Partial Legendre transforms of non-linear equations
aa r X i v : . [ m a t h . A P ] O c t PARTIAL LEGENDRE TRANSFORMS OF NON-LINEAR EQUATIONS
PENGFEI GUAN AND D. H. PHONG
Abstract.
The partial Legendre transform of a non-linear elliptic differential equationis shown to be another non-linear elliptic differential equation. In particular, the partialLegendre transform of the Monge-Amp`ere equation is another equation of Monge-Amp`eretype. In 1+1 dimensions, this can be applied to obtain uniform estimates to all orders forthe degenerate Monge-Amp`ere equation with boundary data satisfying a strict convexitycondition. Introduction
The maximum rank and regularity properties of degenerate fully non-linear equations arestill largely unexplored, despite their considerable interest for many geometric problems.For example, it is still an unresolved problem raised by Donaldson [9, 10] to determine theprecise regularity of geodesics in the spaces of K¨ahler potentials and of volume forms on aRiemannian manifold. These are given respectively by solutions of a degenerate complexMonge-Amp`ere equation and an equation introduced by Donaldson [10]. Many existenceand regularity properties have now been established for these equations (see e.g. [4, 16, 7, 5,15, 17, 3, 18, 19] and references therein), but it is not known how close they are to optimal.In [14], the maximum rank property has been established for several special cases of thedegenerate real Monge-Amp`ere and Donaldson equations, for Dirichlet data satisfying strictconvexity conditions. Thus the natural question arises of whether the regularity of solutionsof fully non-linear elliptic equations can be established, assuming that it is already knownthat they have maximum rank.A potentially useful feature of the maximum rank property is that it allows the use of apartial Legendre transform. In fact, the partial Legendre transform was already exploitedby D. Guan [12] in determining geodesics for the space of K¨ahler potentials on toric varieties,and by P. Guan [13] and Rios, Sawyer, and Wheeden [20, 21] in their study of the localregularity of certain degenerate Monge-Amp`ere equations. A first goal of this paper is torefine their analysis and show that, even though the partial Legendre transform f of afunction u is not a local expression of u , the partial Legendre transform of an elliptic PDEin u is another PDE in f which is again elliptic. In particular, the original Monge-Amp`ereequation can be transformed globally into another dual equation, again of Monge-Amp`eretype, but which does not seem to have been encountered before in the literature and maybe of independent interest (see Theorem 1, (b)). The partial strict convexity properties ofone equation are then equivalent to C estimates for its dual, and one can expect a more Research of the first author was supported in part by NSERC Discovery Grant. Research of the secondauthor was supported in part by the National Science Foundation grant DMS-07-57372. eneral correspondence between bounds for their derivatives. The second goal of this paperis to apply this principle in the simplest case of the 1 + 1 real Monge-Amp`ere equation(which coincides with the 1 + 1 Donaldson equation). As a consequence, we obtain C ∞ bounds for this equation which depend only on the Dirichlet data, and in particular whichremain uniform as the equation degenerates (c.f. Theorem 2). We note that in this case,by [12], the solution of the limiting equation is already known to be smooth, so the realinterest of the result lies in the uniform validity of the approximation.2. Legendre transforms
The main goal of this section is to work out the partial Legendre transforms of fullynon-linear elliptic PDE’s in some generality. We shall find that, just as in the case of thefull Legendre transform, they are given by elliptic PDE’s.2.1.
The full Legendre transform.
We begin by re-visiting the standard Legendre trans-form. Let u ( x ) be a strictly convex function on R n . Then u ( x ) defines a Legendre changeof variables x → y = ∂u∂x ( x ) . (2.1)Clearly, the Jacobian of this change of variables is ∂y j ∂x k = ∂ u∂x j ∂x k ≡ u jk . The strict convexityimplies that the map x → y from R n to its image is invertible. The Jacobian of the inverse y → x is given by the inverse u jk of the Jacobian u jk of x → y . Associated to the function u ( x ) is also its Legendre transform f ( y ), defined by f ( y ) = xy − u ( x ) , y = ∂u∂x . (2.2)Differentiating this relation with respect to y shows that the Legendre change of variablesdefined by f ( y ) is the inverse map y → xy → x = ∂f∂y , (2.3)and we have the following exact analogues of the earlier formulas for u , ∂y j ∂x k = ∂ f∂x j ∂x k = u jk .A partial differential equation of the form F ( u jk ) = 0 can be viewed as a partial differentialequation in f . Its linearization has principal symbol ∂F∂u jk = f jp f kq ξ p ξ q . Thus the ellipticityof the equation in u implies the ellipticity of the equation in f . In particular, a Monge-Amp`ere equation for u , det ( ∂ u∂x j ∂x k ) = K (2.4)is equivalent to a Monge-Amp`ere equation for f det ( ∂ f∂y j ∂y k ) = K − . (2.5)We note that the changes of variables x → y and y → x in (2.1, 2.3) are unaffected if u and/or f are shifted by independent constants. The relation (2.2) can be viewed as acanonical way of fixing the relative normalization of f and u . .2. The partial Legendre transform.
We come now to the situation of main interestto us, namely partial Legendre transforms of functions u ( x, t ) which are periodic and satisfya a strict convexity condition in x . More specifically, let e i be the basis vectors for R n , thatis, e i has component 1 in the i -th position, and all its other components are 0. We considerfunctions u ( x, t ) on R n × I , I = (0 , u ( x + e i ) = u ( x ) , x ∈ R n , i n, (2.6)and the strict convexity condition ∂ u∂x j ∂x k + δ jk > . (2.7)Thus u can also be viewed as a function on X × I , where X is the torus X = ( R / Z ) n .We define the following Legendre change of variables x → y = ( y k ) , y k = ∂u∂x k + x k . (2.8)The inverse map y → x is well-defined and unique as a map from R n to R n . To see this,set v = u + | x | , and note that v → ∞ as x → ∞ . Thus v admits a minimum. After atranslation if necessary, we may assume that this minimum is at 0, and in particular, that ∇ v vanishes there. It suffices then to see that, for any y , the equation ∂v∂x k ( x t ) = ty k , t v jk ) is invertible implying opennessof the set of t ’s for which the equation is solvable. And since |∇ v | → ∞ as | x | → ∞ , wealso find that | x t | is bounded, and the set of such t ’s is also closed.The maps x → y and y → x satisfy the following transformation laws, y k ( x + e i ) = y k ( x ) + δ ik , x k ( y + e i ) = x k ( y ) + δ ik . (2.10)The condition for y k follows immediately from its definition and the fact that u ( x ) is peri-odic. To establish the condition for y , just observe that y k + δ ik = ∂u∂x k ( x ) + x k + δ ik = ∂u∂x k ( x + e i ) + ( x + e i ) k (2.11)and the assertion follows by the uniqueness of the inverse map y → x .Clearly, the Jacobian of the map x → y is given by ∂y k ∂x j = ∂ u∂x k ∂x j + δ jk ≡ g ij . (2.12)Consequently, the Jacobian of the inverse map y → x is given by ∂x j ∂y k = g jk . (2.13)All these expressions are periodic, and descend to equations on the torus X . o far, we have discussed the Legendre maps defined by the function u . We now definethe Legendre transform f of u itself by the following formula, f ( y ) = − | x − y | − u ( x ) , y j = ∂u∂x j + x j (2.14)for y ∈ R n . We note that, in view of the transformation laws (2.10) for x and y under periodshifts, the function f ( y ) is actually periodic, and thus can be identified with a function onthe torus X . Again, the inverse map y → x can be viewed as the map associated with thefunction f , y → x j = ∂f∂y j + y j (2.15)and its Jacobian is given by ∂x k ∂y j = ∂ f∂y j ∂y k + δ jk = g jk ≡ h jk . (2.16)In particular, the Legendre transform f satisfies the strict convexity condition D y f + I > ∂ u∂x j ∂x k + δ jk ) = det − ( ∂ f∂y j ∂y k + δ jk ) . (2.18)Consider now the change in variables R n × I → R n × I defined by( x, t ) → ( y, s ) , y j = ∂u∂x j + x j , s = t. (2.19)The Jacobian of the inverse change of variables is given by t s = 1 , t y p = 0 , ( x k ) y p = g kp , ( x k ) s = − g kp u x p t . (2.20)It follows that the rule for differentiating a function F ( x, t ) with respect to the variables( y, s ) is given by F y p = F x j g jp , F s = − F x j g jp u x p t + F t . (2.21)The dependence of the partial Legendre transform on the additional variables s and t isnow conveniently described by the following three equations ∂ y j u t = − ∂ s x j , ∂ s u t = K (det g ) − , f s = − u t . (2.22)Here K is the ( n + 1) × ( n + 1) determinant det( D xt u + I x ), K = det ( D xt u + I x ) = u tt (det g ) − G jp u tx j u tx p , (2.23)where G jp = (det g ) g jp is the matrix of co-factors of the metric g ij . To see the first equation,we compute both sides. On the left, we have ∂ y p u t = u tx j ( x j ) y p = u tx j g jp . On the right,we have − ∂ s x p = − ( − g pj u tx j ) = g pj u tx j also, as required. Next, we apply the rule fordifferentiation of the previous paragraph and obtain ∂ s u t = − u tx j g jp u tx p + u tt = (det g ) − ( u tt det g − G jp u tx j u tx p )(2.24) s claimed. Finally, differentiating the defining formula (2.14) for f gives(2.25) ∂ s f = ( y − x ) · x s − ( u x · x s + u t ) = ( y − x − u x ) · x s − u t = − u t . All three identities in (2.22) have been proved. They imply readily the following two iden-tities, which we also need later u tx j = − f sy k h kj , u tt = − f ss + f sy j f sy k h jk . (2.26)2.3. The partial Legendre transform of non-linear PDE’s.
We consider now a fullynon-linear equation of the form F ( D u ) = 0(2.27)on X × I , where the unknown u is required to satisfy the strict convexity condition D x u + I x >
0, and the equation is assumed to be elliptic. We would like to view this equation as anequation for the Legendre transform f of u . Note that f is a non-local quantity in u .Nevertheless, we have Theorem 1.
Let X = ( R / Z ) n be the n -dimensional torus, and let I = (0 , . Let u ( x, t ) be a function on X × I satisfying the strict convexity condition (2.7). Let ( x, t ) → ( s = t, y ) be the partial Legendre transform as defined by (2.3), and let f be the partial Legendretransform of the function u as defined by (2.14). (a) If F ( D u ) = 0 is a second-order elliptic PDE for u , then it can also be viewed as asecond-order elliptic PDE for the partial Legendre transform f of u .(b) If particular, u satisfies the Monge-Amp`ere equationdet ( D x,t u + I x ) = K (2.28)if and only if its partial Legendre transform f satisfies the following equation on X × I alsoof Monge-Amp`ere type, ∂ f∂s + K det ( D y f + I ) = 0 . (2.29) Proof : From the discussion in the preceding section, the partial Legendre transform f of u is a well-defined function on X × I , ∂ j ∂ k u + δ jk are given by the inverse of the matrix ∂ j ∂ k f + δ jk , and u tx j and u tt are given by the expressions (2.26) in the second derivativesof f . Thus the equation F ( D u ) is automatically a second order non-linear equation for f .To verify the ellipticity of the equation viewed as an equation for f , we work out first thelinearized operator of F ( D u ), keeping variations in δu , δF = ∂F∂u tt ( δu ) tt + ∂F∂u tx j ( δu ) tx j + ∂F∂u x j x k ( δu ) x j x k . (2.30)We need to express this quantity in terms of the derivatives of δf . In view of the expression(2.26) for δu tt and δu tx j , we have δu tt = − δf ss + 2 δf sy j f sy k h jk − f sy j f sy k δf jk δu tx j = − δf sy k h kj + f sy j δf jk , (2.31) here the indices are raised or lowered using the metric h jk . Thus we have δF = − ∂F∂u tt δf ss + ( − ∂F∂u tx j + 2 ∂F∂u tt f sy j ) h jk δf sy k +( − ∂F∂u tt f sy j f sy k + ∂F∂u tx j f sy k − ∂F∂u x j x k ) δf jk . (2.32)This means that, if τ and ξ j are respectively the variables dual to t and x j , the symbol σ ( τ, ξ ) of the linearized operator is given by σ ( τ, ξ ) = ∂F∂u tt τ + ( ∂F∂u tx j − ∂F∂u tt f sy j ) h jk τ ξ k +( ∂F∂u tt f sy j f sy k − ∂F∂u tx j f sy k + ∂F∂u x j x k ) ξ j ξ k . (2.33)We shall show that σ ( τ, ξ ) is positive definite if the original equation F ( D u ) = 0 is elliptic.Introduce the variable η j ≡ h jk ξ k for convenience. Then completing the square in τ gives σ ( τ, ξ ) = [ τ r ∂F∂u tt + ( 12 ∂F∂u tx j − ∂F∂u tt f sy j ) η j q ∂F∂u tt ] + 14 ∂F∂u tt [4 ∂F∂u tt ∂F∂u x j x k η j η k − ( ∂F∂u tx j η j ) ] . (2.34)To prove part (a) of the Theorem, it suffices then to show that that the second term onthe right is strictly positive for η = 0 when F is elliptic. Since the symbol of the linearizedoperator when the unknown is u is given by ∂F∂u tt τ + ∂F∂u tx j τ ξ j + ∂F∂u x j x k ξ j ξ k (2.35)its ellipticity does imply the positivity of the second term on the right hand side of (2.34).Part (a) is proved.Part (b) follows immediately from the identities in (2.22), f ss = − ∂ s u t = − K (det g ) − = − K det ( D y f + I ) . (2.36)The proof of Theorem 1 is complete. Q.E.D.We conclude this section with a few remarks. • Part (b) of Theorem 1 can be viewed as a refinement of several earlier results in theliterature using the partial Legendre transform: when K = 0, it reproduces the result ofD. Guan [12]. For general K , and when the considerations are local (instead of on a torusas here), then P. Guan [13] in dimension n = 1 and Rios-Sawyer-Wheeden [20] for generaldimension n have shown that the coordinates x j , viewed as functions of ( y, s ), satisfy thefollowing elliptic, non-linear system of equations ∂ s x j + ∂∂y j ( K det( ∂x k ∂y m )) = 0 , j n. (2.37) his system of equations follows immediately from differentiation of the equation (2.29)with respect to y j . • The presence of the background symmetric form δ jk is very similar to the presence ofthe K¨ahler form ø for the complex Monge-Amp`ere equation (ø + i ∂ ¯ ∂φ ) n = F ø n . • The correspondence between an equation in u and its “dual” equation in f can providenon-trivial information. For example, it is not evident that the equation (2.29) admitssmooth solutions for given Dirichlet data, even when K is a strictly positive constant. Onthe other hand, the existence of such solutions is an immediate consequence of the existenceof smooth solutions for the dual equation (2.28), which can be established by the theory ofCaffarelli-Nirenberg-Spruck [8], with the improved barrier arguments of B. Guan [11].Similarly, lower bounds for D x u + I are equivalent to upper bounds for D y f + I and viceversa, and the problems of partial C estimates and partial strict convexity are in this sense“dual”. For example, the C estimates for the original Monge-Amp`ere equation (2.28) canbe established by traditional methods as in [8], or as a consequence of the convexity resultsfor the dual equation (2.29), using for example the recent results of Bian-Guan [1, 2]. • Beyond the Monge-Amp`ere equation, the partial Legendre transforms of Hessian equa-tions may be of interest. By (2.26), D x,t u + I = − ˜ K (det ˜ g ) − − ˜ u sy λ − · · · − ˜ u sy n λ − n − ˜ u sy λ − λ − · · · · · ·− ˜ u sy n λ − n · · · λ − n (2.38)where λ i are the eigenvalues of ˜ g = (˜ u ij + δ ij ), and we have denoted for convenience allquantities associated with the partial Legendre transform by a ˜ (e.g. f is now denoted ˜ u ,and ˜ K is the Monge-Amp`ere determinant of D y,s ˜ u + I ). The standard formula for the k -thsymmetric function σ k of the eigenvalues of a matrix is(2.39) σ k ( V ) = 1 k ! X δ i ··· i k j ··· j k v i j · · · v i k j k We apply this formula to the the symmetric function ˆ σ of the n + 1-dimensional matrix D x,t u + I . We findˆ σ k ( D x,t u + I ) = σ k (˜ g − ) − X i ··· i k − λ − i · · · λ − i k − ( k − X i j =1 ˜ u sy ij λ − i j + ˜ K (det ˜ g ) − ) . (2.40)From here, it is easily seen that the Laplace equation u tt + ∆ u = K gets transformed into(2.41) ˜ K + K det ˜ g = σ n − (˜ g )For general k , the equation ˆ σ k ( u ) = K k gets transformed into(2.42) K k = σ n − k (˜ g ) σ n (˜ g ) − X i ··· i k − λ − i · · · λ − i k − ( k − X i j =1 ˜ u sy ij λ − i j + ˜ K ( det ˜ g ) − )We note that when k = ( n + 1), the above identity recovers (2.29). . The
Monge-Amp`ere equation
In this section, we consider more specifically the case n = 1 of the Monge-Amp`ere equa-tion. In this case, the equation becomes the following equation on X × I , u tt (1 + u xx ) − u xt = ε (3.1)where X = R / Z is a circle and ε > u is required to satisfy D xt u + I x >
0, and we impose the Dirichlet condition u ( t,
0) = u ( x ), u ( t,
1) = u ( x ), with u i ∈ C ∞ ( X ) satisfying the strict convexity condition, u ixx + 1 > λ (3.2)for i = 0 , λ > Theorem 2.
Let u be the solution of the equation (3.1) on X × I , with D xt u + I x > andsmooth Dirichlet data satisfying (3.2). Then for any non-negative integer N , there exists M ( N ) and a constant C N depending only on λ > and the C M ( N ) norms of the Dirichletdata u , u so that X a + b N k ∂ ax ∂ bt u k C ( X × I ) C N . (3.3) In particular, the constants C N are independent of ε . The rest of this section is devoted to the proof of Theorem 2. To apply the partialLegendre transform as in §
2, we need the strict partial convexity of u . This follows fromTheorem 1 of [14]. But since the present case is particularly simple, we can supply the shortproof for the convenience of the reader: Lemma 1.
Let u ( x, t ) be the solution of the equation (3.1), as specified in the statement ofTheorem 2. Then u xx ( x, t ) + 1 > λ for all x ∈ X .Proof of Lemma 1 : Let λ = min ( x,t ) ∈ X × ¯ I ( u xx + 1), and set ϕ ( x, t ) = u xx + 1 − λ . Weestablish a strong maximum principle for ϕ . If ϕ vanishes on the boundary, the lemmais proved. We shall show that in a neighborhood of any interior zero of ϕ , ϕ satisfies anelliptic differential inequality equation of the form F ij ϕ ij C |∇ ϕ | . (3.4)Here we have denoted by F the function F ( D xt u + I x ) with F ( M ) = det( M ij ), for M anysymmetric 2 × F ij = ∂F∂M ij . The constant C is required to be independentof the point ( x, t ) but may depend on everything else. By the strong maximum principle,this would imply that ϕ vanishes in a neighborhood of any interior zero. If the set of suchzeroes is not empty, then ϕ vanishes identically. By continuity ϕ would again vanish on theboundary, and the lemma is proved in all cases.The equation (3.1) can be written as F ( D xt u + I x ) = ε . Differentiating the equationsuccessively gives F ij u ijx = 0 , F ij u ijxx + F ij,kl u ijx u klx = 0 . (3.5)Thus F ij ϕ ij = − F ij,kl u ijx u klx , and more explicitly, F ij ϕ ij = − u ttx u xxx − u xxt = − u ttx ϕ x − ϕ x . (3.6) he inequality (3.4) follows. Q.E.D.Let f ( y, s ) be now the partial Legendre transform of u , as defined in section. When n = 1, the equation for f simplifies to Lf ≡ f ss + εf yy = 0 on X × I, (3.7)and f ( y,
0) and f ( y,
1) are given by the Legendre tranforms f ( y ) and f ( y ) of the functions u ( x ) and u ( x ). General linear elliptic theory says that any derivative of f can be boundedin terms of the boundary data and the ellipticity constant ε . However, we require estimateswhich are uniform as ε →
0, and such estimates do not seem to have been written down inthe literature. We provide below a brief and explicit derivation of estimates uniform in ε ,exploiting the simple form of the equation and of the boundary in the present case. Moreprecisely, we shall establish the following lemma: Lemma 2.
Consider the Dirichlet problem for the Laplacian L in (3.7) with ε a constantsatisfying < ε < . Then for any m, k we have k ∂ my ∂ ks f k C ( ˜ X × I ) C m,k (3.8) where C m,k are constants which depend only on the Dirichlet data (and on m and k ). Inparticular, C m,k are independent of ε .Proof : Clearly ∂ my ∂ ks f satisfies the same Laplace equation. By the maximum principle, wehave then k ∂ my ∂ ks f k C ( ˜ X × I ) = k ∂ my ∂ ks f k C ( ˜ X × ∂I ) . (3.9)We shall show the right hand sides can be estimated in terms of the Dirichlet data alone,for arbitrary m and k = 0 or k = 1. Assuming this, it follows that the left hand side isalso bounded in terms of the Dirichlet data alone for these values of m and k . Since theequation implies that ∂ my ∂ ks f = − ε∂ m +2 y ∂ k − s f for k >
2, it follows that uniform bounds for ∂ my ∂ ks f for arbitrary k follow from the special cases k = 0 and k = 1, and the lemma wouldbe proved.We return to the proof of bounds for k ∂ my ∂ ks f k C ( ˜ X × ∂I ) when k = 0 or k = 1. When k = 0, they are obvious, so we concentrate on the case k = 1. Let ˜ f be a function with thesame boundary values as f (e.g., ˜ f = tf ( y,
1) + (1 − t ) f ( y, w ( y, s ) = − As + Bs (3.10)for constants A, B > L ( ∂ my f ) = 0, we have L ( ∂ my ( f − ˜ f )) − L ( ∂ my ˜ f ) > c (3.11)where c is a constant depending only on the Dirichlet data and independent of 0 < ε < Lw = − A (3.12)so we can choose A large enough so that Lw L ( ∂ my ( f − ˜ f )) on ˜ X × I . Now ∂ my ( f − ˜ f ) vanishes identically on the boundary, so if we choose B large enough so that w > verywhere, we shall also have ∂ my ( f − ˜ f ) w on the boundary. Thus, by the comparisonprinciple, ∂ my ( f − ˜ f ) w on ˜ X × I . Since both of these functions vanish at s = 0, we obtain ∂ s ∂ my f − As + B + ∂ s ∂ my ˜ f , (3.13)which is an upper bound for ∂ s ∂ my f . Applying the argument to − f instead of f , we obtain alower bound for ∂ s ∂ my f at the boundary points ( y, y,
1) is identical, the proof of the lemma is complete.Next, we show that C N bounds for x (viewed as a function of ( y, s ), together with astrict partial convexity bound, imply C M bounds for the original function u : Lemma 3.
For any non-negative integer M , we have X m + b N k ∂ mx ∂ bt u k C ( X × I ) C M (3.14) where C M is a constant depending only on the Dirichlet data u , u , the lower bound λ > ,and the C norm of a finite number N ( M ) of spatial derivatives ∂ my x of the function x .Proof of Lemma 3 : First, we show that bounds for ∂ my x imply bounds for ∂ ax u . This isan easy consequence of the following formula, which is itself a consequence of the chain ruleestablished in the previous section: ∂ my x = − u xx + 1) m ∂ m +1 x u + P ( u, · · · , ∂ mx u )( u xx + 1) m − (3.15)where P is a generic notation for a polynomial in all its entries for all m > m = 2, the bounds of ∂ mx u in terms of boundary data alone are a special caseof the C estimates for the Dirichlet problem for the Monge-Amp`ere equation [8]. Notethat these bounds do not require a strictly positive lower bound for the Monge-Amp`eredeterminant, and thus give bounds which are uniform in ε in our case.Assume that bounds depending only on the Dirichlet data and a strictly positive lowerbound λ for u xx + 1 have been established for m . In view of the above formula for ∂ my x ,it follows that such bounds for ∂ m +1 x u reduce to such bounds for ∂ my x on ˜ X × I . By themaximum principle for ∂ my x , this reduces in turn to bounds for ∂ my x only on the boundary.But the same formula (3.15) for higher derivatives above shows that ∂ my x on the boundary˜ X × ∂I is determined completely by the boundary data. This establishes the desired boundsfor k ∂ ax u k C ( X × I ) .Next, we consider mixed derivatives of the form ∂ ax ∂ t u . It is again easy to establish thefollowing general formula linking ∂ my ∂ s x and ∂ ax ∂ t u for all m > ∂ my ∂ s x = − ∂ m +1 x ∂ t u ( u xx + 1) m +1 + P ( u, · · · , ∂ m +2 x u, ∂ t u, · · · , ∂ mx ∂ t u )( u xx + 1) m +1 , where P is again a polynomial in all its entries. In view of Lemma 2, the left hand side canbe bounded uniformly in terms of the Dirichlet data and the lower bound λ for u xx + 1.Thus the formula implies that ∂ m +1 x ∂ t u can be similarly bounded if ∂ mx ∂ t u is. Since u xt is bounded by the Dirichlet data in view of the C estimates, we obtain by induction theuniform boundedness of ∂ mx ∂ t u for all m . inally, by differentiating the original Monge-Amp`ere equation, we can show inductivelyon the number b of t derivatives in ∂ ax ∂ bt u that they are in turn bounded. First, we showthis for b = 2. Differentiating the equation m times with respect to x gives(3.17) ∂ mx u tt = 1 u xx + 1 P ( ∂ x u, · · · , ∂ m +2 x u, ∂ t ∂ x u, · · · , ∂ t ∂ m +1 x u, ∂ t u, · · · , ∂ t ∂ m − x u )with P a polynomial in all its entries. Since ∂ mx u tt is bounded for m = 0 by the C estimates,and ∂ ax u , ∂ ax ∂ t u are now known to be bounded, the formula allows us to show by inductionon m that ∂ mx ∂ t u is bounded. Next, assume that ∂ mx ∂ bt u is bounded for an integer b > m . We shall show that this remains true if b is replaced by b + 1. Differentiating the equation b − ∂ b +1 t u = 1 u xx + 1 P ( ∂ x ∂ t , · · · , ∂ x ∂ bt u, ∂ x ∂ t u, · · · , ∂ x ∂ bt u )(3.18)where P is a polynomial in all its entries. This shows that ∂ mx ∂ b +1 t u is bounded for m = 0.Thus it suffices to show that if ∂ ℓx ∂ b +1 t u is bounded for all non-negative integers ℓ m , thenthe same is true if m is replaced by m + 1. But this follows readily by differentiating theprevious formula,(3.19) ∂ m +1 x ∂ b +1 t u = ∂ m +1 x (cid:8) u xx + 1 P ( ∂ x ∂ t , · · · , ∂ x ∂ bt u, ∂ x ∂ t u, · · · , ∂ x ∂ bt u ) (cid:9) . Since the right hand side involves terms with at most b derivatives in t , and all derivativesin x of such expressions are bounded, the desired bound for ∂ m +1 x ∂ b +1 t u follows. Q.E.D.Putting together Lemmas 2-4, we obtain Theorem 2. References
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Department of Mathematics, McGill University, Montreal, Quebec. H3A 2K6, Canada.
E-mail address : [email protected] Department of Mathematics, Columbia University, New York, NY 10027, USA.
E-mail address : [email protected]@math.columbia.edu