Optimal regularity and control of the support for the pullback equation O. KNEUSS Department of Mathematics, Federal University of Rio de Janeiro Rio de Janeiro, Brazil olivier.kneuss@gmail.com August 14, 2017 Abstract Given f, g two C r,α either symplectic forms or volume forms on a bounded open set R n with 0 < α < 1 and r 0, we give natural conditions for the existence of a map ϕ Diff r+1,α (; ) satisfying ϕ (g) = f in and supp(ϕ id). 1 Introduction The pullback equation ϕ (g) = f where g and f are both symplectic forms or both volume forms has been studied a lot. One could consult [5] for an extensive survey for the pullback equation in general. We start by giving a very brief summary for the symplectic case: Darboux [8] proved that any two symplectic forms can be pulled back locally one to another. This result has been reproved by Moser [11] using an elegant flow method. These two proofs do not produce any gain in regularity: the map ϕ is at most as regular as the data g and f. Later Bandyopadhyay-Dacorogna [2] established in particular in a local existence result with optimal regularity in the Hölder spaces C r,α, 0 < α < 1. Since the pullback equation is a system of first order PDE s, optimality means here that for g, f C r,α there exists a solution ϕ C r+1,α. For the global case coupled with a Dirichlet condition, ϕ (g) = f in and ϕ = id on (1) the following (quasi) optimal result has been proved in [6] (see also [2] for a slightly weaker result): given ω C r,α ([0, 1] ; Λ 2 ) an homotopy of symplectic forms between g and f such that, for every t [0, 1], ω t ω 0 is exact in and ω t ν = ω 0 ν C r+1,α ( ; Λ 3 ) 1
then there exists ϕ Diff r+1,α (; ) solving (1), where ν denotes the outward unit normal of some smooth bounded open set and is identified with a 1 form. Note that, for a solution to (1) to exist, we necessarily have g ν = f ν on. Note also that the only non natural condition (whose necessity is still an open problem) is the extra regularity of ω t ν on the boundary. Concerning the case of volume forms (in which case, identifying volume forms with functions, ϕ (g) = f reads as the single equation g(ϕ) det ϕ = f) the first existence result (with no gain in regularity) for (1) is due to Moser [11]. Afterwards Dacorogna and Moser proved in [7] that given any g, f C r,α () strictly positive where is a smooth connected bounded open set with g = f, (2) there exists ϕ Diff r+1,α (; ) satisfying (1). Note that (2) is obviously necessary to solve (1). Other proofs of this optimal regularity result have been established in [1], [4] and [13]. In this paper we give conditions to solve the pullback equation ϕ (g) = f in with optimal regularity in Hölder space and imposing that ϕ = id near the boundary (and not only on as in (1)). An obvious necessary condition for this problem is then supp(g f). (3) We prove (cf. Theorems 1 and 3) that the above condition is to some extent also sufficient: Theorem. (i) given a bounded open set in R n star-shaped with respect with some open ball and ω a continuous homotopy of C r,α (; Λ 2 ) symplectic forms between g and f such that supp(ω t f) for every t [0, 1], there exists ϕ Diff r+1,α (; ) verifying ϕ (g) = f in and supp(ϕ id). (4) (ii) given g, f two non vanishing C r,α () functions in some bounded connected open set verifying (2) and (3), there exists ϕ Diff r+1,α (; ) verifying (4). Note that, in the symplectic case, we no longer need the extra regularity of ω t ν on the boundary as mentioned above to solve (1). For another proof of the above result for volume forms (Theorem 3) but with the additional constraint g 1 see [12] and for a proof of Theorem 3 in the annulus one can also consult [10]. 2
The proofs of Theorems 1 and 3 follow similar arguments as in [2] and [6] : we exhibit an appropriate smoothing of f, resp. g, denoted f ɛ, resp. g ɛ, show that we can pullback with optimal regularity f ɛ to f with ϕ 1 as well as g ɛ to g with ϕ 2 and then, by the usual flow method g ɛ to f ɛ with ϕ 3. The desired solution is then ϕ = (ϕ 2 ) 1 ϕ 3 ϕ 1. To ensure that supp(ϕ id) we additionally first enforce that supp(f ɛ g ɛ ) and, afterwards, that supp(ϕ 1 ϕ 2 ) and ϕ 3 = id near ϕ 1 ( ) = ϕ 2 ( ). Note that in [2] and [6] the condition ϕ = id on was done by imposing ϕ i = id on and thus, necessarily, f ν = f ɛ ν and g ν = g ɛ ν on leading to the extra regularity of the tangential parts of f and g on the boundary. Moreover in [2] and [6] the smoothing of f and g was done by convolution with a special kernel lacking compact support. In this paper we show that the convolution with any kernel with compact support (implying directly that supp(f ɛ g ɛ ) ) still produces an appropriate smoothing (cf. Lemma 5 and Remark 6). 2 Notation In this paper we use the following notation and refer to [5] for them. (i) The space C r,α (; Λ k ) with its norm C r,α (), where r 0, 0 α 1, 0 k n and is a bounded open set of R n, denotes the set of C r,α k forms in. When a k form ω depends on some parameter, i.e. ω C r,α ([0, 1] ; Λ k ) we will often use the notation ω(t, x) = ω t (x). (ii) The set Diff r,α (U; ), where U and are two bounded open set of R n, is the set of maps ϕ such that ϕ C r,α (U; ) and, ϕ 1 C r,α (; U) (iii) A set is said to be star-shaped with respect to a set W if for every x and y W the segment [x, y] := {(1 t)x + ty : t [0, 1]} is contained in. (iv) The usual exterior product is denoted by. The interior product of a k form g, with a vector field u is the (k 1)-form denoted by u g. The exterior differential of a k form g is the (k + 1)-for denoted by dg. The coexterior differential of a k form g is the (k 1)-form denoted by δg. A k form g is said to be closed, resp. co-closed in if dg = 0, resp. δg = 0, in. The set H(; Λ k ) denotes the sets of closed and co-closed k forms in. (v) Let ω C 0 (; Λ 2 ). The map ω C 0 (; R n n ) is defined by (where the index below denotes the column and the index above the row) (ω) j i = ω ij where we have used again the convention ω ij = ω ji. Hence for every x the matrix ω(x) R n n is skew-symmetric and thus its rank is even. Note that, for a vector field u, u ω = ω u. When the rank of ω is equal to n (the dimension of ) we have, for vector fields u and v, v = u ω u = (ω) 1 v. (5) 3
A 2 form ω is called symplectic in if dω = 0 and if rank(ω) = n in. (vi) For a k form g defined in and a C 1 map ϕ : U the pullback of g by ϕ is the k form defined in U denoted by ϕ (g). One will constantly use the following property for the pullback 3 Main results We now state our two main results. (ϕ ψ) (g) = ψ (ϕ (g)). (6) Theorem 1. Let R n be a bounded open set star-shaped with respect to some open ball, r 0 and 0 < α < 1. Let f and g be two C r,α symplectic forms in such that there exists ω C 0 ([0, 1] ; Λ 2 ) verifying, for every t [0, 1], ω t is a C r,α symplectic form in, supp(ω t ω 0 ) and ω 0 = f, ω 1 = g. Then there exists ϕ Diff r+1,α (; ) such that ϕ (g) = f and supp(ϕ id). Remark 2. Theorem 1 can be extended (with exactly the same proof) to any bounded open set assuming additionally that the homotopy ω t is such that there exists, for every t [0, 1], F t Cc r+1,α (, Λ 1 ) with df t = ω t ω 0 in. Note that the above property is automatically satisfied as soon as is star-shaped with respect with some open ball (cf. Proposition 7). Theorem 3. Let R n be a bounded connected open set, r 0 and 0 < α < 1. Let f, g C r,α () be such that f g > 0 in, f = g and supp(g f). Then there exists ϕ Diff r+1,α (; ) such that ϕ (g) = f and supp(ϕ id). 4
4 Intermediary results We start be recalling some classical results concerning Hölder spaces. proof see e.g. Theorems 16.26, 16.28 and Corollary 16.25, 16.30 in [5]. For a Theorem 4. Let R n be a bounded open set with Lipschitz boundary. Let r 0 and 0 α 1. The following four assertions are then verified. (i) [Product] There exists a constant C = C(r, ) such that for every f, g C r,α (), ) fg C r,α () ( f C C r,α () g C 0 () + f C 0 () g C r,α (). (ii) [Division] Let A C r,α (; R n n ) and c > 0 be such that 1, A det A C0 () c. C0 () Then there exists a constant C = C(c, r, ) such that A 1 C r,α () C A C r,α (). (iii) [Interpolation] There exists a constant C = C(s, ) such for every integers s, t and every 0 β, γ 1 with one has where λ [0, 1] is such that t + γ r + α s + β f C r,α () C f λ C t,γ () f 1 λ C s,β () λ(t + γ) + (1 λ)(s + β) = r + α. (iv) [Equivalence of norms] The norms C r,α () and C 0 () +[ r ] C 0,α () are equivalent. We next give some elementary estimates for the usual smoothing by convolution of a function. Although these estimates are essentially contained in Theorem 16.43 in [5] (see also Remark 6) we reprove them for the convenience of the reader. Lemma 5. Let, U R n be two bounded open sets with Lipschitz boundary such that U Let r 1, 0 α 1 and ρ C c (R n ) be such that R n ρ = 1. Then, for every f C r,α (U) and every ɛ small enough, f ɛ C (U) defined by where ρ ɛ ( ) = 1/ɛ n ρ( /ɛ), satisfies f ɛ = f ρ ɛ, ɛ f ɛ C 0 () C f C 1 (U) (7) 5
and, for every 0 γ 1, f ɛ C r+1,γ (), ɛf ɛ C r,γ () where C is a constant depending only on r, ρ and U. C ɛ 1+γ α f C r,α (U), (8) Remark 6. Note that in Theorem 16.43 in [5] we obtained a wider range of estimates but for that we had to use a specific kernel which does not have compact support. The compactness of the support of the smoothing kernel will be crucial in the proofs of Theorems 1 and 3. Proof. First since ρ has compact support it is obvious that f ɛ is well defined and C in for any ɛ small enough. Moreover an elementary calculation gives (7). In what follows C will denote a generic constant depending only on r, ρ and U that may change from appearance to appearance. Step 1. We prove the first inequality in (8). First, a direct caculation gives that, for any integer t and any 0 δ 1 with t + δ r + α, f ɛ C t,δ () C f C r,α (U). (9) Next, since 2 f ɛ = 1 ɛ 2 f ( 2 ρ) ɛ we get that (as in (9)) 2 f ɛ C r,α () C ɛ 2 f C r,α (U). Thus, combining the previous inequality and (9) for t = δ = 0 with Theorem 4 (iv), we get f ɛ C r+2,α () C ɛ 2 f C r,α (U). Finally, using the previous inequality, (9) with t = r and δ = α and Theorem 4 (iii), we get where λ is such that f ɛ C r+1,γ () C f ɛ λ C r+2,α () f ɛ 1 λ C r,α () C ɛ 2λ f C r,α (U) = C ɛ 1+γ α f C r,α (U), This proves the first part of (8). λ(r + 2 + α) + (1 λ)(r + α) = r + 1 + γ. Step 2. We prove the second inequality in (8). Note that ɛ f ɛ = 1 ɛ f η ɛ where η(z) = nρ(z) + z; ρ(z). 6
For γ α we hence have, proceeding exactly as in Step 1, ɛ f ɛ C r,γ () = 1 ɛ f η ɛ C r,γ () C ɛ 1+γ α f C r,α (U). (10) It hence only remains to prove the second inequality in (8) for γ < α. First, noticing that η = 0, we have R n r ɛ f ɛ (x) = 1 ɛ r f η ɛ (x) = 1 η(y) [ r f(x ɛy) f(x)] dy, ɛ R n which yields to r ɛ f ɛ C0 () ɛα 1 f C r,α (U). Hence, combining the previous inequality and (10) with Theorem 4 (iii) we get r ɛ f ɛ C 0,γ () C r ɛ f ɛ λ C 0 () r ɛ f ɛ 1 λ C 0,α () C ɛ f (1 α)λ+1 λ C r,α (U) = C ɛ 1+γ α f C r,α (U), where λ is such that (1 λ)α = γ. Combining the previous inequality with (7) we get (8) by Theorem 4 (iv). We now give a version of Poincaré Lemma with optimal regularity for forms with compact support. This result is an extension of a result of Bogovski [3] (see also [9]) for the divergence operator and is essentially contained in [14]. Proposition 7. Let r 0, 0 < α < 1, 1 k n and R n be a bounded open connected set additionally assumed to be star-shaped with respect to some open ball when 1 k < n. Then for every f Cc r,α (; Λ k ) such that { df = 0 in if k < n f = 0 if k = n there exists F Cc r+1,α (; Λ k 1 ) with df = f in. Remark 8. Note that the result is false in general (even without the gain in regularity) in the case 1 k < n when is not star-shaped with respect to some open ball. We give an example of this fact when k = 1 and when = {x R n : 1/2 < x < 1}: taking ω = dg where G is any smooth function being 1 near { x = 1/2} and 0 near { x = 1}, it does not exist a function F with compact support in and with df = ω. Indeed, if such a function existed, one would get, integrating by parts, 0 = df = dg 0, a contradiction. 7
Proof. For the proof of the result when k = n we refer to [3] (see also e.g. [9]). For the case k < n we refer to [14] where an explicit formula for F is given. Note that [14] deals with Sobolev spaces but exactly the same proof (involving Calderon-Zygmund singular operator theory) works for Hölder spaces. We finally recall a regularity result for the flow essentially due to Rivière-Ye [13], see also Theorem 12.4 in [5]. Proposition 9. Let r 0 and 0 < α < 1. Let R n be a bounded open set with Lipschitz boundary, η, c > 0 and be such that for every ɛ (0, η] u ɛ C r+1,γ () u C ((0, η] ; R n ) c ɛ 1+γ α for every γ [0, 1]. Let U be a smooth open set such that U. Then for any ɛ small enough the solution ϕ ɛ of ɛ ϕ ɛ = u ɛ ϕ ɛ and ϕ 0 = id. exists in U with ϕ ɛ (U) and ϕ ɛ Diff r+1,α (U; ϕ ɛ ()). Remark 10. Note that, since u ɛ C r+1,γ L 1 (0, η) for any γ < α, then by classical results we directly have that ϕ ɛ Diff r+1,γ. The above proposition asserts that ϕ ɛ Diff r+1,α making use of the special estimate on u ɛ. 5 Proof of the main results We are now in position to prove Theorems 1 and 3. Proof of Theorem 1. Step 1 (simplification). First, taking slightly smaller we can assume that ω is continuous in [0, 1] and that ω t is symplectic and C r,α in for every t [0, 1]. Moreover we can assume that the homotopy ω is linear, i.e. ω t = (1 t)f + tg. Indeed by continuity of ω there exists an integer M big enough so that, for every 1 i M and every t [0, 1], (1 t)ω (i 1)/M + tω i/m is symplectic in. Since (by (6)) to pullback g to f it is sufficient to pullback ω i/m to ω (i 1)/M for every 1 i M we have the claim. Step 2 (smoothing of ω t ). By Theorem 6.12 in [5] we know that, recalling that df = 0, f = da + h 8
where a C r+1,α (; Λ 1 ) and h H(; Λ 2 ) C (; Λ 2 ) (cf. e.g. Theorem 6.3 in [5] for the previous inclusion). By Proposition 7 there exist b Cc r+1,α (, Λ 1 ) such that db = g f in. Let ρ Cc (R n ) be such that ρ = 1. For every ɛ small enough, taking R n slightly smaller, define a ɛ C (; Λ 1 ) and b ɛ Cc (; Λ 1 ) as a ɛ = a ρ ɛ and b ɛ = b ρ ɛ and thus, appealing to Lemma 5, for every ɛ small enough and every γ [0, 1], and a ɛ C r+2,γ () + bɛ C r+2,γ () ɛ a ɛ C r+1,γ () + ɛb ɛ C r+1,γ () c ɛ 1+γ α (11) c ɛ 1+γ α (12) ɛ a ɛ C0 () + ɛb ɛ C0 () c, (13) where c is a constant independent of ɛ. Finally, for every t [0, 1] and ɛ small enough, define ω ɛ t C (; Λ 2 ) by ω ɛ t = da ɛ + h + tdb ɛ. Since ω 0 t = ω t it is clear that ω ɛ t is symplectic in for any ɛ small enough. Step 3 (estimate). Define, for every ɛ small enough and t [0, 1], u ɛ t C (; R n ) by (see (5)) u ɛ t = (ω ɛ t) 1 ɛ [a ɛ + tb ɛ ] d(u ɛ t ω ɛ t) = ɛ ω ɛ t. (14) We claim that there exists a constant c independent of ɛ and t such that u ɛ t C r+1,γ () c ɛ 1+γ α for every ɛ small enough and every 0 γ 1. (15) In what follows c will denote a generic constant independent of ɛ and t that may change from appearance to appearance. Using Theorem 4 (i),(ii) as well as (11), (12) and (13), we obtain u ɛ t C r+1,γ () c ɛ a ɛ + ɛ b ɛ C r+1,γ () (ωɛ t) 1 C 0 () + c ɛa ɛ + ɛ b ɛ C 0 () (ωɛ t) 1 C r+1,γ () c ɛ 1+γ α + c ωɛ t C r+1,γ () c ɛ 1+γ α + c daɛ + h + tdb ɛ C r+1,γ () c ɛ 1+γ α + c aɛ C r+2,γ () + c bɛ C r+2,γ () c ɛ 1+γ α, showing the claim. 9
Step 4. (pulling back ω ɛ 0 to ω 0 = f and ω ɛ 1 to ω 1 = g). Let 1 and 2 be smooth open sets so that, for any ɛ small enough, supp(g f) supp b ɛ 2 2 1 1. (16) By Proposition 9 and (15), the solution ϕ ɛ and ψ ɛ of ɛ ϕ ɛ = u ɛ 1 ϕ ɛ and ϕ 0 = id and ɛ ψ ɛ = u ɛ 0 ψ ɛ and ϕ 0 = id exists on 1 for ɛ small enough with ϕ ɛ ( 1 ) ϕ 1 ɛ ( 1 ) and ϕ ɛ Diff r+1,α ( 1 ; ϕ ɛ ( 1 )) as similarly for ψ ɛ. Moreover, by (16) and (14), we have supp(u ɛ 0 u ɛ 1) 2 for ɛ small enough. Hence, choosing ɛ smaller if necessary one deduce that as well as ϕ 1 ɛ = ψɛ 1 on 1 \ 2 and thus ϕ 1 ɛ ( 1 ) = ψɛ 1 ( 1 ) (17) 2 ϕ 1 ɛ ( 1 ). (18) Finally, using (14) we get that, by the classical flow method (see eg. Theorem 12.7 in [5]), (ϕ ɛ ) (ω ɛ 0) = f and (ψ ɛ ) (ω ɛ 1) = g in 1. (19) Step 5 (pulling back ω1 ɛ to ω0). ɛ For any ɛ small enough and t [0, 1] define vt ɛ Cc ( 3 ; R n ) by v ɛ t = (ω ɛ t) 1 u ɛ d(v ɛ t ω ɛ t) = t ω ɛ t. Again by the classical flow method we get that the solution φ t of t φ t = v t φ t and φ 0 = id satisfy, for every t [0, 1] φ t Diff (; ) and (φ t ) (ω ɛ t) = ω ɛ 0 in and supp(φ t id) 2. (20) Step 6 (conclusion). We claim that, for ɛ small enough, { ψ 1 ɛ φ ϕ = 1 ϕ ɛ in ϕ 1 ɛ ( 1 ) id in \ ϕ 1 ɛ ( 1 ) has all the wished properties. First, by (17) and since (combining (18) and (20)) we directly get that ψ 1 ɛ φ 1 (ϕ 1 ɛ ( 1 )) = ϕ 1 ɛ ( 1 ), φ 1 ϕ ɛ Diff r+1,α (ϕ 1 ɛ ( 1 ); ψɛ 1 ( 1 )) 10
Moreover combining (17), (18) and (20) we deduce that ψ 1 ɛ φ 1 ϕ ɛ is the identity near ϕ 1 ɛ ( 1 ) implying trivially that ϕ Diff r+1,α (; ). It only remains to show that ϕ pulls back g to f in. First, using (6), (19) and (20), we get that ψɛ 1 φ 1 ϕ ɛ pulls back g to f in ϕ 1 ɛ ( 1 ). Finally, recalling that (cf. (16) and (18)) we trivially have that supp(g f) 2 ϕ 1 ɛ ( 1 ), ϕ (g) = id (g) = f in \ ϕ 1 ɛ ( 1 ). This proves that ϕ pulls back g to f in and concludes the proof. Proof of Theorem 3. Since the proof of Theorem 3 is very similar to one of Theorem 1 we only briefly summarize it. Consider the linear homotopy ω t = (1 t)f + tg and consider its smoothing with respect to x ω ɛ t = λ ɛ t ω t ρ ɛ where ρ ɛ is as before and where λ ɛ t is so that ωt ɛ = f = We then proceed exactly as in the proof of Theorem 1 to conclude. g. References [1] Avinyó A., Solà-Morales J., and València M., A singular initial value problem to construct density-equalizing maps, Journal of Dynamics and Differential Equations, Vol. 24 (2012), 51 59. [2] Bandyopadhyay S. and Dacorogna B., On the pullback equation ϕ(g) = f, Ann. Inst. H. Poincare Anal. Non Lineaire, Vol. 26 (2009), 1717 1741. [3] Bogovski M.E., Solution of the first boundary value, problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl., Vol. 20 (1979), 1094 1098. [4] Carlier G. and Dacorogna B., Réesolution du problème de Dirichlet pour l équation du jacobien prescrit via l équation de Monge-Ampère, C. R. Math. Acad. Sci. Paris, Vol. 350 (2012), 371 374, [5] Csató G., Dacorogna B. and Kneuss O., The pullback equation for differential forms, Birkhäuser/Springer, New York, 2012. 11
[6] Dacorogna B. and Kneuss O., A global version of Darboux theorem with optimal regularity and Dirichlet condition, Advan. Differ. Eq., Vol. 16 (2011), 325 360. [7] Dacorogna B. and Moser J., On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincare Anal. Non Lineaire, Vol. 7 (1990), 1-26. [8] Darboux G., Sur le probl eme de Pfaff, Bull Sci.Math., Vol. 6 (1882), 14 36, 49-68. [9] Galdi G., An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-state problems, Springer, New York, 2011. [10] Matheus C., https://matheuscmss.wordpress.com/2013/07/06/a-remarkon-the-jacobian-determinant-pde, blog. [11] Moser J., On the volume elements on a manifold, Trans. Amer. Math. Soc., Vol. 120 (1965), 286 294. [12] Teixeira P., Dacorogna-Moser theorem with control of support, arxiv:1608.06213. [13] Rivière T. and Ye D., Resolutions of the prescribed volume form equation, Nonlinear Differ. Eq. Appl., Vol. 3 (1996), 323 369. [14] Takahashi S., On the Poincaré-Bogovski lemma on differential forms, Proc. Japan Acad. Ser. A Math. Sci., Vol. 68 (1992), 1 6. 12