ω ( − v ) det D u dx. Therefore, f ′ (0) = ( n + 1) Z Ω (cid:2) λ [Ω] | u | n − det D u (cid:3) v dx. Since f has a minimum value at 0 on [0 , ∞ ), we have f ′ (0) ≥
0. As a consequence, we find that(5.2) Z Ω (cid:2) λ [Ω] | u | n − det D u (cid:3) v dx ≥ v ∈ C (Ω) ∩ K . Using an approximation argument and recalling that u ∈ K ∩ C , (Ω), wefind that the inequality (5.2) also holds for all v ∈ K .Therefore, the bounded function λ [Ω] | u | n − det D u belongs to the dual cone of the cone of convexfunctions K . By a result of Lions [14, p. 1389] (see also Carlier [6, Theorem 2]), there exist boundedRadon measures ( µ ij ) ≤ i,j ≤ n on Ω such that µ ij = µ ji , ( µ ij ) ≥ λ [Ω] | u | n − det D u = ∂ µ ij ∂x i ∂x j in the sense of distributions, that is, for all v ∈ C (Ω) with v = 0 on ∂ Ω(5.3) Z Ω (cid:0) λ [Ω] | u | n − det D u (cid:1) v dx = Z Ω v ij dµ ij . We can say more about the measures ( µ ij ). From the proof of Theorem 2 in [6], there exist C ∞ c (Ω)functions ψ ( k ) ij (1 ≤ i, j ≤ n ) such that for each k = 1 , , · · · , the matrix ( ψ ( k ) ij ) is symmetric and TERATIVE SCHEME FOR THE MONGE-AMPERE EIGENVALUE PROBLEM 11 nonnegative definite, and the weak limit of ( ψ ( k ) ij ), as k → ∞ , is ( µ ij ). In particular, for anycompact set K ⊂ Ω with nonempty interior ◦ K and any nonnegative definite matrix ( a ij ) withentries a ij ∈ C ( K ), we have(5.4) lim k →∞ Z ◦ K a ij ψ ( k ) ij dx = Z K a ij dµ ij . Using the boundedness of ( µ ij (Ω)) and approximations, we find that (5.3) also holds for v ∈ C , (Ω) ∩ C (Ω) with v = 0 on ∂ Ω.In particular, for v = u , using R ( u ) = λ [Ω], we obtain(5.5) Z Ω u ij dµ ij = Z Ω (cid:0) λ [Ω] | u | n − det D u (cid:1) u dx = − Z Ω (cid:0) λ [Ω] | u | n − det D u (cid:1) | u | dx = 0 . Let E = { x ∈ Ω : D u > } . We show that(5.6) λ [Ω] | u | n − det D u = 0 on E. Indeed, let K be any compact subset of E with nonempty interior ◦ K . Then, D u ∈ C ( K ) and thereis a positive constant c K > D u − c K I n ≥ K where I n denotes the identity n × n matrix. Therefore, for each k , using the fact that the matrix ( ψ ( k ) ij ) is symmetric and nonnegativedefinite, we have(5.7) Z ◦ K u ij ψ ( k ) ij dx ≥ c K Z ◦ K Trace( ψ ( k ) ij ) dx ≥ . From (5.5), we deduce that 0 = Z Ω u ij dµ ij ≥ Z K u ij dµ ij . Now, letting k → ∞ in (5.7) and using (5.4), we obtain0 ≥ Z K u ij dµ ij ≥ c K Z K d Trace( µ ij ) ≥ . Hence, Trace( µ ij )( K ) = 0 . Since this holds for all compact subsets K of E , it follows that(5.8) Trace( µ ij )( E ) = 0 . Now, let v ∈ C ∞ c ( E ). Then, there is a positive constant C v such that − C v I n ≤ D v ≤ C v I n in E .Using (5.8) and (5.4), we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z E v ij dµ ij (cid:12)(cid:12)(cid:12)(cid:12) ≤ C v Z E d Trace( µ ij ) = C v Trace( µ ij )( E ) = 0 . Recalling (5.3), we find that Z E (cid:0) λ [Ω] | u | n − det D u (cid:1) v dx = 0 for all v ∈ C ∞ c ( E )from which we obtain (5.6) as claimed.From (5.6), we find that the last equation of (5.5) reduces to Z Ω \ E (cid:0) λ [Ω] | u | n − det D u (cid:1) | u | dx = 0 . Since | u | > D u = 0 in Ω \ E , we find that the Lebesgue measure of Ω \ E is 0. Thus,from (5.6), we in fact have λ [Ω] | u | n − det D u = 0 on Ω . Hence u is a Monge-Amp`ere eigenfunction of Ω. (cid:3) Acknowledgements.
I would like to thank Farhan Abedin and Jun Kitagawa for their criticalcomments and helpful suggestions on the note.
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