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Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 20, 2002, 63–75 PERTURBING FULLY

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Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 20, 2002, 63–75

PERTURBING FULLY NONLINEAR SECOND ORDER ELLIPTIC EQUATIONS

¨ Philippe Delanoe

Abstract. We present two types of perturbations with reverse effects on some scalar fully nonlinear second order elliptic differential operators: on the other hand, first order perturbations which destroy the global solvability of the Dirichlet problem, in smooth bounded domains of Rn ; on the other hand, an integral perturbation which restore the local solvability, on compact connected manifolds without boundary.

Introduction Perturbing scalar second order elliptic equations can bring both bad news and good news. The bad news (Section 1) is that positivity, hence in some cases ellipticity, can be destroyed by a first order perturbation. Let us illustrate this phenomenon with an example. Denote by B(0, 1) the open unit ball centered at the origin in R2 ; there exists a smooth (in fact radial) solution of the Dirichlet problem: uxx uyy − u2xy = 1 in B(0, 1), u = 0 on ∂B(0, 1). By Theorem 2 below, for any small enough real ε 6= 0, there exists a smooth function f positive on B(0, 1) with f ≡ 1 + εux outside an arbitrarily small ball in B(0, 1), such that 2 the perturbed problem: zxx zyy − zxy + εzx = f in B(0, 1), z = 0 on ∂B(0, 1), 2 admits no smooth solution in the connected component of {zxx zyy − zxy > 0} 2000 Mathematics Subject Classification. 35J60, 35B20, 35A07. Key words and phrases. Gradient perturbation, degenerate elliptic, ill-posed, integral perturbation. Supported by the CNRS. c

2002 Juliusz Schauder Center for Nonlinear Studies

63

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where u lies. A similar result holds e.g. with the laplacian uxx + uyy instead of the Monge–Amp`ere operator, but it does not affect the ellipticity of the solution (just the positivity of the laplacian). The idea of the proof first arose in [8] in connection with a particular geometric equation in dimension 4. The good news (Section 2) concern the local solvability of a generic (scalar second order elliptic) fully nonlinear equation without zeroth-order term posed on a compact manifold. Here the difficulty lies in the fact that the local image of the differential operator is expected to have codimension 1, but no equation is known for it. We provide an integral perturbation device, first used in [5], to cope with this situation. We treat also zeroth-order perturbations regardless of monotonicity. 1. Non-existence via a first order perturbation 1.1. Assumptions. Let D be a domain of Rn , n > 1. On the second jetbundle J 2 D → D we are given a smooth real function f positive on a strict subset P(f ) of J 2 D which still projects onto D, with f vanishing on the boundary of P(f ). We assume that the zero section lies in the boundary of P(f ) and that, for any X ∈ P(f ), there exists a point in the kernel through X of the natural projection J 2 D → J 1 D which lies in the boundary of P(f ). In other words, if r denotes the variable of ker (J 2 D → J 1 D) and (x, z, p) the J 1 D variables (with x ∈ D, z ∈ R and p ∈ Tx∗ D), then we have: f (x, 0, 0, 0) = 0 and (1)

∀(x, z, p, r) ∈ P(f ), ∃r0 , f (x, z, p, r + r0 ) = 0.

Let Ω be a smooth bounded domain of Rn with closure contained in D, F , the differential operator associated to f on Ω and P (F ), a connected component of the counterset of P(f ) by the second-jet map u ∈ C ∞ (Ω) 7→ j 2 u ∈ J 2 Ω. We assume that P (F ) is convex, the operator F , elliptic on P (F ) and that, for any z ∈ P (F ), if the principal symbol of dF [z] is positive (resp. negative) definite, then its zeroth-order coefficient dF [z](1) is non-positive (resp. non-negative) in other words ∂f /∂z ≤ 0 (resp. ∂f /∂z ≥ 0). In particular then, the maximum principle ([10]) implies that dF [z] is one-to-one whenever z ∈ P (F ). The preceding set of assumptions is typically fulfilled for a k-hessian operator F [z] = σk [λ(Ddz)] (D stands for the canonical flat connection of Rn ; see [4]). 1.2. A non-existence theorem. Under the preceding assumptions, we shall prove the following result: Theorem 1. Let G be a first order differential operator on D and u ∈ P (F ). Assume there exists x0 ∈ Ω such that G[u](x0 ) > 0. Then, for any compact subset K of P (F ), there exists a real ε > 0 such that, for any s ∈ (0, ε), setting Fs := F + sG, there exists a function ψ ∈ C ∞ (Ω) positive on Ω such that

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ψ ≡ Fs [u] outside an arbitrarily small ball centered at x0 and that the Dirichlet problem: Fs [z] = ψ in Ω, z = u on ∂Ω, admits no solution in K. The sign of s is of course essential in this statement. Let us differ the proof to the next section and concentrate on the basic case when the first order operator G is a fixed directional derivative. Proposition 1. Let u ∈ P (F ) be non-constant. Then there exists a unit vector ξ ∈ Rn such that Theorem 1 holds with G[z] = dz(ξ). Proof. Let u ∈ P (F ) be non-constant. Since F [0] = 0, the function u satisfies in Ω the second order linear equation Lu = v where v = F [u] and R1 L = 0 dF [tu] dt. But 0 ∈ ∂P (F ) and P (F ) is convex, so L is elliptic. Let y0 ∈ ∂Ω be such that u(y0 ) = max ∂Ω u. Take for ξ the outward unit normal to ∂Ω at y0 . Since v > 0 and u is non-constant, Hopf–Oleinik’s lemma (see [10]) implies du(ξ)(y0 ) > 0. Taking x0 ∈ Ω close enough to y0 proves the proposition. From this proof, one readily infers the Corollary 1. Let u ∈ P (F ) be non-constant, and constant on ∂Ω. Then, for any unit vector ξ ∈ Rn , Proposition 1 holds. Under the additional assumption ∂f ≡ 0, ∂z one can strengthen the preceding results as follows: Theorem 2. Let u ∈ P (F ), then there exists a unit vector ξ ∈ Rn and a real number ε > 0 such that, for any s ∈ (0, ε), there exists a function ψ ∈ C ∞ (Ω) positive on Ω with ψ = F [u] + s du(ξ) outside an arbitrarily small ball centered at x0 , such that the Dirichlet problem: F [z]+s dz(ξ) = ψ in Ω, z = u on ∂Ω, admits no solution in P (F ). Furthermore, if u is constant on ∂Ω, then the preceding statement holds with the unit vector ξ ∈ Rn arbitrary and with s ∈ (−ε, ε). Theorem 2, whose proof follows closely that of Theorem 1 (see below), takes a considerable strength when the Dirichlet map associated to F , sends P (F ) onto {ψ ∈ C ∞ (Ω), ψ > 0} × C ∞ (∂Ω) and the ellipticity of F may fail on ∂P (F ), as it is the case for k-hessian operators when Ω is a (k − 1)-convex domain, k > 1 (see [4]), in particular, for the example given in the introduction. Proof of Theorem 1. We need a few auxiliary lemmas. Lemma 1. Let u ∈ P (F ) and x0 ∈ Ω. For any small real ρ > 0, there exists a function u0 ∈ ∂P (F ) with the following properties: (i) u0 coincides with u outside the euclidean ball B(x0 , ρ), (ii) the C 1 [B(x0 , ρ)] norm of (u − u0 ) is O(ρ).

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Proof. Fix r > 0 such that B(x0 , r) ⊂ Ω and let φ be a smooth cut-off function satisfying: φ = 1 in B(x0 , r/2), φ = 0 outside B(x0 , r). By (1) we can find a quadratic polynomial q0 satisfying: q0 (x0 ) = 0, dq0 (x0 ) = 0 and f [x0 , du(x0 ), Dd(u + q0 )(x0 )] = 0. Setting y = x − x0 , let us define: w(y) = φ(x)q0 (x) and, for any real R > 1, zR (x) = R−2 w(Ry). The function zR belongs to C ∞ (Ω) and it is supported in B(x0 , r/R). Furthermore, since dzR (x) = R−1 dw(Ry), the C 1 [B(x0 , r/R)] norm of zR is O(R−1 ). However, at x0 , DdzR (x0 ) ≡ Ddq0 (x0 ), therefore the smaller positive real a0 such that the function (u + a0 zR ) =: u0 belongs to ∂P (F ) is well-defined and satisfies a0 ≤ 1. For R large enough (depending on ρ) the function u0 fulfills all the requirements of Lemma 1. Lemma 2. Let u ∈ P (F ) and u0 be as in Lemma 1, with x0 as in Theorem 1. There exists a real number ε0 > 0 such that, for any s ∈ (0, ε0 ) and any small enough ρ > 0 (as in Lemma 1), the function ψ = Fs [u0 ] is positive on Ω. Proof. For ρ > 0 small enough, setting 2δ = G[u](x0 ), we have G[u0 ] ≥ δ on B(x0 , ρ) by Lemma 1(ii). Moreover, F [u0 ] ≥ 0 because u0 ∈ ∂P (F ). Therefore Fs [u0 ] ≥ sδ > 0 on B(x0 , ρ). Outside B(x0 , ρ) we have Fs [u0 ] = Fs [u] by Lemma 1(i). So there exists ε0 > 0 such that, for any s ∈ (0, ε0 ), the function Fs [u0 ] is positive outside B(x0 , ρ). Altogether, the function ψ = Fs [u0 ] is positive on Ω as claimed. Lemma 3. For any u0 ∈ ∂P (F ) and any compact subset K ⊂ P (F ), there exists a real ε1 > 0 such that, for any s ∈ (−ε1 , ε1 ) and any u ∈ K, setting ut = tu + (1 − t)u0 for t ∈ [0, 1], the Dirichlet map: z ∈ C ∞ (Ω) 7→ (Lu,s [z], z|∂Ω ) ∈ C ∞ (Ω) × C ∞ (∂Ω) R1 associated to the linear operator Lu,s = 0 dFs [ut ] dt is an isomorphism. Proof. Since P (F ) is convex, the function ut lies in P (F ) for t > 0, so the operator Lu,s is elliptic. For each z ∈ P (F ), the Dirichlet map associated to the linear operator dF [z] is an isomorphism. Indeed, by ellipticity it is Fredholm and it can readily be

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deformed continuously into an isomorphism, so it has zero index (e.g. by [11, Theorem IV, 5.17]). By the maximum principle [10] it is one-to-one (recalling our sign assumption on ∂f /∂z), it is thus also onto, by the Fredholm alternative theory (e.g. [3, p. 464]), hence an isomorphism, by the open mapping theorem (e.g. [12, Chapter 1]). Let us consider the Fr´echet space L∞ 2 of linear maps of second order L from C0∞ := {z ∈ C ∞ (Ω), z|∂Ω = 0} to C ∞ (Ω) such that, for each integer j and, for some fixed α ∈ (0, 1), the norm kLkj = sup{|Lz|C j,α (Ω) , z ∈ C0∞ , |z|C j+2,α (Ω) = 1} is finite. Recall L∞ 2 can be endowed with the metric (e.g. [12, Chapter 1]): d(L, L0 ) :=

∞ X j=0

2−j

kL − L0 kj 1 + kL − L0 kj

.

Let L02 be the completion of L∞ 2 for the norm k · k0 . The canonical imbedding 0 J0 : L∞ → L is continuous and the set Isom02 of isomorphisms in L02 is open (e.g. 2 2 −1 0 by [11, Theorem IV, 1.16]), hence so is Isom∞ 2 = J0 {Isom2 }. Moreover, given any small real δ > 0, the map (u, s) ∈ P (F ) × [−δ, δ] 7→ Lu,s ∈ L∞ 2 is continuous, hence uniformly continuous on K × [−δ, δ], and e = {Lu,0 | u ∈ K} K is a compact subset of Isom∞ 2 . Therefore, on the one hand, there exists a tubular e contained in Isom∞ , on neighbourhood V (for the metric d) of the compact K, 2 the other hand, given this neighbourhood V, there exists ε1 ∈ (0, δ) such that: Lu,s ∈ V Lemma 3 is proved.

for all s ∈ (−ε1 , ε1 ) and all u ∈ K.

Proof of Theorem 1. We are given u, x0 and K. Let u0 be as in Lemma 1 and ε1 , as in Lemma 3. Take ε0 and ψ as in Lemma 2, with ε0 ≤ ε1 . Let us argue by contradiction and assume the existence of u1 ∈ K satisfying: Fs [u1 ] = ψ in Ω, u1 = u0 on ∂Ω. For t ∈ [0, 1], set ut = tu1 + (1 − t)u0 . The function u = u1 − u0 R1 satisfies L[u] = 0 in Ω, u = 0 on ∂Ω, with L = 0 dFs [ut ] dt. By Lemma 3, it implies u ≡ 0, which is absurd since u0 ∈ ∂P (F ). So Theorem 1 holds. Proof of Theorem 2. First of all, when ∂f /∂z = 0 necessarily F [z] = 0 if z is constant; so u ∈ P (F ) cannot be constant. Given u and x0 , take u0 as in Lemma 1, ε0 and ψ as in Lemma 2, and argue again by contradiction, now with an arbitrary function u1 fixed in P (F ). Since ∂f /∂z = 0 and G[z] = dz(ξ), the

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R1 operator L = 0 dFs [ut ] dt has no zeroth-order term. Moreover, it is elliptic by the convexity of P (F ). So L is one-to-one, by the maximum principle (see [10]), which is enough to conclude as above. 2. Local existence via integral perturbation In this section, we are given a second order differential operator F0 on a compact connected manifold M of dimension n (without boundary), satisfying: (2)

F0 [u + constant] ≡ F0 [u].

Let u0 ∈ C ∞ (M ) be a smooth real function on M at which F0 is elliptic. Given ψ ∈ C ∞ (M ) close to ψ0 = F0 [u0 ], we want to solve the equation F0 [u] = ψ with u ∈ C ∞ (M ) close to u0 . 2.1. The local image problem. Let us start with a couple of elementary observations. Lemma 4. Given any neighbourhood U of ψ0 in C ∞ (M ), there is no neighbourhood of u0 ∈ C ∞ (M ) mapped onto U by F0 . Proof. Let us argue by contradiction and assume the existence of a nonzero real number c, arbitrarily small, such that the equation F0 [u] = ψ0 + c admits a solution u1 ∈ C ∞ (M ) close to u0 . Setting ut = tu1 + (1 − t)u0 for t ∈ [0, 1], we infer that v = u1 − u0 satisfies on M the second order linear equation Lv = c, R1 where L = 0 dF0 [ut ] dt. For u1 close to u0 , this equation is elliptic; moreover, condition (2) readily implies that L has no zeroth-order term. The maximum principle [10] thus implies that v is constant, contradicting c 6= 0. Lemma 5. For any u ∈ C ∞ (M ) close enough to u0 , the kernel of dF0 [u] coincides with the functions on M which are constant: ker dF0 [u] = R. Proof. For any u ∈ C ∞ (M ), the kernel of dF0 [u] certainly contains the constant functions, due to (2). Conversely, if dF0 [u] is elliptic (as it is the case for u close to u0 ), then ker dF0 [u] ⊂ R due to the maximum principle. Fixing an auxiliary Lebesgue measure dλ on M , it is easy to see that the restriction of dF0 [u] to the subspace ∞ R⊥ dλ = {v ∈ C (M ), hvi = 0}

(where hvi stands for the dλ-average of v over M ), is one-to-one when u is close to u0 in C ∞ (M ). Therefore the restriction of F0 to the affine subspace u0 +R⊥ dλ is ∞ an immersion near u0 into C (M ). Moreover, the local image of that immersion coincides with that of F0 due to condition (2). The problem which we are now facing consists in identifying an equation for the local image of F0 . In other words, in order to solve locally the equation

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F0 [u] = ψ, we look for an a priori constraint on F0 [u], for u near u0 , telling us where ψ should lie (near ψ0 ) for the equation to be solvable. 2.2. A self-adjointness ansatz? Whenever F0 is linear, the Fredholm alternative theory (cf. e.g. [3, p. 464]) solves the problem. Specifically then, there exists a riemannian metric g and a vector field ξ on M such that, up to sign: F0 [z] = ∆z + dz(ξ), where ∆ stands for the (positive) laplacian of g. Now ψ lies in the image of F0 if and only if it is L2 orthogonal to the 1-dimensional subspace: coker F0 = {v ∈ C ∞ (M ), ∆v + div(vξ) = 0}, where L2 and div are both relative to (the Lebesgue measure of) g. In the fully nonlinear case, which we are considering here for F0 , we can first complement Lemma 5 with Lemma 6. For any u close to u0 in C ∞ (M ), the image of dF0 [u] has codimension 1. Proof. We can speak of the (formal) adjoint of dF0 [u] in L2 (M, dλ). The elliptic operator dF0 [u] is Fredholm, of index zero because it can be deformed continuously into a (second-order elliptic) self-adjoint operator. So it has a 1dimensional cokernel, whose L2 (M, dλ)-orthogonal coincides with the image of dF0 [u] according to Fredholm theory. The lemma is proved. The local image problem thus amounts to integrating near ψ0 in C ∞ (M ) the codimension 1 distribution (coker dF0 [u])⊥ . The simplest way to do it is to find a Lebesgue measure dλ with respect to which dF0 [u] is identically self-adjoint. Indeed then, we have the following result (pointed out to us by Pengfei Guan): Proposition 2. Let F0 be as above, satisfying (2), and dλ be a Lebesgue measure on M . If dF0 [u] is self-adjoint in L2 (M, dλ) for all u close to u0 in C ∞ (M ), then the local image of F0 near ψ0 consists of the codimension 1 affine submanifold: Z Z Σ0 = ψ ∈ C ∞ (M ), ψ close to ψ0 , ψ dλ = ψ0 dλ . M

M

R Proof. Near u0 in C ∞ (M ), consider the map u 7→ M F0 [u] dλ. Under the self-adjointness assumption, recalling Lemma 5, we see that its derivative at R u, given by v 7→ M dF0 [u](v) dλ, vanishes identically. So the map is constant, proving that the local image of F0 lies in Σ0 .

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It remains to prove that F0 is onto Σ0 near u0 . To do so, we use the elliptic inverse function theorem with constraints of [7, Theorem 2, p. 686] applied at u0 to the map (for u close to u0 ): u ∈ (u0 + R⊥ dλ ) 7→ F0 [u] ∈ Σ0 . Under the self-adjointness assumption, the derivative of this map at u0 is readily seen to be an automorphism of R⊥ dλ by the Fredholm alternative theory. So [7, Theorem 2] implies: ∀ψ ∈ Σ0 near ψ0 , ∃u ∈ (u0 + R⊥ dλ ) near u0 , F0 [u] = ψ. The proof is complete.

Nontrivial examples for Proposition 2 are provided by the Calabi–Yau operator on compact K¨ ahler manifolds, dλ being the riemannian measure (cf. e.g. [2]), and by the almost-K¨ ahler version of it (as easily verified) ([8]). Can Proposition 2 serve as an ansatz to solve our image problem? In other words, given (F0 , u0 ), can one always find a Lebesgue measure dλ such that Proposition 2 holds? The answer is no, as shown by the following counterexample (Proposition 3 below). Pick a riemannian metric g on M , with Levi–Civita connection ∇, and take for F0 [z] the second elementary symmetric function σ f2 [λ(z)] of the eigenvalues with respect to g of (g + ∇dz), with σ f2 normalized by F0 [0] = σ f2 (1, . . . , 1) = 1. This is indeed a second order fully nonlinear operator satisfying (2). Moreover (see [9]), it is elliptic on the open convex set P (F0 ) = {z ∈ C ∞ (M ), F0 [z] > 0}. If | · | stands for the g-norm and ∆, for the (positive) g-laplacian, we have: F0 [z] = 1 −

1 2 2 ∆z + [(∆z)2 − |∇dz| ], n n(n − 1)

and a routine computation yields the identity: Z Z 1 − → − → (3) (F0 [z] − 1) dµ ≡ Ricci( ∇ z, ∇ z) dµ, n(n − 1) M M − → where dµ (resp. Ricci, ∇ ) denotes the Lebesgue measure of g (resp. its Ricci tensor, its gradient operator). Therefore, whenever g is Ricci-flat, the image of F0 lies a priori in the following smooth codimension 1 submanifold: Z ψ ∈ C ∞ (M ), (ψ − 1)dµ = 0 . M

Actually then, F0 and dµ also fulfill the assumptions of Proposition 2 near u0 = 0 (routine exercise).

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Proposition 3. If g is not Ricci-flat, there exists no Lebesgue measure dλ on M such that, for any u close enough to u0 = 0 in C ∞ (M ), the operator dF0 [u] is formally self-adjoint with respect to dλ. Proof. Let us argue by contradiction and pick a Lebesgue measure dλ for which dF0 [u] is self-adjoint at each u close enough to u0 = 0 in C ∞ (M ). By Proposition 2, the functional Z φ(u) = (F0 [u] − 1) dλ M ∞

vanishes identically in C (M ) near u0 = 0. Therefore dφ(0)(v) = 0 for all v ∈ C ∞ (M ), which reads Z dF0 [0](v) dλ = 0 M

or else, Z ∆v dλ = 0. M

In other words, the Radon–Nikodym derivative ρ of dλ with respect to dµ, satisfies ∆ρ = 0 in the distribution sense on M . Since ∆ is elliptic, ρ must be smooth [2, p. 85] and the maximum principle [10] implies that ρ is constant. Recalling φ(u) ≡ 0 and (3), we reach a contradiction unless g is Ricci-flat. From Proposition 3 we conclude that the image problem remains open for a generic fully nonlinear second-order differential operator F0 satisfying (2) on M compact. 2.3. An integral perturbation device. To cope with the preceding situation and restore a local solvability, the idea is to break the invariance of F0 expressed by (2), at a somewhat lower cost (loosing the locality of the operator). Let hzi still denote the average on M of a function z with respect to a fixed Lebesgue measure dλ. Theorem 3. Let (F0 , u0 , ψ0 ) be as above, with F0 satisfying (2). Without loss of generality, assume: hu0 i = 0. Then, given any nonzero real number s, the perturbed operator Fs [z] := F0 [z] + shzi is a smooth diffeomorphism from a neighbourhood of u0 in C ∞ (M ) onto a neighbourhood of ψ0 in C ∞ (M ). Remark 1. If Fs [zs ] = Ft [zt ] with st 6= 0 and both zs and zt close enough to u0 in C ∞ (M ), then Theorem 3 implies: 1 zt = zs + (s − t)hzs i. t One may thus use the normalization s = 1 without loss of generality.

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Remark 2. The idea of adding an average term goes back to [5, Theorem 1] (see also [6, p. 426]) where it is used to invert (globally) in C ∞ (M ) the elliptic riemannian Monge–Amp`ere operator: det (g + ∇du) u 7→ F0 [u] = log , det (g) g standing for a smooth riemannian metric and ∇, for its Levi–Civita connection. Another global application is drawn in [9]. Remark 3. Fix s 6= 0 and set: ψ 7→ S(ψ) for the local solution map defined by Theorem 3, Gs for the open subset of functions ψ1 close to ψ0 in C ∞ (M ) satisfying hS(ψ1 )i 6= 0. Granted the next corollary, we now know an equation for the image by F0 of a neighbourhood of u0 in C ∞ (M ), namely: hS(ψ)i = 0. The proof of Theorem 3, given below, relies on an inverse function theorem argument. With Theorem 3 at hand, we can characterize the ill-posedness of the original equation F0 [z] = ψ with ψ near ψ0 as follows: Corollary 2. Let s, F0 , u0 (and ψ0 ) be as in Theorem 3, and let ψ1 be given in Gs (cf. Remark 3). Then the equation F0 [u1 ] = ψ1 admits no solution u1 ∈ C ∞ (M ) such that the operator F0 remains elliptic along the path t ∈ [0, 1] 7→ ut := tu1 + (1 − t)u0 . Proof. Let us argue by contradiction and take ψ1 ∈ Gs and u1 as stated. By (2), we may assume hu1 i = 0. So u1 solves Fs [u1 ] = ψ1 as well. If u1 is close enough to u0 we reach a contradiction by the very definition of Gs ; if not, we need to argue further. Since ψ1 is close to ψ0 , Theorem 3 provides a solution z1 of Fs [z1 ] = ψ1 close to u0 in C ∞ (M ). It follows that F0 remains elliptic along the path t ∈ [0, 1] 7→ vt = tu1 + (1 − t)z1 . Now v = u1 − z1 satisfies on R1 M the linear elliptic equation L[v] = 0, where L = 0 dFs [vt ] dt. Noting that L[v + constant] ≡ L[v], we conclude from the maximum principle that v = 0, which is absurd since hvi = −hz1 i = 6 0. Corollary 2 yields the global ill-posedness of elementary hessian equations σ fk [λ(z)] = ψ > 0 (with notations used for Proposition 3 above) on a compact riemannian manifold, since these equations are a priori elliptic and their ellipticity set is convex (cf. [9, Section 1]). 2.4. Proof of Theorem 3. For any fixed nonzero real s, the operator Fs is an elliptic map in the sense of [7] from a neighbourhood of u0 in C ∞ (M ), to C ∞ (M ). Theorem 3 thus follows from the elliptic inverse function Theorem [7, Theorem 2] provided we can prove the following linear result: Theorem 4. For s 6= 0, the linear operator dFs [u0 ] is an automorphism of C (M ). ∞

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Proof. Since F0 is elliptic at u0 , satisfying (2), there exist a riemannian metric g and a vector field ξ, both smooth on M , such that the linear operator Ls = dFs [u0 ] reads, up to sign: (4)

Ls [z] = ∆z + dz(ξ) ± shzi,

where ∆ stands for the (positive) laplacian of g. Without loss of generality, we may take +shzi in the right-hand side (the sign of s here is unimportant). Clearly, Ls is a continuous linear map from C ∞ (M ) to itself; it is one-toone by the maximum principle (easy check). According to the open mapping Theorem [12, Chapter 2], it remains only to show that Ls is onto, which we now do with an argument inspired from [5, Lemma 2, p. 346]. Let dµg be the canonical Lebesgue measure of the metric g, and dλ, the Lebesgue measure used to define the average h · i in Theorem 3; let ρ ∈ L1 (M, dµg ) be the density of dλ with respect to dµg . Set: Z Z Z Z 1 V = z dµg dλ ≡ ρ dµg , Vg = dµg , hzig = Vg M M M M for a generic real function z ∈ L1 (M, dµg ). Lemma 7. The formal L2 (M, dµg ) adjoint of the operator Ls is given by Vg hzig . V In particular, the adjoint of the differential operator L0 is given by

(5)

L∗s [z] = ∆z + div(zξ) + sρ

L∗0 z = ∆z + div(zξ). The latter has a 1-dimensional null space and, if w ∈ ker L∗0 , then w 6= 0 ⇒ hwig 6= 0. Formula (5) is routinely obtained, integrating by parts on M (compact without boundary) with the measure dµg . The assertion on the dimension was proved in Lemma 6; for the last assertion, we argue by contradiction: if w 6= 0 satisfies hwig = 0 and spans ker L∗0 then, according to Fredholm Theorem [3, p. 464], one can solve on M the equation ∆z + dz(ξ) = 1, because its right-hand side is orthogonal to w in L2 (M, dµg ). But the maximum principle ([10]) implies that the solution z must be constant, which is absurd. With Lemma 7 at hand, we can complete the proof of Theorem 4 as follows. Given any ψ ∈ C ∞ (M ) and s 6= 0, consider the ratio Z Z −1 c(s, ψ) := wψ dµg s w dµg M

M

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where w stands for a nonzero element of ker L∗0 ; clearly, c(s, ψ) does not depend on a particular choice of such a w. The function [ψ − s c(s, ψ)] is orthogonal to ker L∗0 in L2 (M, dµg ), so by the Fredholm Theorem [3, p. 464] one can solve on M the equation L0 z = ψ − s c(s, ψ). Moreover, since the solution z is defined up to an additive constant (by Lemma 5), we can define z by imposing: hzi = c(s, ψ). Now z ∈ C ∞ (M ) satisfies Ls z = ψ as required, so Ls is indeed onto.

Remark 4. When the density ρ lies only in L1 (M, dµg ), the operator Ls provides an example of an automorphism of C ∞ (M ) whose formal L2 (M, dµg )adjoint, given by (5), maps C ∞ (M ) to L1 (M, dµg ) only. 2.5. Zeroth-order perturbation. Let F0 , u0 and ψ0 be as in Theorem 3. We wish to deduce from Theorem 4 a local existence result for the equation: (6)

F0 [z] = ψ + sz

near u0 , where ψ ∈ C ∞ (M ) is close to ψ0 and s 6= 0 is a small real parameter. Although the sign of s is irrelevant for our result, let us stress that it can be on the resonant side of zero, where s could interfere with the spectrum of dF0 [z] (whereas for s on the other side of zero, equation (6) is a priori locally invertible). Theorem 5. Let F0 , u0 and ψ0 be as in Theorem 3. Then there exists a neighbourhood V of ψ0 in C ∞ (M ) and a real number s0 > 0 such that, for any ψ ∈ V and any nonzero s ∈ (−s0 , s0 ), equation (6) admits a unique solution u close to u0 in C ∞ (M ). Moreover, this solution depends smoothly on the data (ψ, s), for s 6= 0. Proof. We first consider, near the origin in R × C ∞ (M ) × C ∞ (M ), the smooth map (s, φ, v) 7→ G(s, φ, v) ∈ C ∞ (M ) defined by G(s, φ, v) = F0 [u0 + v] + hvi − (ψ0 + φ) − s(u0 + v). It satisfies G(0, 0, 0) = 0 and, by Theorem 4, ∂G (0, 0, 0) ≡ dF1 [u0 ] ∂v is an automorphism of C ∞ (M ). An implicit function theorem argument, using [7, Theorem 2], yields near zero the existence of a smooth solution-map (s, φ) 7→ v = S(s, φ) such that: G[s, φ, S(s, φ)] = 0. The theorem follows for s 6= 0, by letting ψ = ψ0 + φ and 1 u = u0 + S(s, φ) − hS(s, φ)i. s

Remark 5. The idea of pushing the parameter s toward the resonant side accross the value s = 0 by means of the equation G(s, φ, v) = 0 goes back to [5,

Perturbing Nonlinear Equations

75

Theorem 2] (where an existence result is proved for any value of s 6= 0, for the riemannian Monge–Amp`ere equation). Using it for the complex Monge–Amp`ere equation, Th. Aubin could take up the c1 > 0 case of the Calabi’s conjecture (cf. [1, footnote p. 148]). Acknowledgments. This note was prepared for a fully nonlinear workshop held in April 2001 at the Newton Institute (Cambridge, UK); my warmest thanks go to Neil Trudinger for his invitation. I am also grateful to Pengfei Guan for his remark on self-adjointness (cf. Proposition 2).

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Manuscript received December 4, 2001

Philippe Delano¨ e Universit´ e de Nice-Sophia Antipolis Math´ ematiques, Parc Valrose F-06108 Nice CEDEX 2, FRANCE E-mail address: [email protected]

TMNA : Volume 20 – 2002 – No 1