Weak and strong density results for the Dirichlet energy

LetY be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in Bn×Y with equibounded Dirichlet energies, Bn being the unit ball inRn. More precisely, weak limits of graphs of smooth maps uk : Bn → Y with equibounded Dirichlet integral give rise to elements of the space cart 2,1(Bn×Y) (cf. [4], [5], [6]). In this paper we prove that every element T in cart2,1(Bn×Y) is the weak limit of a sequence {uk} of smooth graphs with equibounded Dirichlet energies. Moreover, in dimension = 2, we show that the sequence {uk} can be chosen in such a way that the energy of uk converges to the energy of T . 1. Notation and preliminary results In this section we recall some facts from the theory of Cartesian currents with finite Dirichlet energy. We refer to [6] and [4] for proofs and details. LetB be the unit ball inR and letY be a smooth oriented Riemannian manifold of dimensionM ≥ 2. By the Nash theorem we can suppose that Y is isometrically embedded in R for someN ≥ 3. We shall assume that Y is compact, connected, without boundary and that its integral 2-homology group H2(Y,Z) has no torsion, so that H2(Y, X) = H2(Y,Z)⊗X for X = R,Q. Note that the last condition automatically holds if M = 2. Dn,2-currents. Every differentialn-formω∈D(B×Y) splits as a sumω = ∑n k=0ω , n := min(n,M), where theω’s aren-forms that contain exactly k differentials in the verticalY variables. We denote by Dn,2(Bn×Y) the subspace of D(B×Y) of n-forms of the typeω = ∑2 k=0ω , and byDn,2(B×Y) the dual space of Dn,2(Bn×Y). Every (n,2)-currentT ∈ Dn,2(B×Y) splits asT = ∑2 k=0 T(k), whereT(k)(ω) = T (ω ). For example, ifu ∈ W1,2(Bn,Y), i.e.,u ∈ W1,2(Bn,RN ) with u(x) ∈ Y for a.e.x ∈ B, thenGu ∈ Dn,2(B × Y), where in an approximate sense Gu := (Id FG u)#[[ B ]], (Id FG u)(x) := (x, u(x)) (cf. [6]). M. Giaquinta: Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56100 Pisa; e-mail: giaquinta@sns.it D. Mucci: Dipartimento di Matematica dell’Universit à di Parma, Via D’Azeglio 85/A, I-43100 Parma; e-mail: domenico.mucci@unipr.it 96 Mariano Giaquinta, Domenico Mucci D-norm. Forω ∈ Dn,2(Bn × Y) we set ‖ω‖D := max { sup x,y |ω(0)(x, y)| 1 + |y|2 , ∫ Bn sup y |ω(1)(x, y)|2 dx, ∫ Bn sup y |ω(2)(x, y)| dx } , ‖T ‖D := sup{T (ω) | ω ∈ Dn,2(Bn × Y), ‖ω‖D ≤ 1}. It is not difficult to show that‖T ‖D is a norm on{T ∈ Dn,2(B × Y) | ‖T ‖D < ∞}. WeakDn,2-convergence.If {Tk} ∈ Dn,2(B×Y), we say that {Tk} converges weakly in Dn,2(B × Y), Tk ⇀ T , if Tk(ω) → T (ω) for everyω ∈ Dn,2(Bn × Y). Now, the class Dn,2(B × Y) is closed under weak convergence and ‖ · ‖D is weakly lower semicontinuous. Moreover, if sup k ‖Tk‖D < ∞, then there is a subsequence which weakly converges to someT ∈ Dn,2(B × Y) with ‖T ‖D < ∞. Boundaries. The exterior differential, d splits into a horizontal and a vertical differential, d = dx + dy . Clearly∂xT (ω) := T (dxω) defines a boundary operator ∂x : Dn,2(B ×Y) → Dn−1,2(B × Y). Now, for anyω ∈ Dn−1,2(Bn × Y), dyω belongs toDn,2(Bn × Y) if and only if dyω = 0. Then∂yT makes sense only as an element of the dual space of Zn−1,2(Bn × Y) := {ω ∈ Dn−1,2(Bn × Y) | dyω = 0}. D-graphs. The study of weak limits of sequences of maps with equibounded Dirichlet energy, minimization problems and concentration phenomena (see [6]) drew the authors of [5] to introduce the subclass D-graph(B × Y) given by the(n,2)-currents T ∈ Dn,2(B × Y) with ‖T ‖D < ∞ and such that T = GuT + ST (1.1) for some functionuT ∈ W1,2(Bn,Y) and someST ∈ Dn,2(B × Y) with ST (0) = ST (1) = 0, i.e.ST is completely vertical, so that ∂xT = 0 onDn−1,2(Bn × Y), ∂yT = 0 onZn−1,2(Bn × Y). They also showed that: (i) the decomposition (1.1) is unique; (ii) weak limits inDn,2 of sequences of graphs of smooth maps uk : B → Y, with equibounded Dirichlet energy, belong to D-graph(B × Y); (iii) if T ∈ D-graph(B × Y), then in general ∂GuT 6= 0, but ∂GuT = 0 onD n−1,1(Bn×Y), ∂GuT (ω (2)) = 0 if ω(2) = dη and sptη ⊂ B×Y and ∂yGuT = 0 onZ n−1,1(Bn × Y), ∂GuT = ∂xGuT onZ n−1,2(Bn × Y); in particular ∂yST (ω (2)) = 0 if ω(2) = dη and sptη ⊂ B×Y, ∂xST = 0 onDn−1,2(Bn×Y); (iv) ‖GuT ‖D = ‖uT ‖W1,2 ≤ ‖T ‖D, and consequently ‖ST ‖D ≤ 2‖T ‖D; Density results for the Dirichlet energy 97 (v) D-graph(B × Y) is closed under weak convergence in Dn,2 with equiboundedDnorm. The 2-dimensional case.If n = 2, obviouslyDn,2(B × Y) = D2(B × Y) and∂T is the usual boundary of currents, whereas M(T ) ≤ c‖T ‖D for some absolute constant. Consequently, weak limits of smooth graphs with equibounded Dirichlet energy are integer multiplicity (briefly i.m.) rectifiable currents in R2(B ×Y), andD-graph(B2 ×Y)∩ R2(B × Y) is closed under weak convergence with equibounded D-norm. It was proved in [5] and [6] that every T in D-graph(B2 × Y) ∩R2(B × Y) decomposes as T = GuT + ST , ST = I ∑ i=1 δxi × Ci + ST ,sing, (1.2) whereδx is the Dirac mass at x, xi ∈ B2, Ci ∈ Z2(Y) are integral cycles with nontrivial homology andST ,sing is a completely vertical, homologically trivial, i.m. rectifiable current supported on a set not containing {xi} × Y, i = 1, . . . , I . More precisely, for every Borel setA ⊂ B2 we have∂(ST A × R ) = 0. Moreover, ifπ : R2 × R → R2 andπ̂ : R2 × R → R denote the orthogonal projections onto the first and the second factor, respectively, then for any bounded Borel function φ i B2 we have ST ,sing(π #φ ∧ π̂#σ) = 0 for every element [ σ ] in the second de Rham cohomology group H 2 dR(Y). Finally, ‖ST ,sing‖({x1, . . . , xI } × Y) = 0, ‖ · ‖ denoting the total variation. As a consequence, we have ST ,sing(ω) 6= 0 only on formsω ∈ D2(B2 × Y) such thatdyω 6= 0. In particular, ifY has dimension 2, then ST ,sing = 0, whereas ifY = S2, the unit 2-sphere inR3, thenCi = zi [[ S2 ]] for some integerzi . Definition 1.1. We say that an integral 2-cycleC ∈ Z2(Y) is of spherical typeif its homology class contains a Lipschitz image of the 2-sphereS2; more precisely, if there existZ ∈ Z2(Y), R ∈ R3(Y) and a Lipschitz functionφ : S2 → Y such that C − Z = ∂R and φ#[[ S 2 ]] = Z. Spherical cycles come into play since, as proved in [5], [6], if T is in the sequential weak closure of smooth graphs with equibounded Dirichlet energies, then every Ci is of spherical type. This fact leads to the following Definition 1.2. If n = 2, we denote by cart2,1(B2×Y) the class of i.m. rectifiable currents T in D-graph(B2 ×Y) which decompose as in (1.2), where theCi ’s are of spherical type. It turns out (see [4], [5]) that cart 2,1(B2 × Y) is closed under weak convergence, with equiboundedD-norm, and contains the weak limits of sequences of smooth graphs with equiboundedD-norm. The n-dimensional case.As before, letπ : R × R → R denote the orthogonal projection onto the first factor. Let P be an oriented 2-plane in R, andPt := 98 Mariano Giaquinta, Domenico Mucci P + ∑n−2 i=1 tiνi the family of oriented 2-planes parallel to P , t = (t1, . . . , tn−2) ∈ R n−2, span(ν1, . . . , νn−2) being the orthogonal subspace to P . Similarly to the case of normal currents, for everyT ∈ Dn,2(B × Y) with ‖T ‖D < ∞, for Hn−2-a.e.t the slice T π(Pt ) of T overπ(Pt ) is a well defined current in D2((B∩Pt )×Y) with finite D-norm. Moreover, ifTk ⇀ T with equiboundedD-norm, forHn−2-a.e.t , passing to a subsequence we have Tk π(Pt ) ⇀ T π(Pt ) with equiboundedD-norm. Finally, if T ∈ D-graph(B×Y), forHn−2-a.e.t we haveT π(Pt ) ∈ D-graph((B∩Pt )×Y). Therefore in any dimensionthe following definition was introduced in [4]: Definition 1.3. We say thatT is in cart2,1(Bn × Y) if T ∈ D-graph(B × Y) and for any 2-planeP and forHn−2-a.e. t the 2-dimensional currentT π(Pt ) belongs to cart2,1((Bn ∩ Pt )× Y). It turns out that the class cart 2,1(Bn × Y) is closed under weak convergence with equiboundedD-norm and, in caseY = S2, that the class cart 2,1(Bn × S2) coincides with D-graph(B × S2), ST ,sing = 0 and T = GuT + LT × [[ S 2 ]] , (1.3) whereLT ∈ Rn−2(B) is an i.m. rectifiable current. Definition 1.4. We say that a Sobolev map u ∈ W1,2(Bn,Y) is in cart2,1(Bn,Y) if the currentGu associated to its graph is in cart2,1(Bn × Y). Therefore, aW1,2 mapu is in cart2,1(Bn,Y) if its graph has no inner boundary, i.e., ∂xGu = 0 onDn−1,2(Bn × Y), ∂yGu = 0 onZn−1,2(Bn × Y). Remark 1.5.If u : B → Y is a continuous map inW1,2(Bn,Y), by a standard convolution and projection argument it can be approximated in W1,2-strong sense by a smooth sequence inC(B,Y). This implies in particular that u ∈ cart2,1(Bn,Y). The Dirichlet energy in cart2,1. Denote by ∧ n R the space of n-vectors inR . Moreover, ifG : R → R is a linear transformation, and with the same notation G := (Gji ) n,M i,j=1 is the associated (M × n)-matrix, we let M(G) := (e1 +Ge1) ∧ · · · ∧ (en +Gen) ∈ ∧ n R , (ei) n i=1 being the canonical basis in R . ThenM(G) determines the plane graph of G in R , and in fact orients such an n-plane. IfT ∈ D-graph(B × Y), we define the Dirichlet density as the function of y ∈ Y, ξ ∈ ∧ n R given by F(y, ξ) := sup{φ(ξ) | φ : ∧ n R → R linear,φ(M(G)) ≤ 1 2|G| 2 for all linear mapsG : R → TyY}, TyY being the tangent M-space toY aty. The Dirichlet integral then extends to D-graphs T (cf. [6]) as D(T ) := ∫ F(y, E T ) d‖T ‖D, Density results for the Dirichlet energy 99 E T being the Radon–Nikodym derivative dT /d‖T ‖D, and if (1.1) holds, one has