On the naturality of the exterior differential
aa r X i v : . [ m a t h . A P ] J a n On the naturality of the exterior differential
Vladimir Gol’dshtein and Marc TroyanovNovember 18, 2018
Abstract
We give sufficient conditions for the naturallity of the exterior differential underSobolev mappings. In other words we study the validity of the equation d f ∗ α = f ∗ dα for a smooth form α and a Sobolev map f . The main results of the paper are Theorems6.3 and 7.1.AMS Mathematics Subject Classification: 46E35, 58DxxKeywords: Sobolev mappings, differential forms. One of the main properties of calculus with differential forms is the naturallity of the exteriorderivative, that is the fact that for any smooth map f : U → R n , where U is a boundeddomain in R m , and any smooth differential form α in R n , we have df ∗ α = f ∗ dα. (1.1)Note that this equation is just an avatar of the chain rule; its proof can be found in anytextbook on differential forms.For applications in the calculus of variation, non linear elasticity or geometric analysis, it isimportant to extend this result to non smooth situations. If the map f is smooth and α isa Sobolev differential form, then the pull back f ∗ α is also a locally Sobolev differential formand the naturality (1.1) can be proved by standard arguments. If both the differential form α and the map f belong to W , loc , then the problem is not well posed and it is not clear underwhat conditions, should the equation (1.1) make sense and be proved.If the differential form α is smooth, then the situation is better and it is our goal in this paperto give sufficient condition for a Sobolev map f : U → R n to satisfy the naturality of theexterior derivative for smooth forms. Our main results are Theorem 6.3 and Theorem 7.1.As consequences of these theorem, we can formulate the following special results (corollaries6.4 and 7.2): • Let U be a bounded domain in R m and f ∈ W ,k +1 ( U, R n ) . Then the chain rule (1.1) holdsfor any smooth k -forms α on R n . • Suppose that f ∈ W ,k ( U, R n ) . If all the k × k minors of the Jacobian matrix (cid:16) ∂f ν ∂x µ (cid:17) belongto the space L k/ ( k − ( U ) , then the chain rule (1.1) holds for any smooth k -forms α on R m . emarks 1.) The first results says in particular that if f ∈ W ,m ( U, R n ), then the naturality(1.1) holds for a smooth form of any degree. See [4] for more on this case. The case k = n − k = n − f ∈ W ,n − ( U ; R n ) and | Λ k ( f ) | ∈ L q ( U ) for some q ≥ n/ ( n −
1) (instead of q ≥ p/ ( p − For convenience, we formulate our results for maps from a bounded domain into euclideanspace. However, the chain rule (1.1) is a local formula and our results also apply to the caseof mappings between smooth manifolds.
Let U ⊂ R m be a domain in m -dimensional euclidean space. A measurable differential form of degree k in U is a measurable function θ : U → Λ k ( R m ). If x , x , . . . , x m is a system ofsmooth coordinates in U , then any measurable differential k -form writes as θ = X i
A sequence { θ j } ⊂ L ( U, Λ k ) is said to converge weakly to θ ∈ L ( U, Λ k ) ifand only if for every ω ∈ C ( U, Λ m − k ) , we have Z U θ j ∧ ω → Z U θ ∧ ω . It is clear that strong convergence in L implies weak convergence. The converse is not true. Definition 2.2.
Let θ ∈ L loc ( U, Λ k ) be a k − form. If there exists a ( k + 1) − form ψ ∈ L loc ( M, Λ k +1 ) for which the equality Z U θ ∧ dω = ( − k +1 Z U ψ ∧ ω holds for any ω ∈ C ( U, Λ m − k − ) , then ψ is called the weak exterior derivative of θ (orthe exterior derivative of θ in the sense of currents) and is denoted by ψ = dθ . The form θ ∈ L loc ( M, Λ k ) is weakly closed if dθ = 0 in the weak sense, that is if Z U θ ∧ dω = 0 holds for any ω ∈ C ( U, Λ m − k − ) . emma 2.1. Let α ∈ L loc ( U, Λ k ) and β ∈ L loc ( U, Λ k +1 ) . If there exists a sequence { α j } ⊂ C ( U, Λ k ) such that α j → α and dα j → β weakly, then dα = β in the weak sense. Proof
For any ω ∈ C ( U, Λ m − k − ), we have Z U α ∧ dω = lim j →∞ Z U α j ∧ dω = ( − k +1 Z U dα j ∧ ω = ( − k +1 Z U β ∧ ω . Lemma 2.2.
Let h : U → R be a bounded function such that dh ∈ L p ′ ( R n ) and β ∈ L p ( U, Λ k ) such that dβ ∈ L ∞ ( U, Λ k ) where p ′ = p/ ( p − . Then h · β ∈ L p ( U, Λ k ) and d ( h · β ) = dh ∧ β + h · dβ. (2.1) Proof
The equation (2.1) is classic for smooth forms. Use now the density of smooth formsin L p and the H¨older inequality to obtain the equality (2.1) in the general case. Definition
A map f : U → R n is said to be bounded if f ( U ) ⊂ R n is relatively compact.It belongs to W ,p ( U, R n ) if all its component ( f , f , . . . , f n ) belong to the Sobolev space W ,p ( U, R ).Given a map f ∈ W ,p ( U, R n ), one defines the pullback of a smooth differential form α ∈ C ( R n , Λ k ) by the following formula: if α = X i
Let us denote by F k ( U, R n ) the class of maps f : U → R n defined as follow: f ∈ F k ( U, R n ) ⇔ f ∈ W , ( U, R n ) and Λ k ( f ) ∈ L ( U ) . This definition is motivated by the obvious fact that for any map f ∈ F k ( U, R n ), the pullback α f ∗ α defines a bounded operatorΛ k f = f ∗ : C ( R n , Λ k ) → L ( U, Λ k ) . Observe that the F ( U, R n ) = W , ( U, R n ) and that F k ( U, R n ) is not a vector space for2 ≤ k ≤ m .We denote by τ k the initial topology on F k ( U, R n ) induced by the inclusion F k ( U, R n ) ⊂ W , ( U, R n ) and the family of functions λ α,ω : F k ( U, R n ) → R , λ α,ω ( f ) = Z U f ∗ α ∧ ω where α ∈ C ( R n , Λ k ) and ω ∈ C ( U, Λ m − k ). In other words τ k is the coarsest topology forwhich the inclusion F k ( U, R n ) ⊂ W , ( U, R n ) is continuous, as well as all functions λ α,ω .Observe that if a sequence f j ∈ F k ( U, R n ) converges to a map f in the topology τ k , then f ∗ k α converges weakly to f ∗ α by definition.An explicit sufficient condition for the τ k -convergence in F k ( U, R n ) is given in the next result: Lemma 4.1.
Let { f j } ⊂ W , ( U, R n ) be a sequence of mappings which converges to a map f ∈ F k ( U, R n ) in the W , -topology. Assume that {| Λ k f j |} is equi-integrable, i.e. there existsa function w ∈ L ( U, R ) such that | Λ k f j | ≤ w ( x ) a.e. x ∈ U for any j ∈ N . Then f j → f inthe τ k topology. roof Let α ∈ C ( R n , Λ k ) be an arbitrary smooth k -form on R n and ω ∈ C ( U, Λ m − k ).Since f j → f in W , , we havelim j →∞ ( f ∗ j α ) ∧ ω = lim j →∞ (Λ k f j ) α ∧ ω = f ∗ α ∧ ω almost everywhere. Furthermore, we have at every point x ∈ U | ( f ∗ j α ) x ∧ θ x | ≤ | Λ k f j ( x ) | | α x | | θ x | ≤ Q · | Λ k f j ( x ) | ≤ Q · w ( x )for some constant Q . Because w ∈ L ( U, R ), the Lebesgue dominated convergence theoremimplies that lim j →∞ Z U ( f ∗ j α ) ∧ ω = Z U ( f ∗ α ) ∧ ω. (4.1) Proposition 4.2.
Let f ∈ W , ( U, R n ) be a map such thata.) The m -dimensional Hausdorff measure of the image f ( U ) ⊂ R n is finite;b.) f has essentially finite multiplicity , i.e. if there exists a constant Q < ∞ and a set E ⊂ U with measure zero such that for every point y ∈ R n , Card { x ∈ U \ E (cid:12)(cid:12) f ( x ) = y } ≤ Q. Then f ∈ F m ( U, R n ) . This proposition applies e.g. if f is a homeomorphism onto a bounded domain. Proof
In that case, | Λ k ( f ) | belongs to L by the area formula (see e.g. [2, page 220]). Remark
In [2, page 229], Giaquinta introduce a class of maps A ( U, R n ) which is very similarto our class F m ( U, R n ) (where m = dim( U )). The main difference is that the condition f ∈ W , ( U, R n ) is relaxed to the assumption that f is approximately differentiable almosteverywhere. In any case, we have a continuous embedding W , ( U, R n ) ⊂ A ( U, R n ) . k -stable maps in F k ( U, R n ) Definition
A map f ∈ F k ( U, R n ) is said to be k - stable if it belongs to the closure of C ( U, R n )in the τ k topology, i.e. there exists a sequence of smooth maps converging to f in the τ k topology. We denote by S k ( U, R n ) ⊂ F k ( U, R n ) the set of k -stable maps : S k ( U, R n ) = C ( U, R n ) τ k ⊂ F k ( U, R n ) . observe that W ,k ( U, R n ) ⊂ S k ( U, R n ).The pullback of a closed form by a stable map is again a closed form: Proposition 5.1.
Let f ∈ S k ( U, R n ) be k -stable map and α ∈ C ( R n , Λ k ) . If α is closed,then f ∗ α is weakly closed. roof Because f ∈ S k ( U, R n ), there exists a sequence { f j } of smooth maps converging to f in the τ k -topology. Assume that dα = 0, then for any φ ∈ C ( U, Λ m − k − ) we have Z U ( f ∗ j α ) ∧ dφ = ( − k +1 Z U d ( f ∗ j α ) ∧ φ = ( − k +1 Z U f ∗ j ( dα ) ∧ φ = 0 . We thus have Z U ( f ∗ α ) ∧ dφ = lim j →∞ Z U ( f ∗ j α ) ∧ dφ = 0 , (5.1)for any φ ∈ C ( U, Λ m − k − ). This means that f ∗ α is weakly closed. Proposition 5.2.
Let f ∈ W , ( U, R n ) be a map such that inf { f j } Z U sup j | Λ k f j | ! dx < ∞ , where the infimum is taken over the set of all sequences { f j } of smooth maps such that k f j − f k W , → . Then f ∈ S k ( U, R n ) . Proof
By mollification, we know that the set sequences { f j } of smooth maps such that k f j − f k W , → k † -stable maps Definition 6.1.
We define the space F k † ( U, R n ) by F k † ( U, R n ) = F n ( U, R n ) if k = n, F k ( U, R n ) ∩ F k +1 ( U, R n ) if ≤ k < n. The τ k † topology is defined for k < n to be the initial topoply for which both inclusions F k † ( U, R n ) ⊂ F k ( U, R n ) and F k † ( U, R n ) ⊂ F k +1 ( U, R n ) are continuous. For k = n , we simply define τ k † = τ k . We then say that a map f : U → R n is k † -stable if it belongs to the closure of C ( U, R n ) inthe space F k † ( U, R n ) for the τ k † topology.Observe the following elementary Lemma 6.1.
A map f : U → R n is k † -stable if and only if there exists a sequence { f j } ⊂ C ( U, R n ) of smooth maps which weakly converges to f in both spaces F k ( U, R n ) and F k +1 ( U, R n ) . roposition 6.2. Let f ∈ W , ( U, R n ) be a map such that for some k < n , inf { f j } Z U sup j ( | Λ k f j | + | Λ k +1 f k | ) ! dx < ∞ , where the infimum is taken over all sequences { f j } of smooth maps such that k f j − f k W , → . Then f ∈ S k † ( U, R n ) . Proof
This follows directly from Proposition 5.2 and the previous lemma.One can rephrase this Proposition as follow. Let f ∈ W , ( U, R n ), and assume that thereexists a sequence of smooth maps { f j } ⊂ C ( U, R n ) and a function w ∈ L ( U, R ) such that | Λ k f j ( x ) | + | Λ k +1 f j ( x ) | ≤ w ( x )a.e. x ∈ U for any j ∈ N . Then f is k † -stable.The naturality of the exterior differential holds for k † -stable maps: Theorem 6.3.
Let f ∈ S k † ( U, R n ) be k † -stable map, and let α ∈ C ( R n , Λ k ) be a smooth k -form in R m , then f ∗ α ∈ L ( U, Λ k ) , f ∗ dα ∈ L ( U, Λ k +1 ) and the equation df ∗ α = f ∗ dα holds in the weak sense. Proof
By hypothesis, there exists a sequence of smooth mappings f j ∈ C ( U, R n ) whichconverges to f in F k ( U, R n ) and F k +1 ( U, R n ) for both the τ k and τ k +1 topologies.Let α ∈ C ( R n , Λ k ) be an arbitrary smooth k -form on R m and θ ∈ C ( U, Λ m − k ). Byhypothesis, we have lim j →∞ Z U ( f ∗ j α ) ∧ θ = Z U ( f ∗ α ) ∧ θ. (6.1)We also have lim j →∞ Z U ( f ∗ j β ) ∧ φ = Z U ( f ∗ β ) ∧ φ (6.2)for any β ∈ C ( R n , Λ k +1 ) and φ ∈ C ( U, Λ m − k − ).Let us now choose β = dα and θ = dφ , we then have df ∗ j α = f ∗ j dα for any j ∈ N becauseboth α and f j are of class C , this imples that Z U ( f ∗ j α ) ∧ dφ = ( − k +1 Z U d ( f ∗ j α ) ∧ φ = ( − k +1 Z U ( f ∗ j dα ) ∧ φ. Applying (4.1) and (6.2) one gets then Z U ( f ∗ α ) ∧ dφ = lim j →∞ Z U ( f ∗ j α ) ∧ dφ = lim j →∞ ( − k Z U f ∗ j ( dα ) ∧ φ = ( − k Z U f ∗ ( dα ) ∧ φ for any φ ∈ C ( U, Λ n − k − ), this means precisely that d ( f ∗ α ) = f ∗ ( dα ) in the weak sense.7 orollary 6.4. Let U be a domain in R m and f ∈ W ,k +1 ( U, R n ) . Then the naturality (1.1)holds for any smooth k -forms α on R n . Proof
This follows from the fact that W ,k +1 ( U, R n ) ⊂ S k † ( U, R n ). We denote by S kq,p ( U, R n ) the class of maps f ∈ S k ( U, R n ) such that | df | ∈ L p ( U ) and | Λ k ( f ) | ∈ L q ( U ) . Observe that S kq,p ( U, R n ) ⊂ W ,p ( U, R n ). Theorem 7.1.
Let f ∈ S kq,p ( U, R n ) , and assume ≤ p ≤ ∞ , q = p/ ( p − .Let α ∈ C ( R n , Λ k ) be a smooth k -form in R n , then f ∗ α ∈ L ( U, Λ k ) , f ∗ dα ∈ L ( U, Λ k +1 ) and the chain rule df ∗ α = f ∗ dα holds in the weak sense. Proof
Observe that f ∗ γ is weakly closed for any closed k -form γ ∈ C ( R m , Λ k ) by Proposition5.1.Suppose first that α = a · γ where γ ∈ C ( R m , Λ k ) is a closed k − form and that a ∈ C ( R n ) isa function. Then f ∗ a = a ◦ f ∈ W , ( U ) and df ∗ a = f ∗ da (see e.g. [3, Theorem 7.8]). Because f ∈ S kq,p ( U, R n ), we have in fact | df ∗ a | ∈ L p ( U ) and | f ∗ ( γ ) | ≤ | Λ k +1 f j ( x ) | · | γ | ∈ L p ′ ( U ).Therefore we have by Lemma 2.2: df ∗ α = df ∗ ( a · γ )= d ( f ∗ a · f ∗ γ )= d ( f ∗ a ) ∧ f ∗ γ + ( f ∗ a ) · ( df ∗ γ ) | {z } =0 = d ( f ∗ a ) ∧ f ∗ γ = ( f ∗ da ) ∧ f ∗ γ = f ∗ ( da ∧ γ )= f ∗ ( dα )Consider now an arbitrary smooth k -form on R n . It can be written as a sum α = X i
Suppose that f ∈ W ,k ( U, R m ) and Λ k ( f ) ∈ L k/ ( k − ( U ) , then the naturality(1.1) holds for any smooth k -forms α on R m . Proof.
The hypothesis imply that f ∈ S kq,p ( U, R n ) .8 eferences [1] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity.
Arch. Ra-tional Mech. Anal. 63 (1976/77), no. 4, 337–403.[2] M. Giaquinta, G. Modica, and J. Souˇcek
Cartesian currents in the calculus of variations.I.
Springer-Verlag, 1998.[3] D. Gilbarg & N. Trudinger.
Elliptic partial differential equations of second order , Secondedition. Grundlehren der Mathematischen Wissenschaften
Springer-Verlag, Berlin,1983.[4] V. Gol’dshtein and M. Troyanov
A Conformal de Rham Complex arXiv:0711.1286[5] S. M¨uller, T. Qi & B.S. Yan, O n a new class of elastic deformations not allowing forcavitation.
Ann. Inst. Henri Poincar´e 11 (1994), 217–243.[6] V. ˇSver´ak