Singular Abreu equations and minimizers of convex functionals with a convexity constraint
aa r X i v : . [ m a t h . A P ] D ec SINGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEXFUNCTIONALS WITH A CONVEXITY CONSTRAINT
NAM Q. LE
Abstract.
We study the solvability of second boundary value problems of fourth order equationsof Abreu type arising from approximation of convex functionals whose Lagrangians depend on thegradient variable, subject to a convexity constraint. These functionals arise in different scientificdisciplines such as Newton’s problem of minimal resistance in physics and monopolist’s problem ineconomics. The right hand sides of our Abreu type equations are quasilinear expressions of secondorder; they are highly singular and a priori just measures. However, our analysis in particular showsthat minimizers of the 2D Rochet-Chon´e model perturbed by a strictly convex lower order term,under a convexity constraint, can be approximated in the uniform norm by solutions of the secondboundary value problems of singular Abreu equations. Introduction
In this paper, we study the solvability and convergence properties of second boundary valueproblems of fourth order equations of Abreu type arising from approximation of several convexfunctionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint.Our analysis in particular shows that minimizers of the 2D Rochet-Chon´e model perturbed by astrictly convex lower order term, under a convexity constraint, can be approximated in the uniformnorm by solutions of the second boundary value problems of Abreu type equations. An intriguingfeature of our Abreu type equations is that their right hand sides are quasilinear expressions ofsecond order derivatives of a convex function. As such, they are highly singular and a priori justmeasures. The main results consist of Theorems 2.1, 2.3, 2.6 and 2.8 to be precisely stated in Section2. In the following paragraphs, we motivate the problems to be studied and recall some previousresults in the literature.Let Ω be a bounded, open, smooth, convex domain in R n ( n ≥ F ( x, z, p ) : R n × R × R n → R be a smooth function which is convex in each of the variables z ∈ R and p = ( p , · · · , p n ) ∈ R n . Let ϕ be a convex and smooth function defined in a neighborhood of Ω . In several problems in differentscientific disciplines such as Newton’s problem of minimal resistance in physics and monopolist’sproblem in economics (see for example [3, 6, 26]), one usually encounters the following variationalproblem with a convexity constraint:(1.1) inf u ∈ ¯ S [ ϕ, Ω ] Z Ω F ( x, u ( x ) , Du ( x )) dx where(1.2) ¯ S [ ϕ, Ω ] = { u : Ω → R | u is convex,u admits a convex extension ϕ in a neighborhood of Ω } . Due to the convexity constraint, it is in general difficult to write down a tractable Euler-Lagrangeequation for the minimizers of (1.1) [5, 7, 23]. Lions [23] showed that, in the sense of distributions,
Mathematics Subject Classification.
Key words and phrases.
Singular Abreu equation, convex functional, convexity constraint, second boundary valueproblem, Monge-Amp`ere equation, linearized Monge-Amp`ere equation, Rochet-Chon´e model.The research of the author was supported in part by the National Science Foundation under grant DMS-1764248. the Euler-Lagrange equation for a minimizer u of (1.1) in a limiting case of the constraint (1.2) isof the form(1.3) ∂F∂z ( x, u ( x ) , Du ( x )) − n X i =1 ∂∂x i (cid:18) ∂F∂p i ( x, u ( x ) , Du ( x )) (cid:19) = n X i,j =1 ∂ ∂x i ∂x j µ ij for some symmetric non-negative matrix µ = ( µ ij ) ≤ i,j ≤ n of Radon measures; see also Carlier [5] fora new proof of this result and related extensions.The structure of the matrix µ in (1.3), to the best of the author’s knowledge, is still mysteriousup to now. Thus, for practical purposes such as implementing numerical schemes to find minimizersof (1.1), it is desirable to find suitably explicit approximations of µ in particular and minimizers of(1.1) in general. This has been done by Carlier and Radice [8] when the Lagrangian F does notdepend on the gradient variable p ; see Sect. 1.1 for a quick review. In this paper, we tackle the morechallenging case when F depends on the gradient variable. This case is relevant to many realisticmodels in physics and economics such as ones described in [3, 26].1.1. Fourth order equations of Abreu type approximating convex functionals with a con-vexity constraint.
When ϕ is strictly convex in a neighborhood of Ω , the Lagrangian F ( x, z, p )does not depend on p , that is, F ( x, z, p ) = F ( x, z ), and uniform convex in its second argumentand ∂∂z F ( x, z ) is bounded uniformly in x for each fixed z , Carlier and Radice [8] show that one canapproximate the minimizer of(1.4) inf u ∈ ¯ S [ ϕ, Ω ] Z Ω F ( x, u ( x )) dx by solutions of second boundary value problems of fourth order equations of Abreu type. Moreprecisely, for each ε >
0, consider the following second boundary value problem for a uniformlyconvex function u ε on an open Euclidean ball B containing Ω :(1.5) ε n X i,j =1 U ijε ∂ w ε ∂x i ∂x j = g ε ( · , u ε ) in B,w ε = (det D u ε ) − in B,u ε = ϕ on ∂B,w ε = ψ on ∂B, where ψ := (det( D ϕ )) − on ∂B , g ε ( x, u ) = (cid:26) ∂F ∂z ( x, u ) x ∈ Ω , ε ( u ( x ) − ϕ ( x )) x ∈ B \ Ω , and U ε = ( U ijε ) is the cofactor matrix of D u ε = (cid:16) ∂ u ε ∂x i ∂x j (cid:17) ≤ i,j ≤ n ≡ (( u ε ) ij ) of the uniformly convexfunction u ε , that is U ε = (det D u ε )( D u ε ) − . Carlier and Radice [8, Theorems 4.2 and 5.3] show that (1.5) has a unique uniformly convex solution u ε ∈ W ,q ( B ) (for all q < ∞ ) which converges uniformly on Ω to the unique minimizer of (1.4)when ε → U ij ) = (det D u )( D u ) − and u ij := ∂ u∂x i ∂x j forany function u . The Abreu equation [1] for a uniformly convex function u n X i,j =1 U ij [(det D u ) − ] ij = f INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 3 first arises in differential geometry [1, 11, 12] where one would like to find a K¨ahler metric of constantscalar curvature. Its related and important cousin is the affine maximal surface equation [30, 31, 32]in affine geometry: n X i,j =1 U ij [(det D u ) − n +1 n +2 ] ij = 0 . We call (1.5) the second boundary value problem because the values of the function u ε and itsHessian determinant det D u ε are prescribed on the boundary ∂B . This is in contrast to the firstboundary value problem where one prescribes the values of the function u ε and its gradient Du ε on ∂B . The fourth order equation in (1.5) arises as the Euler-Lagrange equation of the functional Z Ω F ( x, u ( x )) dx + 12 ε Z B \ Ω ( u − ϕ ) dx − ε Z B log det D udx. At the functional level, the penalization ε R B log det D udx involving the logarithm of the Hessiandeterminant acts as a good barrier for the convexity constraint in problems like (1.4); see also [2] forrelated rigorous numerical results at a discretized level. At the equation level, the results of Carlierand Radice [8] show that, when the Lagrangian F does not depend on p , the matrix µ in (1.3) iswell approximated by ε ( D u ε ) − ≡ ε ( u ijε ) where u ε is the solution of (1.5). To see this, we just notethat the cofactor matrix U ε of D u ε is divergence-free, that is P nj =1 ∂∂x j U ijε = 0 for all i = 1 , · · · , n and hence, noting that U ijε w ε = u ijε , we can write the left hand side of the first equation in (1.5) as ε n X i,j =1 U ijε ∂ w ε ∂x i ∂x j = n X i,j =1 ∂ ∂x i ∂x j ( εU ijε w ε ) = n X i,j =1 ∂ ∂x i ∂x j εu ijε . Gradient-dependent Lagrangians and the Rochet-Chon´e model.
The analysis of Carlier-Radice [8] left open the question of whether one can approximate minimizers of (1.1) by solutionsof second boundary value problems of fourth order equations of Abreu type when the Lagrangian F depends on the gradient variable p . This case is relevant to physics and economic applications. Webriefly describe here the Rochet-Chon´e model in economics. In the Rochet-Chon´e model [26] of themonopolist problem in product line design where the cost of producing product q is the quadraticfunction | q | , the monopolist’s profit as a functional of the buyers’ indirect utility function u isΦ( u ) = Z Ω { x · Du ( x ) − | Du ( x ) | − u ( x ) } γ ( x ) dx. Here Ω ⊂ R n is the collection of types of agents and γ is the relative frequency of different typesof agents in the population. For a consumer of type x ∈ Ω , the indirect utility function u ( x ) iscomputed via the formula u ( x ) = max q ∈ Q { x · q − p ( q ) } where Q ⊂ R n is the product line and p : Q → R is a price schedule that the monopolist needsto design to maximize her overall profit. Since u is the maximum of a family of affine functions,it is convex. Maximizing Φ( u ) over convex functions u is equivalent to minimizing the followingfunctional J over convex functions u : J ( u ) = Z Ω F RC ( x, u ( x ) , Du ( x )) dx where F RC ( x, z, p ) = 12 | p | γ ( x ) − x · pγ ( x ) + zγ ( x ) . As mentioned in [16], even in this simple looking variational problem, the convexity is not easyto handle from a numerical standpoint. M´erigot and Oudet [24] were among the first to makeinteresting progress in this direction. Here we analyze this problem, and its generalization, from anasymptotic analysis standpoint.
NAM Q. LE
In this paper, we are interested in using the second boundary value problems of fourth orderequations of Abreu type to approximate minimizer(s) of the following variational problem(1.6) inf u ∈ ¯ S [ ϕ, Ω ] J ( u )where(1.7) J ( u ) = Z Ω F ( x, u ( x ) , Du ( x )) dx, with F ( x, z, p ) = F ( x, z ) + F ( x, p ) . The choice of form of F in (1.7) simplifies some of our arguments and is clearly motivated by theanalysis of the Rochet-Chon´e model.Similar to the analysis of (1.5) carried out by Carlier-Radice [8], our analysis leads us to two verynatural questions concerning the following second boundary value problem of a highly singular, fullynonlinear fourth order equation of Abreu type for a uniformly convex function u :(1.8) n X i,j =1 U ij w ij = f δ ( · , u, Du, D u ) in Ω ,w = (det D u ) − in Ω ,u = ϕ on ∂ Ω ,w = ψ on ∂ Ω . Here ( U ij ) ≤ i,j ≤ n is the cofactor matrix of the Hessian matrix D u = ( u ij ), δ >
0, Ω is a bounded,open, smooth, uniformly convex domain containing Ω and(1.9) f δ ( x, u ( x ) , Du ( x ) , D u ( x )) = ( ∂∂z F ( x, u ( x )) − P ni =1 ∂∂x i (cid:16) ∂F ∂p i ( x, Du ( x )) (cid:17) x ∈ Ω , δ ( u ( x ) − ϕ ( x )) x ∈ Ω \ Ω . Question 1.
Given ϕ, ψ, F , and F , can we solve the second boundary value problem (1.8)-(1.9)? Question 2.
Are minimizers of (1.6)-(1.7) well approximated by solutions of (1.8)-(1.9) when δ → F and F –see Theorems 2.1 and 2.3 respectively–via analysis of singular Abreu equations.1.3. Singular Abreu equations.
By now, the second boundary value problem for the Abreuequation is well understood [9, 19, 20, 21, 32]. In particular, from the analysis in [20], we know thatif f ∈ L q (Ω) where q > n then we have a unique uniformly convex W ,q (Ω) solution to the secondboundary value problem of a more general form of the Abreu equation:(1.10) n X i,j =1 U ij w ij = f in Ω ,w = G ′ (det D u ) in Ω ,u = ϕ on ∂ Ω ,w = ψ on ∂ Ωwhere ϕ ∈ W ,q (Ω), ψ ∈ W ,q (Ω) with inf ∂ Ω ψ >
0, and G belongs to a class of concave functionswhich include G ( t ) = t θ − θ where 0 < θ < /n and G ( t ) = log t. On the other hand, if the right handside f is only in L q (Ω) with q < n then solutions to (1.10) might not be in W ,q (Ω).In [8], the authors established an a priori uniform bound for solutions of (1.5) thus confirmingthe solvability of (1.5) in all dimensions, where g ε is now bounded, by using the solvability resultsfor (1.10). INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 5
The second boundary value problem of Abreu type in (1.8) has highly singular right hand sideeven in the simple but nontrivial setting of F ( x, z ) = 0 and F ( x, p ) = | p | . Among the simplestanalogue of the first equation in (1.8) is(1.11) U ij [(det D u ) − ] ij = − ∆ u in Ω . To the best of our knowledge, the Abreu type equation of the form (1.11) has not appeared beforein the literature. There are several serious challenges in establishing the solvability of its secondboundary value problem. We highlight here two aspects among these challenges:(C1) It is not known a priori if we can establish the lower bound and upper bound for det D u .Thus, for a convex function u , ∆ u can be only a measure.(C2) Even if we can establish the positive lower bound λ and upper bound λ for det D u ,that is λ ≤ det D u ≤ λ in Ω, we can only deduce from the regularity results for the Monge-Amp`ere equation of De Philippis-Figalli-Savin [10], Schmidt [29] and Savin [27] that ∆ u ∈ L ε (Ω) where ε = ε ( n, λ , λ ) > ε ( n, λ , λ ) → λ /λ → ∞ . In other words, the right hand side of (1.11) has low integrability a prioriwhich can be less than the dimension n . Thus, the results on the solvability of the secondboundary value problem of the Abreu equation in [9, 19, 20, 21, 32] do not apply to the secondboundary value problems of (1.11) and (1.8).In this paper, we are able to overcome these difficulties for both (1.8) and (1.11) in two dimensionsunder suitable conditions on the convex functions F and F ; see Theorem 2.1 which asserts thesolvability of (1.8)-(1.9). This is done via a priori fourth order derivatives estimates and degreetheory. For the a priori estimates, the structural conditions on F and F allow us to establish that λ ≤ det D u ≤ λ in Ω for some positive constants λ , λ and that f δ ( · , u, Du, D u ), as explainedin (C2) for − ∆ u , belongs to L ε (Ω) for some possibly small ε >
0. We briefly explain here howwe can go beyond second order derivatives estimates and why the dimension is restricted to 2.Note that (1.8) consists of a Monge-Amp`ere equation for u in the form of det D u = w − anda linearized Monge-Amp`ere equation for w in the form of U ij w ij = f δ ( · , u, Du, D u ) because thecoefficient matrix ( U ij ) comes from linearization of the Monge-Amp`ere operator: U = ∂ det D u∂u ij . Forthe solvability of second boundary problems such as (1.8) and (1.11), as in [9, 19, 20, 21, 32], a keyingredient is to establish global H¨older continuity of the linearized Monge-Amp`ere equation withright hand side having low integrability. In our case, the integrability exponent is 1 + ε for a small ε > q for the right hand side of thelinearized Monge-Amp`ere equation (with Monge-Amp`ere measure just bounded away from 0 and ∞ ) for which one can establish a global H¨older continuity estimate is q > n/
2. This fact was provedin the author’s paper with Nguyen [22]. The constraint 1 + ε > n/ ε > n to be2. It is exactly this reason that we restrict ourselves in this paper to considering thecase n = 2 . Using the bounds on the Hessian determinant for u and the global H¨older estimates in[22], we can show that w is globally H¨older continuous. Once we have this, we can apply the global C ,α estimates for the Monge-Amp`ere equation in [28, 32] to conclude that u ∈ C ,α (Ω). We updatethis information to U ij w ij = f δ ( · , u, Du, D u ) to have a second order uniformly elliptic equation for w with global H¨older continuous coefficients and bounded right hand side. This gives second orderderivatives estimates for w . Now, fourth order derivative estimates for u easily follows.Under suitable conditions on F and F we can show that solutions to (1.8)-(1.9) converge uni-formly on compact subsets of Ω to the unique minimizer of (1.6)-(1.7); see Theorem 2.3. It impliesin particular that minimizers of the 2D Rochet-Chon´e model perturbed by a highly convex lowerorder term, under a convexity constraint, can be approximated in the uniform norm by solutions ofsecond boundary value problems of singular Abreu equations. NAM Q. LE
Remark 1.1.
Our analysis also covers the case when w = (det D u ) − in (1.8) is replaced by w = (det D u ) θ − where 0 ≤ θ < /n . We will consider these general cases in our main results. Remark 1.2.
As mentioned above, due to the possibly low integrability of the right hand side − ∆ u of (1.11), our analysis is at the moment restricted to two dimensions. However, for the solvabilityof the second boundary value problem, it is quite unexpected that the structure of − ∆ u in twodimensions, that is − ∆ u = − trace ( U ij ), allows us to replace the term (det D u ) − in (1.11) by H (det D u ) for very general functions H including H ( d ) = d θ − for θ ∈ [0 , ∞ ) \ { } ; see Theorem2.6. Note that, it is an open question if (1.10) is solvable for f ∈ L q (Ω) when q > n and G ′ ( d ) = d θ − where θ ∈ [ n , ∞ ) . Remark 1.3.
It should not come as a surprise when Abreu type equations appear in problemsmotivated from economics. On the one hand, in addition to [8] and this paper, Abreu type equationsalso appear in the continuum Nash’s bargaining problem [34]. On the other hand, the monopolist’sproblem can be treated in the framework of optimal transport (see, for example [15, 16]) so it is nottotally unexpected to have deep connections with the Monge-Amp`ere equation. The interesting pointhere is that Abreu type equations involve both the Monge-Amp`ere equation and its linearization.
Notation.
The following notations will be used throughout the paper. Points in R n will bedenoted by x = ( x , · · · , x n ) ∈ R n or p = ( p , · · · , p n ) ∈ R n . I n is the identity n × n matrix. We use ν = ( ν , · · · , ν n ) to denote the unit outer normal vector field on ∂ Ω and ν on ∂ Ω . Unless otherwisestated, repeated indices are summed such as U ij w ij = P ni,j =1 U ij w ij ; f ( x, z ) = ∂F ( x, z ) ∂z ; F p i ( x, p ) = ∂F ( x, p ) ∂p i ; ∇ p F ( x, p ) = ( F p ( x, p ) , · · · , F p n ( x, p )); F p i p j ( x, p ) = ∂ F ( x, p ) ∂p i ∂p j ; F p i x j ( x, p ) = ∂ F ( x, p ) ∂p i ∂x j ; div ( ∇ p F ( x, p )) = n X i =1 ∂∂x i (cid:18) ∂F ( x, p ) ∂p i (cid:19) . We use U = ( U ij ) ≤ i,j ≤ n to denote the cofactor matrix of the Hessian matrix D u = (cid:16) ∂ u∂x i ∂x j (cid:17) ≡ ( u ij ) ≤ i,j ≤ n of a function u ∈ C (Ω). If u is uniformly convex in Ω then U = (det D u )( D u ) − . The rest of the paper is organized as follows. We state our main results in Section 2. In Section 3,we recall tools used in the proofs of our main theorems. In Section 4, we establish a priori estimates.The final section 5 proves the main results in Theorems 2.1, 2.3 2.6 and 2.8.2.
Statements of the main results
Let δ > , Ω be open, smooth, bounded, convex domains in R n such that Ω ⊂⊂ Ω.We study the solvability of the following second boundary value problem of a fully nonlinear,fourth order equation of Abreu type for a uniformly convex function u :(2.1) n X i,j =1 U ij w ij = f δ ( · , u, Du, D u ) in Ω ,w = (det D u ) θ − in Ω ,u = ϕ on ∂ Ω ,w = ψ on ∂ Ω . Here U = ( U ij ) ≤ i,j ≤ n is the cofactor matrix of the Hessian matrix D u = ( u ij ) and(2.2) f δ ( x, u ( x ) , Du ( x ) , D u ( x )) = (cid:26) f ( x, u ( x )) − div ( ∇ p F ( x, Du ( x ))) x ∈ Ω , δ ( u ( x ) − ϕ ( x )) x ∈ Ω \ Ω . We consider the following sets of assumptions for nonnegative constants ρ, c , C ∗ , ¯ c , ¯ C ∗ :(2.3) ( f ( x, z ) − f ( x, ˜ z ))( z − ˜ z ) ≥ ρ | z − ˜ z | ; | f ( x, z ) | ≤ η ( | z | ) for all x ∈ Ω and all z, ˜ z ∈ R INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 7 where η : [0 , ∞ ) → [0 , ∞ ) is a continuous and increasing function.(2.4) 0 ≤ F p i p j ( x, p ) ≤ C ∗ I n ; | F p i x i ( x, p ) | ≤ c | p | + C ∗ for all x ∈ Ω and for each i. (2.5) | F p i ( x, p ) | ≤ ¯ c | p | + ¯ C ∗ for x ∈ ∂ Ω and for each i ; |∇ p F ( x, p ) | ≤ η ( | p | ) for all x ∈ Ω . Our first main theorem is concerned with the solvability of (2.1)-(2.2) in two dimensions.
Theorem 2.1 (Solvability of highly singular second boundary value problems of Abreu type) . Let n = 2 and ≤ θ < /n . Let δ > and let Ω , Ω be open, smooth, bounded, convex domains in R n such that Ω ⊂⊂ Ω . Assume moreover that Ω is uniformly convex. Assume that ϕ ∈ C , (Ω) and ψ ∈ C , (Ω) with inf ∂ Ω ψ > . Assume that (2.3)-(2.5) are satisfied.(i) If either min { c , ¯ c } is sufficiently small (depending only on inf ∂ Ω ψ , Ω and Ω ), or min { ρ, δ } is sufficiently large (depending only on min { c , ¯ c } , Ω and Ω ), then there is a uni-formly convex solution u ∈ W ,q (Ω) to the system (2.1)-(2.2) for all q ∈ ( n, ∞ ) .(ii) if ¯ c = ¯ C ∗ = 0 , then there is a unique uniformly convex solution u ∈ W ,q (Ω) to the system(2.1)-(2.2) for all q ∈ ( n, ∞ ) . Theorem 2.1 will be proved in Section 5. The existence proof uses a priori estimates in Theorem4.1 and degree theory. For the a priori estimates, the technical size conditions in (i) guarantee theuniform bound for u and the L bound for its gradient Du in terms of the data of the problem. Remark 2.2.
Consider the perturbed Rochet-Chon´e model F ( x, z, p ) = F ( x, z ) + F ( x, p ) where F ( x, z ) = γ ( x ) z + ρ | z | , F ( x, p ) = 12 γ ( x ) | p | − x · pγ ( x )where ρ ≥ γ is a Lipschitz function satisfying 0 < γ ≤ C , | Dγ | ≤ C in Ω . Then(2.3)-(2.5) are satisfied with suitable constants c , ¯ c , C ∗ , ¯ C ∗ . If γ = 0 on ∂ Ω then ¯ c = ¯ C ∗ = 0 . More generally, if max ∂ Ω γ ( x ) is small then ¯ c is small. If k Dγ k L ∞ (Ω ) is small then c is small.Our second main theorem asserts the convergence of solutions to (2.1)-(2.2) in two dimensions tothe unique minimizer of (1.6)-(1.7) when the Lagrangian F ( x, z, p ) = F ( x, z ) + F ( x, p ) is highlyconvex in the second variable. Theorem 2.3 (Convergence of solutions of the approximate second boundary value problems ofAbreu type to the minimizer of the convex functional) . Let n = 2 and ≤ θ < /n . Let Ω , Ω be open, smooth, bounded, convex domains in R n such that Ω ⊂⊂ Ω . Moreover, assume that Ω is uniformly convex. Assume that ϕ ∈ C , (Ω) is uniformly convex with inf Ω det D ϕ > and ψ ∈ C , (Ω) with inf ∂ Ω ψ > . Assume that (2.3)-(2.5) are satisfied, and ρ > . For each ε > ,consider the following second boundary value problem: (2.6) ε n X i,j =1 U ijε ( w ε ) ij = f ε ( · , u ε , Du ε , D u ε ) in Ω ,w ε = (det D u ε ) θ − in Ω ,u ε = ϕ on ∂ Ω ,w ε = ψ on ∂ Ω . Here U ε = ( U ijε ) ≤ i,j ≤ n is the cofactor matrix of the Hessian matrix D u ε = (( u ε ) ij ) and (2.7) f ε ( x, u ε ( x ) , Du ε ( x ) , D u ε ( x )) = (cid:26) f ( x, u ε ( x )) − div ( ∇ p F ( x, Du ε ( x ))) x ∈ Ω , ε ( u ε ( x ) − ϕ ( x )) x ∈ Ω \ Ω . Assume that either ¯ c = ¯ C ∗ = 0 or ρ is sufficiently large (depending only on ¯ c + ¯ C ∗ , Ω and Ω ). Let u ε be a uniformly convex solution u ε ∈ W ,q (Ω) to the system (2.6)-(2.7) for all q ∈ ( n, ∞ ) . Then, u ε converges uniformly on compact subsets of Ω to the unique minimizer u ∈ ¯ S [ ϕ, Ω ] (defined in(1.2)) of the problem (1.6) where J is defined by (1.7). NAM Q. LE
Theorem 2.3 will be proved in Section 5.
Remark 2.4.
For the convergence result in Theorem 2.3, we need to establish a uniform bound for u ε independent of ε ; see Lemma 4.4. For this, the uniform convexity of ϕ plays an important role.On the other hand, in Theorem 2.1, we basically use the boundary value of ϕ on ∂ Ω, and therefore ϕ need not be uniformly convex. Remark 2.5.
Several pertinent remarks on Theorem 2.3 are in order.(i) Theorem 2.3 is applicable to the perturbed Rochet-Chon´e model considered in Remark 2.2.Theorem 2.3 implies that minimizers of the 2D Rochet-Chon´e model perturbed by a highlyconvex lower order term, under a convexity constraint, can be approximated in the uniformnorm by solutions of second boundary value problems of Abreu type equations.(ii) The minimization problem (1.6)-(1.7) when F ( x, p ) = γ ( x ) | p | − x · pγ ( x ) with γ ( x ) = 0on ∂ Ω (that is, ¯ c = ¯ C ∗ = 0) was studied by Carlier in [4].(iii) In Theorem 2.3, when ρ > c + ¯ C ∗ >
0, we are unable to prove the uniquenessof uniformly convex solutions u ε to (2.6)-(2.7). Despite this lack of uniqueness, Theorem 2.3says that we have the full convergence of all solutions u ε to the unique minimizer u ∈ ¯ S [ ϕ, Ω ]of the problem (1.6)-(1.7). This is surprising to us.In Theorem 2.1, the function F ( x, p ) grows at most quadratically in p . The following extensiondeals with more general Lagrangian F . Theorem 2.6.
Let Ω ⊂ R be an open, smooth, bounded and uniformly convex domain. Assumethat ϕ ∈ C ∞ (Ω) and ψ ∈ C ∞ (Ω) with inf ∂ Ω ψ > . Let F ( p ) : R → R and H : (0 , ∞ ) → (0 , ∞ ) besmooth. Consider the following second boundary value problem of a fourth order equation of Abreutype for a uniformly convex function u : (2.8) X i,j =1 U ij w ij = − div ( ∇ p F ( Du )) in Ω ,w = H (det D u ) in Ω ,u = ϕ on ∂ Ω ,w = ψ on ∂ Ω . (i) Assume that F is convex, and that F p i p j ( p ) is bounded for p bounded. Assume that H isstrictly decreasing, H ( d ) → when d → ∞ and H ( d ) → ∞ when d → . Then there exists asmooth, uniformly convex solution u ∈ C ∞ (Ω) to (2.8). If H ( d ) = d θ − where ≤ θ < / then the solution is unique.(ii) Assume that ≤ F p i p j ( p ) ≤ C ∗ I . Assume that H is strictly monotone and that H − maps compact subsets of (0 , ∞ ) into compact subsets of (0 , ∞ ) . Then there exists a smooth,uniformly convex solution u ∈ C ∞ (Ω) to (2.8). Theorem 2.6 will be proved in Section 5.
Remark 2.7.
Examples of Lagrangians F in Theorem 2.6 (i) include F ( p ) = 1 s | p | s ( s ≥ , s integer) , or F ( p ) = e | p | . The existence results in Theorem 2.1 and 2.6 can be extended to certain non-convex Lagrangians F . To illustrate the scope of our method, we consider the case of Lagrangian F ( x, z, p ) = 14 ( z − + 12 | p | arising from the study of Allen-Cahn functionals. Our existence result for the singular Abreuequation with Allen-Cahn Lagrangian states as follows. INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 9
Theorem 2.8.
Let Ω ⊂ R be an open, smooth, bounded and uniformly convex domain. Assumethat ϕ ∈ C ∞ (Ω) and ψ ∈ C ∞ (Ω) with inf ∂ Ω ψ > . Then there exists a smooth, uniformly convexsolution u ∈ C ∞ (Ω) to the following second boundary value problem: (2.9) X i,j =1 U ij w ij = u − u − ∆ u in Ω ,w = (det D u ) − in Ω ,u = ϕ on ∂ Ω ,w = ψ on ∂ Ω . Theorem 2.8 will be proved in Section 5.
Remark 2.9.
It would be interesting to establish the higher dimensional versions of Theorems 2.1,2.3, 2.6 and 2.8.
Remark 2.10 (Universal constants) . In Sections 4 and 5, we will work with a fixed exponent q > n ,and we call a positive constant universal if it depends only on n, θ, η , q, δ, c , ¯ c , C ∗ , ¯ C ∗ , ρ , Ω, Ω , k ϕ k W ,q (Ω) , k ψ k W ,q (Ω) and inf ∂ Ω ψ . We use C, C , C , · · · , to denote universal constants and theirvalues may change from line to line.3. Tools used in the proofs of main theorems
In this section, we recall the statements of two main tools used in the proofs of our main theorems.The first tool is the global H¨older estimates for the linearized Monge-Amp`ere equation with righthand side having low integrability. These estimates were established by Nguyen and the author in[22, Theorem 1.7]. They extend in particular the previous result in [19, Theorem 1.4] (see also [21,Theorem 1.13]) where the case of L n right-hand side was treated. Theorem 3.1 (Global H¨older estimates for the linearized Monge-Amp`ere equation) . Let Ω be abounded, uniformly convex domain in R n with ∂ Ω ∈ C . Let φ : Ω → R , φ ∈ C , (Ω) ∩ C (Ω) be aconvex function satisfying < λ ≤ det D φ ≤ Λ < ∞ , and φ | ∂ Ω ∈ C . Denote by (Φ ij ) = (det D φ )( D φ ) − the cofactor matrix of D φ . Let v ∈ C (Ω) ∩ W ,nloc (Ω) be thesolution to the linearized Monge-Amp`ere equation ( Φ ij v ij = f in Ω ,v = ϕ on ∂ Ω , where ϕ ∈ C α ( ∂ Ω) for some α ∈ (0 , and f ∈ L q (Ω) with q > n/ . Then, v ∈ C β (Ω) with theestimate k v k C β (Ω) ≤ C (cid:0) k ϕ k C α ( ∂ Ω) + k f k L q (Ω) (cid:1) where β depends only on λ, Λ , n, q, α , and C depends only on λ, Λ , n, q, α , diam (Ω) , k φ k C ( ∂ Ω) , k ∂ Ω k C and the uniform convexity of Ω . The second tool is concerned with the global W , ε estimates for the Monge-Amp`ere equation.They follows from the interior W , ε estimates in De Philippis-Figalli-Savin [10] and Schmidt [29]and the global estimates in Savin [27] (see also [14, Theorem 5.3]). Theorem 3.2 (Global W , ε estimates for the Monge-Amp`ere equation) . Let Ω ⊂ R n ( n ≥ )be a bounded, uniformly convex domain. Let φ : Ω → R , φ ∈ C , (Ω) ∩ C (Ω) be a convex functionsatisfying < λ ≤ det D φ ≤ Λ in Ω . Assume that ϕ | ∂ Ω and ∂ Ω are of class C . Then, there is a positive constant ε ∈ (0 , dependingonly n, λ, Λ and a positive constant K depending only on n, λ, Λ , Ω , k ϕ k C ( ∂ Ω) and k ∂ Ω k C such that k D ϕ k L ε (Ω) ≤ K. We will frequently use the following estimates for convex functions.
Lemma 3.3.
Let u be a convex function on Ω where Ω ⊂ R n is an open, bounded, and convex set.(i) We have the following L ∞ estimate for u in terms of its boundary value and L norm: k u k L ∞ (Ω) ≤ C ( n, Ω , max ∂ Ω u ) + C ( n, Ω) Z Ω | u | dx. (ii) If Ω ⊂⊂ Ω then (3.1) | Du ( x ) | ≤ max ∂ Ω u − u ( x ) dist ( x, ∂ Ω) ≤ dist (Ω , ∂ Ω) (max ∂ Ω u + k u k L ∞ (Ω) ) for any x ∈ Ω . Proof of Lemma 3.3. (i) The proof is by comparison with cone. We show that if u ≤ ∂ Ω, then(3.2) k u k L ∞ (Ω) ≤ n + 1 | Ω | Z Ω | u | dx. Applying this inequality to the convex function u − max ∂ Ω u , we obtain k u k L ∞ (Ω) ≤ (cid:18) C ( n, Ω) Z Ω | u | dx + C ( n, Ω , max ∂ Ω u ) (cid:19) ≤ C ( n, Ω) Z Ω | u | dx + C ( n, Ω , max ∂ Ω u ) . It remains to prove (3.2) when u ≤ ∂ Ω. Suppose that | u | attains its maximum at x ∈ Ω.Let ˆ C be the cone with base ∂ Ω and vertex at ( x , u ( x )). Then (3.2) follows from the followingestimates 1 n + 1 k u k L ∞ (Ω) | Ω | = 1 n + 1 | u ( x ) || Ω | = Volume of ˆ C ≤ Z Ω | u | dx. (ii) The estimate (3.1) just follows from the convexity of u ; see, for example [21, Lemma 3.11]. (cid:3) A priori estimates for singular Abreu equations
In this section, δ > , Ω are open, smooth, bounded, convex domains in R n such thatΩ ⊂⊂ Ω. We assume moreover that Ω is uniformly convex.The main result of this section is the following global a priori estimates for the second boundaryvalue problem (2.1)-(2.2).
Theorem 4.1.
Let n = 2 , ≤ θ < /n , and q > n . Assume that ϕ ∈ W ,q (Ω) and ψ ∈ W ,q (Ω) with inf ∂ Ω ψ > . Assume that (2.3)-(2.5) are satisfied. Suppose that either min { c , ¯ c } is sufficientlysmall (depending only on inf ∂ Ω ψ , Ω and Ω ), or min { ρ, δ } is sufficiently large (depending only on min { c , ¯ c } , Ω and Ω ). Let u be a smooth, uniformly convex solution of the system (2.1)-(2.2).Then, there is a universal constant C > such that k u k W ,q (Ω) ≤ C. We refer to Remark 2.10 for our convention on universal constants . We now give the outline ofthe proof of Theorem 4.1: • We first prove the L ∞ bound for u (Lemma 4.2) • We next prove the lower bound for the Hessian determinant det D u and then the upperbound for the Hessian determinant det D u (Lemma 4.6) • Finally, we prove the W ,q estimateWe use ν = ( ν , · · · , ν n ) to denote the unit outer normal vector field on ∂ Ω and ν on ∂ Ω .For simplicity, we introduce the following size condition used in statements of several lemmas: INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 11 ( SC ) Either min { c , ¯ c } is sufficiently small (depending only on inf ∂ Ω ψ , Ω and Ω), or min { ρ, δ } is sufficiently large (depending only on min { c , ¯ c } , Ω and Ω).The following lemma establishes the universal L ∞ bound for solutions to the second boundaryvalue problem (2.1)-(2.2) Lemma 4.2.
Let n ≥ , ≤ θ < /n , and q > n . Let u be a smooth solution of the system(2.1)-(2.2). Assume that ϕ ∈ W ,q (Ω) and ψ ∈ W ,q (Ω) with inf ∂ Ω ψ > . Assume that (2.3)-(2.5)are satisfied. Assume that either n ≥ or that ( SC ) holds when n = 2 . Then, there is a universalconstant C > such that k u k L ∞ (Ω) + Z ∂ Ω u nν ≤ C. In the proof of Lemma 4.2, we will use the following basic geometric construction and estimates.
Lemma 4.3. [20, Lemma 2.1 and inequality (2.7)]
Let G : (0 , ∞ ) → R be a smooth, strictlyincreasing and strictly concave function on (0 , ∞ ) . Assume that q > n ≥ and ϕ ∈ W ,q (Ω) . Thereexist a convex function ˜ u ∈ W ,q (Ω) and constants C and C ( G ) depending only on n , q , Ω , and k ϕ k W ,q (Ω) with the following properties:(i) ˜ u = ϕ on ∂ Ω ,(ii) k ˜ u k C (Ω) + k ˜ u k W ,q (Ω) ≤ C, and det D ˜ u ≥ C − > , (iii) letting ˜ w = G ′ (det D ˜ u ) , and denoting by ( ˜ U ij ) the cofactor matrix of (˜ u ij ) , then (cid:13)(cid:13)(cid:13) ˜ U ij ˜ w ij (cid:13)(cid:13)(cid:13) L q (Ω) ≤ C ( G ) , (iv) if u ∈ C (Ω) is a convex function with u = ϕ on ∂ Ω then for u + ν = max(0 , u ν ) , we have k u k L ∞ (Ω) ≤ C + C ( n, Ω) (cid:18)Z ∂ Ω ( u + ν ) n (cid:19) /n . Proof of Lemma 4.2.
The proof is similar to [20, Lemma 2.2]. Let u + ν = max(0 , u ν ). Since u isconvex with boundary value ϕ on ∂ Ω,we have(4.1) u ν ≥ −k Dϕ k L ∞ (Ω) . Our goal is reduced to showing that(4.2) Z ∂ Ω ( u + ν ) n ≤ C because the universal L ∞ bound for u follows from Lemma 4.3 (iv).Let G ( t ) = t θ − θ for t > θ = 0, we set G ( t ) = log t ). Then G ′ ( t ) = t θ − for all t > w = G ′ (det D u ) in Ω.Let ˜ u ∈ W ,q (Ω) be as in Lemma 4.3. The function ˜ G ( d ) := G ( d n ) on (0 , ∞ ) is strictly concavebecause ˜ G ′′ ( d ) = n d n − (cid:20) G ′′ ( d n ) d n + (1 − n ) G ′ ( d n ) (cid:21) < . Using this, G ′ >
0, and the concavity of the map M (det M ) /n in the space of symmetricmatrices M ≥
0, we obtain˜ G ((det D ˜ u ) /n ) − ˜ G ((det D u ) /n ) ≤ ˜ G ′ ((det D u ) /n )((det D ˜ u ) /n − (det D u ) /n ) ≤ ˜ G ′ ((det D u ) /n ) 1 n (det D u ) /n − U ij (˜ u − u ) ij . Since ˜ G ′ ((det D u ) /n ) = nG ′ (det D u )(det D u ) n − n , we rewrite the above inequalities as(4.3) G (det D ˜ u ) − G (det D u ) ≤ wU ij (˜ u − u ) ij . Similarly, for ˜ w = G ′ (det D ˜ u ), we have(4.4) G (det D u ) − G (det D ˜ u ) ≤ ˜ w ˜ U ij ( u − ˜ u ) ij . Adding (4.3) and (4.4), integrating by parts twice and using the fact that ( U ij ) is divergence free,we obtain0 ≥ Z Ω wU ij ( u − ˜ u ) ij + ˜ w ˜ U ij (˜ u − u ) ij = Z ∂ Ω wU ij ( u j − ˜ u j ) ν i + Z Ω U ij w ij ( u − ˜ u ) + Z ∂ Ω ˜ w ˜ U ij (˜ u j − u j ) ν i + Z Ω ˜ U ij ˜ w ij (˜ u − u )= Z ∂ Ω ( ψU ij − ˜ w ˜ U ij )( u j − ˜ u j ) ν i + Z Ω f δ ( · , u, Du, D u )( u − ˜ u ) + Z Ω ˜ U ij ˜ w ij (˜ u − u ) . (4.5)Let us analyze the boundary terms in (4.5). Since u − ˜ u = 0 on ∂ Ω, we have ( u − ˜ u ) j = ( u − ˜ u ) ν ν j ,and hence U ij ( u − ˜ u ) j ν i = U ij ν j ν i ( u − ˜ u ) ν = U νν ( u − ˜ u ) ν where U νν = det D x ′ u with x ′ ⊥ ν denoting the tangential directions along ∂ Ω. Therefore,(4.6) ( ψU ij − ˜ w ˜ U ij )( u j − ˜ u j ) ν i = ( ψU νν − ˜ w ˜ U νν )( u ν − ˜ u ν ) . To simplify notation, we use f δ to denote f δ ( · , u, Du, D u ) when there is no confusion. By Lemma 4.3, the quantities ˜ u , ˜ u ν , ˜ U νν and k ˜ U ij ˜ w ij k L (Ω) are universally bounded. These boundscombined with (4.5) and (4.6) give(4.7) Z ∂ Ω ψU νν u ν ≤ C + C k u k L ∞ (Ω) + C Z ∂ Ω ( | U νν | + | u ν | ) + Z Ω − f δ ( u − ˜ u ) dx. On the other hand, from u − ϕ = 0 on ∂ Ω, we have, with respect to a principle coordinate systemat any point y ∈ ∂ Ω (see, e.g., [17, formula (14.95) in § D ij ( u − ϕ ) = ( u − ϕ ) ν κ i δ ij , i, j = 1 , · · · , n − , where κ , · · · , κ n − denote the principle curvatures of ∂ Ω at y .Let K = κ · · · κ n − be the Gauss curvature of ∂ Ω at y ∈ ∂ Ω. Then, at any y ∈ ∂ Ω, by notingthat det D x ′ u = det( D ij u ) ≤ i,j ≤ n − and taking the determinants of(4.8) D ij u = u ν κ i δ ij − ϕ ν κ i δ ij + D ij ϕ, we obtain, using also (4.1)(4.9) U νν = K ( u ν ) n − + E, where | E | ≤ C (1 + | u ν | n − ) = C (1 + ( u + ν ) n − ) . Now, it follows from (4.7), (4.9) and Lemma 4.3(iv) that Z ∂ Ω Kψu nν ≤ C + C k u k L ∞ (Ω) + C Z ∂ Ω ( u + ν ) n − + Z Ω − f δ ( u − ˜ u ) dx ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) n (cid:19) ( n − /n + Z Ω − f δ ( u − ˜ u ) dx. (4.10)We will analyze the last term on the right hand side of (4.10).By construction, k ˜ u k L ∞ (Ω) ≤ C and k ϕ k L ∞ (Ω) ≤ C . Thus, we have(4.11) − δ ( u − ϕ )( u − ˜ u ) ≤ − | u | δ + C ( δ ) INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 13 where C ( δ ) > A := Z Ω − f ( x, u )( u − ˜ u ) dx ≤ Z Ω − f ( x, ˜ u )( u − ˜ u ) dx − ρ Z Ω | u − ˜ u | dx ≤ Z Ω η ( | ˜ u | ) | u − ˜ u | dx − ρ Z Ω | u − ˜ u | dx ≤ C Z Ω | u | dx + C − ρ Z Ω | u − ˜ u | dx. Step 1: Estimate R Ω − f δ ( u − ˜ u ) dx by expansion. By the convexity of u and F ( x, p ) in p , we have F p i p j u ij ≥
0. Moreover, u ≤ sup ∂ Ω ϕ ≤ C and | ˜ u | ≤ C . Thus, recalling (2.4), we find that(4.13) F p i p j u ij ( u − ˜ u ) ≤ CF p i p j u ij ≤ CC ∗ ∆ u. On the other hand, for any i = 1 , · · · , n , using (2.4) and the first inequality in (3.1), we can bound F p i x i ( x, Du ( x ))( u − ˜ u ) in Ω by(4.14) | F p i x i ( x, Du ( x ))( u ( x ) − ˜ u ( x )) | ≤ ( c | Du ( x ) | + C ∗ ) | u ( x ) − ˜ u ( x ) |≤ ( c C (Ω , Ω) | u ( x ) | + C )( | u ( x ) | + C ) ≤ c C (Ω , Ω) | u ( x ) | + C | u ( x ) | + C. From (4.12), (4.13) and (4.14) together with the divergence theorem, we find that Z Ω − f δ ( u − ˜ u ) dx = Z Ω [ − f ( x, u ( x )) + div ( ∇ p F ( x, Du ( x )))]( u − ˜ u ) dx = A + Z Ω ( F p i x i ( x, Du ( x )) + F p i p j u ij )( u − ˜ u ) dx ≤ A + Z Ω ( c C (Ω , Ω) | u | + C | u | + C ) dx + Z Ω C ∗ C ∆ udx ≤ C k u k L ∞ (Ω) + C + c C (Ω , Ω) Z Ω | u | dx − ρ Z Ω | u − ˜ u | dx + C Z ∂ Ω ( u ν ) + . (4.15) Step 2: Estimate R Ω − f δ ( u − ˜ u ) dx by integration by parts. Note that ˜ u and | D ˜ u | are universallybounded. Thus, using the convexity of F ( x, p ) in p together with (2.5) and (3.1), we have thefollowing estimates in Ω −∇ p F ( x, Du ( x )) · ( Du − D ˜ u ) ≤ −∇ p F ( x, D ˜ u ) · ( Du − D ˜ u )(4.16) ≤ η ( | D ˜ u | )( C (Ω , Ω) | u ( x ) | + C ) ≤ C | u ( x ) | + C. On the other hand, also by (2.5) and (3.1), we have the following estimates on ∂ Ω :( u − ˜ u ) ∇ p F ( x, Du ( x )) · ν ≤ (¯ c | Du | + ¯ C ∗ ) | u − ˜ u |≤ ¯ c C (Ω , Ω) k u k L ∞ (Ω) + (¯ c + ¯ C ∗ ) C (Ω , Ω) k u k L ∞ (Ω) + C. (4.17)Now, integrating by parts and using (4.12) together with (4.16) -(4.17), we obtain(4.18) Z Ω − f δ ( u − ˜ u ) = Z Ω − f ( x, u ( x ))( u − ˜ u ) + Z Ω div ( ∇ p F ( x, Du ( x )))( u − ˜ u )= A + Z ∂ Ω ( u − ˜ u ) ∇ p F ( x, Du ( x )) · ν + Z Ω −∇ p F ( x, Du ( x )) · ( Du ( x ) − D ˜ u ( x )) ≤ C + (¯ c + ¯ C ∗ ) C (Ω , Ω) k u k L ∞ (Ω) + C Z Ω | u | dx − ρ Z Ω | u − ˜ u | dx + ¯ c C (Ω , Ω) k u k L ∞ (Ω) . Step 3: n ≥ . From (4.18) and (4.11), and recalling Lemma 4.3 (iv), we have Z Ω − f δ ( u − ˜ u ) = Z Ω \ Ω − δ ( u − ϕ )( u − ˜ u ) + Z Ω − f δ ( u − ˜ u ) ≤ C + C | u k L ∞ (Ω) + C k u k L ∞ (Ω) ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) n (cid:19) /n . From n ≥
3, (4.10), (4.1) and the above estimates, we obtain Z ∂ Ω Kψ ( u + ν ) n ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) n (cid:19) ( n − /n + Z Ω − f δ ( u − ˜ u ) ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) n (cid:19) ( n − /n . From H¨older inequality, n ≥
3, and inf ∂ Ω Kψ >
0, we easily obtain R ∂ Ω ( u + ν ) n ≤ C which is (4.2). For the rest of the proof of this lemma, we focus on the more difficult case of n = 2 . We will nowuse the condition (SC) . Step 4: when min { c , ¯ c } is sufficiently small (depending only on inf ∂ Ω ψ , Ω and Ω ). From (4.15)and (4.18), and recalling (4.11), we obtain for ˆ c = min { c , ¯ c } Z Ω − f δ ( u − ˜ u ) ≤ C k u k L ∞ (Ω) + C + ˆ c C (Ω , Ω) k u k L ∞ (Ω) + C Z ∂ Ω ( u ν ) + + Z Ω \ Ω − δ ( u − ϕ )( u − ˜ u ) ≤ C k u k L ∞ (Ω) + C + ˆ c C (Ω , Ω) k u k L ∞ (Ω) + C Z ∂ Ω ( u ν ) + . (4.19)From (4.10), (4.19) and n = 2, we deduce from Lemma 4.3 (iv) that Z ∂ Ω Kψu ν ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) (cid:19) / + Z Ω − f δ ( u − ˜ u ) ≤ C (cid:18)Z ∂ Ω ( u + ν ) (cid:19) / + C + ˆ c C (Ω , Ω) Z ∂ Ω ( u + ν ) . (4.20)Suppose that ˆ c is small, say ˆ c C (Ω , Ω) < (1 / ∂ Ω ψ )(inf ∂ Ω K ) . Then it follows from (4.20), H¨older inequality and inf ∂ Ω ψ > R ∂ Ω ( u + ν ) ≤ C . Thus (4.2) isproved. Step 5: When min { ρ, δ } is sufficiently large (depending only on min { c , ¯ c } , Ω and Ω ). Case 1: c ≤ ¯ c . In this case, we use (4.15). Suppose that ρ > c C (Ω , Ω) + 1 >
0. Then,(4.21) − ρ Z Ω | u − ˜ u | dx ≤ − ρ Z Ω | u | dx + C ( ρ ) ≤ − c C (Ω , Ω) Z Ω | u | dx + C. Combining (4.15) and (4.21) with (4.11) and Lemma 4.3(iv), we get(4.22) Z Ω − f δ ( u − ˜ u ) = Z Ω − f δ ( u − ˜ u ) + Z Ω \ Ω − δ ( u − ϕ )( u − ˜ u ) ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) n (cid:19) /n . Thus, for n = 2, we deduce from (4.10) and (4.22) that Z ∂ Ω Kψu ν ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) (cid:19) / + Z Ω − f δ ( u − ˜ u ) ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) (cid:19) / . (4.23) INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 15
From (4.23), the H¨older inequality and inf ∂ Ω ψ >
0, we easily obtain R ∂ Ω ( u + ν ) n ≤ C and (4.2) follows. Case 2: ¯ c ≤ c . From (4.18), Lemma 3.3 (i), and recalling (4.11), we obtain(4.24) Z Ω − f δ ( u − ˜ u ) ≤ C k u k L ∞ (Ω) + C + ¯ c C (Ω , Ω) k u k L ∞ (Ω) − ρ Z Ω | u − ˜ u | dx + Z Ω \ Ω − δ ( u − ϕ )( u − ˜ u ) ≤ C k u k L ∞ (Ω) + C + ¯ c C (Ω , Ω) Z Ω | u | dx − ρ Z Ω | u | dx + Z Ω \ Ω − | u | δ . Thus, if min { ρ, δ } > c C (Ω , Ω), then (4.24) gives(4.25) Z Ω − f δ ( u − ˜ u ) ≤ C k u k L ∞ (Ω) + C. Now, for n = 2, we deduce from (4.10), (4.25) and Lemma 4.3(iv) that(4.26) Z ∂ Ω Kψu ν ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) (cid:19) / + Z Ω − f δ ( u − ˜ u ) ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) (cid:19) / . From (4.26), the H¨older inequality and inf ∂ Ω ψ >
0, we easily obtain R ∂ Ω ( u + ν ) ≤ C . This completesthe proof of (4.2) in all cases. (cid:3) In the following lemma, we establish universal a priori estimates for solutions to (2.6)-(2.7). Theseestimates do not depend on ε . Lemma 4.4.
Let n = 2 , q > n and ≤ θ < /n . Assume that (2.3)-(2.5) are satisfied, and ρ > .Assume that ϕ ∈ W ,q (Ω) with inf Ω det D ϕ > , and ψ ∈ W ,q (Ω) with inf ∂ Ω ψ > . Assume thatone of the following conditions holds:(i) ¯ c = ¯ C ∗ = 0 .(ii) ρ is large and ε is small ( depending only on ¯ c + ¯ C ∗ , Ω and Ω ).Let u ε ∈ W ,q (Ω) be a uniformly convex solution to the system (2.6)-(2.7). Then, there is a universalconstant C > (depending also on inf Ω det D ϕ but independent of ε ) such that Z ∂ Ω ε ( u ε ) ν + ρ Z Ω | u ε − ϕ | dx + Z Ω \ Ω ε | u ε − ϕ | dx ≤ C. (4.27) Proof.
Let ˆ C ∗ := ¯ c + ¯ C ∗ . Let ¯ u = ϕ and ¯ f = ¯ U ij ¯ w ij where ¯ w = (det D ¯ u ) θ − and ¯ U = ( ¯ U ij ) is thecofactor matrix of D ¯ u . Since q > n , and ¯ u ∈ W ,q (Ω) with inf Ω det D ¯ u >
0, we have(4.28) k ¯ f k L (Ω) ≤ C. We redo the estimates in Lemma 4.2 for u ε where we replace ˜ u by ¯ u . First, (4.5) becomes(4.29) Z ∂ Ω w ε U ijε (( u ε ) j − ¯ u j ) ν i + Z Ω U ijε ( w ε ) ij ( u ε − ¯ u )+ Z ∂ Ω ¯ w ¯ U ij (¯ u j − ( u ε ) j ) ν i + Z Ω ¯ U ij ¯ w ij (¯ u − u ε ) ≤ . To simplify notation, we use f ε to denote f ε ( · , u ε , Du ε , D u ε ).From U ijε ( w ε ) ij = ε − f ε , Lemma 4.3(iv), (4.28), and the uniform boundedness of ¯ u, ¯ w, D ¯ u , (4.29)becomes Z ∂ Ω ( u ε ) ν ≤ C (cid:18)Z ∂ Ω ( u ε ) ν (cid:19) / + Z Ω − ε − f ε ( u ε − ¯ u ) dx + C. (4.30)This estimate is similar to (4.10) where now we also use inf ∂ Ω ψ > ∂ Ω to absorb inf ∂ Ω ψ > ∂ Ω to the right hand side of (4.10). From Young’s inequality, we can absorb the first term on the right hand side of (4.30) to its left hand side. Then,multiplying both sides by ε , we get Z ∂ Ω ε ( u ε ) ν ≤ C + C Z Ω − f ε ( u ε − ¯ u ) . (4.31)As in the estimates for A in (4.12), we have A ε := Z Ω − f ( x, u ε ( x ))( u ε − ¯ u ) dx ≤ C + C Z Ω | u ε | dx − ρ Z Ω | u ε − ¯ u | dx. From Lemma 3.3 (i) and the inequality | u ε | ≤ | u ε − ϕ | + | ϕ | ), we have(4.32) k u ε k L ∞ (Ω) ≤ C (Ω) Z Ω | u ε | dx + C ≤ C (Ω) Z Ω | u ε − ϕ | dx + C. Using (4.18) to u ε , f ε , ¯ u and taking into account (4.32), ρ > C ∗ := ¯ c + ¯ C ∗ , we have Z Ω − f ε ( u ε − ¯ u ) ≤ C + C Z Ω | u ε | dx − ρ Z Ω | u ε − ¯ u | dx + ˆ C ∗ C (Ω , Ω) (cid:16) k u ε k L ∞ (Ω) (cid:17) ≤ C − ρ Z Ω | u ε − ¯ u | dx + ˆ C ∗ C (Ω , Ω) Z Ω | u ε − ϕ | dx. (4.33)It follows from (4.31), (4.33), ¯ u = ϕ in Ω, and f ε = ε ( u ε − ϕ ) on Ω \ Ω that Z ∂ Ω ε ( u ε ) ν ≤ C + Z Ω − f ε ( u ε − ¯ u ) dx = C + Z Ω − f ε ( u ε − ¯ u ) + Z Ω \ Ω − f ε ( u ε − ¯ u ) dx ≤ C − ρ Z Ω | u ε − ϕ | dx + Z Ω \ Ω − ε ( u ε − ϕ ) dx + ˆ C ∗ C (Ω , Ω) Z Ω | u ε − ϕ | dx. (4.34) Case 1: ρ > and ¯ c = ¯ C ∗ = 0 . In this case, ˆ C ∗ = 0 in (4.34) and (4.27) follows from this inequality. Case 2: ρ is sufficient large and ε is sufficiently small (depending only on ¯ c + ¯ C ∗ , Ω and Ω ). Whenmin { ε , ρ } > C ∗ C (Ω , Ω) + 2 , then clearly (4.34) gives (4.27). (cid:3) We prove the uniqueness part of Theorem 2.1 in the following lemma.
Lemma 4.5.
Let n ≥ , q > n and ≤ θ < /n . Assume that ϕ ∈ W ,q (Ω) and ψ ∈ W ,q (Ω) with inf ∂ Ω ψ > . Assume that (2.3)-(2.5) are satisfied with ¯ c = ¯ C ∗ = 0 . Then the problem (2.1)-(2.2)has at most one uniformly convex solution u ∈ W ,q (Ω) .Proof. Suppose that u ∈ W ,q (Ω) and ˆ u ∈ W ,q (Ω) are two uniformly convex solutions of (2.1)-(2.2).Let ˆ U = ( ˆ U ij ) be the cofactor matrix of D ˆ u and let ˆ w = G ′ (det D ˆ u ). Here, G ( t ) = t θ − θ for t > θ = 0, we set G ( t ) = log t ). We use the same notation as in the proof of Lemma 4.2. Then,we obtain as in (4.5) the estimate0 ≥ Z Ω wU ij ( u − ˆ u ) ij + ˆ w ˆ U ij (ˆ u − u ) ij = Z ∂ Ω ψ ( U ij − ˆ U ij )( u j − ˆ u j ) ν i + Z Ω ( f δ ( · , u, Du, D u ) − f δ ( · , ˆ u, D ˆ u, D ˆ u ))( u − ˆ u ) dx = Z ∂ Ω ψ ( U νν − ˆ U νν )( u ν − ˆ u ν ) + Z Ω [ f ( x, u ( x )) − f ( x, ˆ u ( x ))]( u − ˆ u ) dx + Z Ω div ( ∇ p F ( x, D ˆ u ( x )) − ∇ p F ( x, Du ( x )))( u − ˆ u ) dx + 1 δ Z Ω \ Ω ( u − ˆ u ) dx. (4.35) INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 17
By (2.3), the integral concerning f in the above expression is nonnegative. Hence, integrating byparts and using ¯ c = ¯ C ∗ = 0, we find that(4.36) 0 ≥ Z ∂ Ω ψ ( U νν − ˆ U νν )( u ν − ˆ u ν ) + Z Ω ( ∇ p F ( x, D ˆ u ( x )) − ∇ p F ( x, Du ( x )))( D ˆ u − Du ) dx. It is clear from (4.8) that if u ν > ˆ u ν then U νν > ˆ U νν . Therefore, from the convexity of F ( x, p ) in p which follows from (2.4), we deduce that u ν = ˆ u ν on ∂ Ω and that (4.36) must be an equality. Thisimplies that (4.35) must be an equality. From the derivation of (4.35) using the strict concavityof G as in (4.3) and (4.4) but applied to det D u and det D ˆ u , and the fact that (4.35) is now anequality, we deduce that det D u = det D ˆ u in Ω. Hence u = ˆ u on Ω. (cid:3) In the next lemma, we establish the universal bounds from below and above for the Hessiandeterminant of solutions to (2.1)-(2.2).
Lemma 4.6.
Let n = 2 , q > n and ≤ θ < /n . Assume that ϕ ∈ W ,q (Ω) and ψ ∈ W ,q (Ω) with inf ∂ Ω ψ > . Assume that (2.3)-(2.5) are satisfied. Suppose that ( SC ) holds. Let u be a smooth,uniformly convex solution of the system (2.1)-(2.2). There is a universal constant C > such that C − ≤ det D u ≤ C in Ω . Proof of Lemma 4.6.
To simplify notation, we use f δ to denote f δ ( · , u, Du, D u ). Step 1: Lower bound for det D u . Let ¯ w = w + C ∗ | x | . Let χ Ω be the characteristic function of Ω ,that is χ Ω ( x ) = 1 if x ∈ Ω and χ Ω ( x ) = 0, if otherwise. Then, using (2.2), (2.4) and the universalbound for u in Lemma 4.2, we find that in Ω the following hold: U ij ¯ w ij = f δ + 2 C ∗ ∆ u ≥ − C − | F p i x i ( x, Du ) | χ Ω ( x ) − F p i p j ( x, Du ) u ij + 2 C ∗ ∆ u ≥ − C − C ∗ | Du | χ Ω := − ˆ f . By combining the universal bound for u in Lemma 4.2 and Lemma 3.3(ii), we obtain that k ˆ f k L ∞ (Ω) ≤ C for a universal constant C . Thus, k ˆ f k L (Ω) ≤ C . In two dimensions, we havedet( U ij ) = (det D u ) n − = det D u = w θ − where we used the second equation of (2.1) for the last equality. Now, we apply the ABP estimate[18, Theorem 2.21] to ¯ w on Ω to obtain k ¯ w k L ∞ (Ω) ≤ sup ∂ Ω ¯ w + C diam(Ω) k ˆ f (det U ij ) / k L (Ω) ≤ sup ∂ Ω ( ψ + C ∗ | x | ) + C k w − θ ) ˆ f k L (Ω) . Clearly, ¯ w ≥ w >
0. Therefore, the above estimates and 0 ≤ θ < / k w k L ∞ (Ω) ≤ C + C k w − θ ) ˆ f k L (Ω) ≤ C + C k w − θ ) k L ∞ (Ω) k ˆ f k L (Ω) ≤ C + C k w k − θ ) L ∞ (Ω) . Because 0 ≤ θ < /
2, we have − θ ) <
1. It follows that w is bounded from above. Sincedet D u = w θ − , we conclude that det D u is bounded from below by a universal constant C > Step 2: Upper bound for det D u . Since det D u = w θ − where 0 ≤ θ < /
2, to prove the universalupper bound for det D u , we only need to obtain a positive lower bound for w . For this, we use theABP maximum principle; see [9, 31] for a slightly different argument.First, we will use the following splitting of f δ :(4.37) U ij w ij = f δ = γ ∆ u + g in Ω , where γ ( x ) = ( − F pipj ( x,Du ) u ij ∆ u x ∈ Ω , x ∈ Ω \ Ω
08 NAM Q. LE and(4.38) g ( x ) = (cid:26) f ( x, u ( x )) − F p i x i ( x, Du ( x )) x ∈ Ω , δ ( u ( x ) − ϕ ( x )) x ∈ Ω \ Ω . From (2.4), we have(4.39) k γ k L ∞ (Ω) ≤ C ∗ . We claim that(4.40) k g k L ∞ (Ω) ≤ C. Indeed, from (2.3) and (2.4), we easily find that for all x ∈ Ω(4.41) | g ( x ) | ≤ (cid:26) η ( k u k L ∞ (Ω) ) + c | Du ( x ) | + C ∗ x ∈ Ω , δ ( k u k L ∞ (Ω) + k ϕ k L ∞ (Ω) ) x ∈ Ω \ Ω . Now, we use the universal bound for u in Lemma 4.2 and Lemma 3.3 (ii) to derive (4.40) from(4.41).Recall from (i) that det D u ≥ C . Thus, w det D u = (det D u ) θ ≥ C θ := c > . From (4.37) and γ ≤
0, we find that f δ ≤ g . By (4.40), we find that f + δ ≤ | g | is bounded by auniversal constant. Let M = | f + δ | L ∞ (Ω) + 12 c < ∞ and v ε = log( w + ε ) − M u ∈ W ,q (Ω)where ε >
0. Then, in Ω, we have u ij v εij = u ij (cid:18) w ij w + ε − w i w j ( w + ε ) − M u ij (cid:19) ≤ u ij w ij w + ε − nM = f δ ( w + ε ) det D u − M ≤ k f + δ k L ∞ (Ω) c − M < . By the ABP estimate (see [18, Theorem 2.21]) for − v ε in Ω, we have v ε ≥ inf ∂ Ω v ε ≥ log(inf ∂ Ω ψ ) − M ϕ ≥ − C in Ω . From v ε = log( w + ε ) − M u and the universal bound for u in Lemma 4.2, we obtain log( w + ε ) ≥ − C .Thus, letting ε →
0, we get w ≥ e − C as desired. (cid:3) Now, we are a in position to prove Theorem 4.1.
Proof of Theorem 4.1.
To simplify notation, we use f δ to denote f δ ( · , u, Du, D u ).From Lemma 4.6, we can find a universal constant C > C − ≤ det D u ≤ C in Ω . From ϕ ∈ W ,q (Ω) with q > n , we have ϕ ∈ C (Ω) by the Sobolev embedding theorem. Byassumption, Ω is bounded, smooth and uniformly convex. From u = ϕ on ∂ Ω and (4.42), we canapply the global W , ε estimates for the Monge-Amp`ere equation in Theorem 3.2 to conclude that(4.43) k D u k L ε (Ω) ≤ C for some universal constants ε > C >
0. Recall the following splitting of f δ in (4.37):(4.44) f δ = γ ∆ u + g. Thus, from (4.43), (4.39) and (4.40), we find that k f δ k L ε (Ω) ≤ C for a universal constant C >
0. From ψ ∈ W ,q (Ω) with q > n , we have ψ ∈ C (Ω) by the Sobolevembedding theorem. Now, we apply the global H¨older estimates for the linearized Monge-Amp`ereequation in Theorem 3.1 to U ij w ij = f δ in Ω with boundary value w = ψ ∈ C ( ∂ Ω) on ∂ Ω toconclude that w ∈ C α (Ω) with(4.45) k w k C α (Ω) ≤ C (cid:16) k ψ k C ( ∂ Ω) + k f δ k L ε (Ω) (cid:17) ≤ C for universal constants α ∈ (0 ,
1) and C >
0. Now, we note that u solves the Monge-Amp`ereequation det D u = w θ − with right hand side being in C α (Ω) and boundary value ϕ ∈ C ( ∂ Ω) on ∂ Ω. Therefore, by theglobal C ,α estimates for the Monge-Amp`ere equation [32, 28], we have u ∈ C ,α (Ω) with universalestimates(4.46) k u k C ,α (Ω) ≤ C and C − I ≤ D u ≤ C I . As a consequence, the second order operator U ij ∂ ij is uniformly elliptic with H¨older continuouscoefficients.Recalling (4.44), and using (4.39) together with (4.40), we obtain(4.47) k f δ k L ∞ (Ω) ≤ C . Thus, from the equation U ij w ij = f δ with boundary value w = ψ where ψ ∈ W ,q (Ω), we concludethat w ∈ W ,q (Ω) and therefore u ∈ W ,q (Ω) with universal estimate k u k W ,q (Ω) ≤ C . (cid:3) Proofs of the main theorems
In this section, we prove Theorems 2.1, 2.3, 2.6 and 2.8.
Proof of Theorem 2.1.
The existence and uniqueness result in (ii) follows from the existence in (i)and the uniqueness result in Lemma 4.5. It remains to prove (i).The proof of (i) uses the a priori estimates in Theorem 4.1 and degree theory as in [9, 31] (seealso [21]). Since the proof is short, we include it here. Assume q > n .Fix α ∈ (0 , R > D ( R ) in C α (Ω)as follows: D ( R ) = { v ∈ C α (Ω) | v ≥ R − , k v k C α (Ω) ≤ R } . For t ∈ [0 , t : D ( R ) → C α (Ω) as follows. Given w ∈ D ( R ), define u ∈ C ,α (Ω) to be the unique uniformly convex solution to(5.1) ( det D u = w θ − in Ω ,u = ϕ on ∂ Ω . The existence of u follows from the boundary regularity result of the Monge-Amp`ere equationestablished by Trudinger and Wang [32]. Next, let w t ∈ W ,q (Ω) be the unique solution to theequation(5.2) ( U ij ( w t ) ij = tf δ ( · , u, Du, D u ) in Ω ,w t = tψ + (1 − t ) on ∂ Ω . Because q > n , w t lies in C α (Ω). We define Φ t to be the map sending w to w t .We note that:(i) Φ ( D ( R )) = { } , and in particular, Φ has a unique fixed point. (ii) The map [0 , × D ( R ) → C α (Ω) given by ( t, w ) Φ t ( w ) is continuous.(iii) Φ t is compact for each t ∈ [0 , t ∈ [0 , w ∈ D ( R ) is a fixed point of Φ t then w / ∈ ∂D ( R ).Indeed, part (iii) follows from the standard a priori estimates for the two separate equations (5.1)and (5.2). For part (iv), let w > t . Then w ∈ W ,q (Ω) and hence u ∈ W ,q (Ω).Next we apply Theorem 4.1 to obtain w > R − and k w k C α (Ω) < R for some R sufficiently large,depending only on the initial data but independent of t ∈ [0 , t is well-defined for each t and is constant on [0 ,
1] (see[25, Theorem 2.2.4], for example). Φ has a fixed point and hence Φ must also have a fixed point w , giving rise to a uniformly convex solution u ∈ W ,q (Ω) of our second boundary value problem(2.1)-(2.2). (cid:3) Proof of Theorem 2.3.
If ¯ c = ¯ C ∗ = 0, then the existence of a unique uniformly convex solution u ε ∈ W ,q (Ω) to the system (2.6)-(2.7) for all q ∈ ( n, ∞ ) follows from Theorem 2.1 (ii). If ρ issufficiently large (depending only on ¯ c + ¯ C ∗ , Ω and Ω), then the existence of a uniformly convexsolution u ε ∈ W ,q (Ω) to the system (2.6)-(2.7) for all q ∈ ( n, ∞ ) follows from Theorem 2.1 (i);moreover, since we are interested in the limit of { u ε } when ε →
0, we can assume that ε is sufficientlysmall (depending only on ¯ c + ¯ C ∗ , Ω and Ω) so that Lemma 4.4 applies.In all cases, by Lemma 4.4, there is a universal constant C independent of ε such that(5.3) Z ∂ Ω ε ( u ε ) ν + ρ Z Ω | u ε − ϕ | dx + Z Ω \ Ω ε | u ε − ϕ | dx ≤ C. Step 1: A subsequence of { u ε } converges. First, we show that, up to extraction of a subsequence, u ε converges uniformly on compact subsets of Ω to a convex function u ∈ ¯ S [ ϕ, Ω ] where ¯ S [ ϕ, Ω ] isdefined as in (1.2). Indeed, by Lemma 3.3 (i), u ε = ϕ on ∂ Ω and (5.3) where ρ >
0, we have k u ε k L ∞ (Ω) ≤ C ( n, Ω , max ∂ Ω u ε ) + C ( n, Ω) Z Ω | u ε | dx ≤ C for a universal constant C >
0. It follows that the sequence { u ε } is uniformly bounded. By Lemma3.3 (ii), | Du ε | is uniformly bounded on compact subsets of Ω. Thus, by the Arzela–Ascoli theorem,up to extraction of a subsequence, u ε converges uniformly on compact subsets of Ω to a convexfunction u . Moreover, we can assume that u ε converges to u on W , (Ω ). Using (5.3), we get u ∈ ¯ S [ ϕ, Ω ].Let G ( t ) = t θ − θ for t > θ = 0, we set G ( t ) = log t ).Next, consider the following functional J ε over the set of convex functions v on Ω:(5.4) J ε ( v ) = Z Ω [ F ( x, v ( x )) + F ( x, Dv ( x ))] dx + 12 ε Z Ω \ Ω ( v − ϕ ) dx − ε Z Ω G (det D v ) dx. By the Rademacher theorem (see [13, Theorem 2, p.81]), v is differentiable a.e. By the Alexandrovtheorem (see [13, Theorem 1, p.242]), v is twice differentiable a.e and at those points of twicedifferentiability, we denote, with a slight abuse of notation, D v its Hessian matrix. Thus, thefunctional J ε is well defined with this convention.Let U ννε = U ijε ν i ν j be as in the proof of Lemma 4.2. Let τ be the tangential direction along ∂ Ω.Let K be the curvature of ∂ Ω. Since u ε = ϕ on ∂ Ω, in two dimensions, we have as in (4.8)(5.5) U ννε = ( u ε ) ττ = K ( u ε ) ν − Kϕ ν + ϕ ττ . Step 2: Almost minimality property of u ε . We show that if v is a convex function in Ω with v = ϕ in a neighborhood of ∂ Ω then(5.6) J ε ( v ) − J ε ( u ε ) ≥ ε Z ∂ Ω ψU ννε ∂ ν ( u ε − ϕ ) + Z ∂ Ω ( v − u ε ) ∇ p F ( x, Du ε ( x )) · ν dS. INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 21
The proof of (5.6) uses mollification to deal with general convex functions v . For h >
0, letΩ h = { x ∈ Ω | dist ( x, ∂ Ω) > h } and v h ( x ) = h − n Z Ω φ ( x − yh ) v ( y ) dy for x ∈ Ω h where φ ≥ , φ ∈ C ∞ ( R n ) , supp φ ⊂ B (0) and R R n φdx = 1 . Clearly, v h → v uniformly on compactsubsets of Ω. Since v = ϕ near ∂ Ω and ϕ ∈ C , (Ω) is uniformly convex in Ω, we can extend v h tobe a uniformly convex C (Ω) function, still denoted by v h , such that(5.7) D k v h → D k v in a neighborhood of ∂ Ω for all k ≤ . By [30, Lemma 6.3], we have lim h → Z Ω h G (det D v h ) = Z Ω G (det D v ) . This together with (5.7) implies that(5.8) lim h → J ε ( v h ) = J ε ( v ) . Now, we estimate J ε ( v h ) − J ε ( u ε ).As in the proof of Lemma 4.2, we have − G (det D v h ) + G (det D u ε ) ≥ w ε U ijε ( u ε − v h ) ij . Integrating by parts twice, recalling U ijε ( w ε ) ij = ε − f ε ( · , u ε , Du ε , D u ε ) , and (4.6), we get(5.9) Z Ω ( − G (det D v h ) + G (det D u ε )) dx ≥ Z Ω w ε U ijε ( u ε − v h ) ij = Z Ω ε − f ε ( · , u ε , Du ε , D u ε )( u ε − v h ) dx − Z ∂ Ω ( w ε ) i U ijε ( u ε − v h ) ν j + Z ∂ Ω ψU ννε ∂ ν ( u ε − v h ) . From the convexity of F , F and ( v − ϕ ) , we have J ε ( v h ) − J ε ( u ε ) ≥ Z Ω [ f ( x, u ε ( x ))( v h − u ε ) + ∇ p F ( x, Du ε ( x ))( Dv h − Du ε )] dx + 1 ε Z Ω \ Ω ( u ε − ϕ )( v h − u ε ) dx + ε Z Ω ( − G (det D v h ) + G (det D u ε )) dx. (5.10)In view of (2.7) and (5.9), we can integrate by parts the right hand side of (5.10) to get, after asimple cancellation,(5.11) J ε ( v h ) − J ε ( u ε ) ≥ − ε Z ∂ Ω ( w ε ) i U ijε ( u ε − v h ) ν j + ε Z ∂ Ω ψU ννε ∂ ν ( u ε − v h )+ Z ∂ Ω ( v h − u ε ) ∇ p F ( x, Du ε ( x )) · ν . By (5.7), the right hand side of (5.11) tends to the right hand side of (5.6) when h →
0. On theother hand, in view of (5.8), the left hand side of (5.11) tends to the left hand side of (5.6) when h →
0. Therefore, (5.6) is proved by letting h → Step 3: Minimality of u . We show that u (in Step 1) is a minimizer of the functional J definedby (1.7) over ¯ S [ ϕ, Ω ]. For all v ∈ ¯ S [ ϕ, Ω ] (extended by ϕ on Ω \ Ω ), we use (5.6) to conclude that J ε ( v ε ) − J ε ( u ε ) ≥ ε Z ∂ Ω ψU ννε ∂ ν ( u ε − ϕ ) − O ( ε )where v ε = (1 − ε ) v + εϕ ∈ ¯ S [ ϕ, Ω ] . Since lim ε → J ( v ε ) = J ( v ), it follows that(5.12) J ( v ) ≥ lim inf ε J ( u ε )+lim inf ε ε [ Z Ω G (det D v ε ) − G (det D u ε )]+lim inf ε ε Z ∂ Ω ψU ννε ∂ ν ( u ε − ϕ ) . From (5.3), we have Z ∂ Ω | ( u ε ) ν | ≤ Cε − / and hence, invoking (5.5), one finds that(5.13) ε Z ∂ Ω ψU ννε ∂ ν ( u ε − ϕ ) ≥ − Cε Z ∂ Ω [1 + | ( u ε ) ν | ] ≥ − Cε / . Observe from 0 ≤ θ < / G ( d ) ≤ C (1 + d / ) for all d > . It follows that Z Ω G (det D u ε ) ≤ C Z Ω (1 + (det D u ε ) / ) dx ≤ C Z Ω (1 + ∆ u ε ) dx = C ( | Ω | + Z ∂ Ω ( u ε ) ν ) ≤ C (1 + ε − / ) . (5.14)Note that D v ε ≥ εD ϕ . Therefore det D v ε ≥ ε det D ϕ ≥ C ε in Ω for C > ε Z Ω G (det D v ε ) ≥ ε Z Ω G ( C ε ) → ε → . From (5.12)–(5.15), we easily obtain(5.16) J ( v ) ≥ lim inf ε J ( u ε ) . Since u ε converges uniformly to u on Ω , by Fatou’s lemma, we have(5.17) lim inf ε Z Ω F ( x, u ε ( x )) dx ≥ Z Ω F ( x, u ( x )) dx. From the convexity of F ( x, p ) in p and the fact that u ε converges to u on W , (Ω ), by lowersemicontinuity, we have(5.18) lim inf ε Z Ω F ( x, Du ε ( x )) dx ≥ Z Ω F ( x, Du ( x )) dx. Therefore, by combining (5.16)–(5.18), we obtain J ( v ) ≥ lim inf ε J ( u ε ) ≥ J ( u ) for all v ∈ ¯ S [ ϕ, Ω ] , showing that u is a minimizer of the functional J defined by (1.7) over ¯ S [ ϕ, Ω ]. Step 4: Full convergence of u ε to the unique minimizer of J . Since ρ >
0, functional J definedby (1.7) over ¯ S [ ϕ, Ω ] has a unique minimizer in ¯ S [ ϕ, Ω ]. Thus, Steps 1 and 3 actually showsthat the whole sequence { u ε } converges to the unique minimizer of J . The proof of the theorem iscompleted. (cid:3) Proof of Theorem 2.6.
When H ( d ) = d θ − where 0 ≤ θ < , the proof of the uniqueness of solutionsin (i) is similar to that of Lemma 4.5 so we omit it. The existence proof uses a priori estimates anddegree theory as in Theorem 2.1. Here, we only focus on proving the a priori estimates. The key isto obtain the positive bound from below and above for det D u :(5.19) C − ≤ det D u ≤ C in Ω . Once (5.19) is established, we can apply the global W , ε (Ω) estimates in Theorem 3.2 for u andargue as in the proof of Theorem 4.1 that u ∈ C ,α (Ω). A bootstrap argument concludes the proof. INGULAR ABREU EQUATIONS AND MINIMIZERS OF CONVEX FUNCTIONALS 23
It remains to prove (5.19).(i) First, by the convexity of F and u , we have U ij w ij = − div ( ∇ p F ( Du )) = − F p i p j ( Du ) u ij = − trace( D F ( Du ) D u ) ≤ . By the maximum principle, the function w attains its minimum value on ∂ Ω. It follows that w ≥ inf ∂ Ω ψ := C > . From the assumptions on H and w = H (det D u ), we deduce thatdet D u ≤ C < ∞ in Ω . From det D u ≤ C in Ω, u = ϕ on ∂ Ω, and the uniform convexity of Ω, we can construct an explicitbarrier to show that | Du | ≤ C in Ω for a universal constant C . This together with the boundednessassumption on F p i p j ( p ) gives F p i p j ( Du ( x )) ≤ C I in Ω . We compute, in Ω, U ij ( w + C | x | ) ij = − F p i p j ( Du ) u ij + 2 C trace ( U ij ) ≥ − C trace ( u ij ) + 2 C ∆ u = C ∆ u ≥ . By the maximum principle, w ( x ) + C | x | attains it maximum value on the boundary ∂ Ω. Recallthat w = ψ on ∂ Ω. Thus, for all x ∈ Ω, we have(5.20) w ( x ) ≤ w ( x ) + C | x | ≤ max ∂ Ω ( ψ + C | x | ) ≤ C. From this universal upper bound for w , we can use w = H (det D u ) and the assumptions on H toobtain det D u ≥ C − > ≤ F p i p j ( p ) ≤ C ∗ I . As above, using the convexity of F and u , we can prove that w ≥ C >
C >
0. From U ij ( w + C ∗ | x | ) ij = − F p i p j ( Du ) u ij + 2 C ∗ trace ( U ij ) ≥ − C ∗ trace ( u ij ) + 2 C ∗ ∆ u = C ∗ ∆ u ≥ w ≤ C in Ω. Consequently,0 < C ≤ w ≤ C < ∞ in Ω . From w = H (det D u ) and the fact that H − maps compact subsets of (0 , ∞ ) into compact subsetof (0 , ∞ ), we find that C ≤ det D u ≤ C in Ω. Therefore, (5.19) is also proved. (cid:3) Proof of Theorem 2.8.
The proof is very similar to that of Theorem 2.1. We focus here on the apriori estimates. The key point is to establish the universal bound for u as in Lemma 4.2. We dothis via proving the estimate of the type (4.2). We use the same notation as in the proof of Lemma4.2 with θ = 0. The estimate (4.10) with n = 2 now becomes(5.21) Z ∂ Ω Kψu ν ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) (cid:19) / + Z Ω [ − ( u − u ) + ∆ u ]( u − ˜ u ) dx. Since ˜ u is universally bounded, there is a universal constant C > − ( u − u )( u − ˜ u ) ≤ C. Moreover, from u ≤ sup ∂ Ω ϕ ≤ C by the convexity of u , we have Z Ω ∆ u ( u − ˜ u ) ≤ C Z ∂ Ω ∆ udx = C Z ∂ Ω u ν . Thus (5.21) gives Z ∂ Ω Kψu ν ≤ C + C (cid:18)Z ∂ Ω ( u + ν ) (cid:19) / + C Z ∂ Ω u ν . A simple application of Young’s inequality to the above inequality together with the fact thatinf ∂ Ω ( Kψ ) > R ∂ Ω u ν ≤ C which is exactly what we need to prove. (cid:3) References [1] Abreu, M. K¨ahler geometry of toric varieties and extremal metrics.
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