Optimal regularity and control of the support for the pullback equation

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1 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 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

2 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

3 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

4 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

5 4 Intermediary results We start be recalling some classical results concerning Hölder spaces. proof see e.g. Theorems 16.26, and Corollary 16.25, 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 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

6 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 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 α) + (1 λ)(r + α) = r 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

7 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

8 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

9 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

10 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 (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

11 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), [2] Bandyopadhyay S. and Dacorogna B., On the pullback equation ϕ(g) = f, Ann. Inst. H. Poincare Anal. Non Lineaire, Vol. 26 (2009), [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), [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), , [5] Csató G., Dacorogna B. and Kneuss O., The pullback equation for differential forms, Birkhäuser/Springer, New York,

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