Quantitative characterization of traces of Sobolev maps

QUANTITATIVE CHARACTERIZATION OF TRACES OF SOBOLEV MAPS arXiv:2101.10934v1 [math.AP] 26 Jan 2021 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN Abstract. We give a quantitative characterization of traces on the boundary of Sobolev maps in Ẇ 1,p (M, N ), where M and N are compact Riemannian manifolds, ∂M 6= ∅: the Borel–measurable maps u : ∂M → N that are the trace of a map U ∈ Ẇ 1,p (M, N ) are characterized as the maps for which there exists an extension energy density w : ∂M → [0, ∞] that controls the Sobolev energy of extensions from ⌊p − 1⌋–dimensional subsets of ∂M to ⌊p⌋–dimensional subsets of M. 1. Introduction Given M a compact Riemannian manifold with non-empty boundary, we consider the homogeneous Sobolev space defined as   Ẇ 1,p (M, R) := U : M → R : U is weakly differentiable and DU ∈ Lp (M) . The classical trace theorem of E. Gagliardo [12] states that for p > 1 there is a well-defined continuous and surjective trace operator tr∂M : Ẇ 1,p (M, R) → Ẇ 1−1/p,p (∂M, R), such that for functions U that are additionally continuous we have tr∂M U = U ∂M . Here, for 0 < s < 1 and p ≥ 1, Ẇ s,p (∂M, R) is the homegeneous Sobolev–Slobodeckij space, or fractional Sobolev space, defined as ˆ ˆ   |u(x) − u(y)|p s,p dx dy < ∞ , (1.1) Ẇ (∂M, R) := u : M → R : m−1+sp ∂M ∂M d∂M (x, y) where m := dim M = dim ∂M + 1 and d∂M is the geodesic distance on ∂M. If N is a compact Riemannian manifold, that by Nash’s embedding theorem [21] can be assumed without loss of generality to be isometrically embedded into some Euclidean space Rν , then the homogeneous spaces of Sobolev mappings can be defined for p ≥ 1 as Ẇ 1,p (M, N ) := U ∈ Ẇ 1,p (M, Rν ) : U (x) ∈ N for almost every x ∈ M  and for 0 < s < 1 and p ≥ 1 Ẇ s,p(∂M, N ) := {u ∈ Ẇ s,p(∂M, Rν ) : u(x) ∈ N for almost every x ∈ ∂M}; these nonlinear Sobolev spaces arise naturally, for example, as domains of functionals in the calculus of variations and of partial differential equations in geometric analysis and physical models. As a consequence of the straightforward vector version of Gagliardo’s trace theorem, the trace operator tr∂M is well-defined and continuous from Ẇ 1,p (M, N ) to Ẇ 1−1/p,p (∂M, N ). 2010 Mathematics Subject Classification. 58D15 (46E35, 46T10, 46T20, 55S35). 1 2 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN The question of surjectivity of the trace operator is however much more delicate: given a map u ∈ Ẇ 1−1/p,p (∂M, N ), the classical linear extension construction gives a function U ∈ Ẇ 1,p (∂M, Rν ) such that tr∂M U = u with no guarantee whatsoever about the range of the extension U . Indeed, the surjectivity of the trace operator can first fail because of global topological obstructions: For instance if p > m, by the Morrey–Sobolev embedding, mappings in the spaces Ẇ 1,p (M, N ) and in Ẇ 1−1/p,p (∂M, N ) are almost everywhere equal to continuous maps, and classical topological obstructions for the extension of continuous maps results in obstructions for the extension of Sobolev mappings. When 1 ≤ p < m, a Lipschitz–continuous map u ∈ Lip (∂M, N ) is known to be a trace of a map in Ẇ 1,p (M, N ) if and only if u has a continuous extension to ∂M ∩ M⌊p⌋ , where M⌊p⌋ is a ⌊p⌋–dimensional skeleton of M [24, Section 4]; here and in the sequel ⌊t⌋ denotes the integer part of the real number t, so that ⌊t⌋ ∈ Z and ⌊t⌋ ≤ t < ⌊t⌋ + 1. Topological obstructions can also prevent locally the surjectivity of the trace operator: If p < m and if the homotopy group π⌊p−1⌋ (N ) is not trivial, by definition there is a map f ∈ C ∞ (S⌊p−1⌋ , N ) which is not homotopic to a constant; define the mapping u : Bm−1 → N for x = (x′ , x′′ ) ∈ Bm−1 ⊂ R⌊p⌋ × Rm−⌊p+1⌋ by u(x) := f (x′ /|x′ |); we have u ∈ Ẇ 1−1/p,p (∂M, N ), whereas there is no U ∈ Ẇ 1,p (Bm−1 × (0, 1), N ) such that trBm−1 U = u [3, Theorem 4; 14, Section 6.3]. Analytical obstructions finally arise locally for the extension problem: There exist maps in Ẇ 1−1/p,p (Bm−1 , N ) that are strong limits of smooth maps from Bm−1 to N but are not traces of maps in Ẇ 1,p (Bm−1 × (0, 1), N ). This is known to happen when either the homotopy group πℓ (N ) is infinite for some ℓ ∈ N with ℓ ≤ max{m, p} − 1 [1] (see also [3, Theorem 6]) and when p ∈ N \ {0, 1} and the homotopy group πp−1 (N ) is nontrivial [19]. These analytical obstructions can be seen in view of nonlinear uniform boundedness principle as a consequence of the failure of linear estimates on extensions for smooth maps [20]; when 2 ≤ p < 3, these analytical obstructions are connected to similar analytical obstructions for the lifting problem in fractional Sobolev spaces [2, 18]. On the other hand, the trace is known to be surjective from Ẇ 1,p (Bm−1 × (0, 1), N ) onto in the following cases: when p ≥ m [3, Theorem 1 & 2], for p > m it is a consequence of the Morrey–Sobolev embedding whereas for p = m it is a consequence of the embedding into maps of vanishing mean oscillation (VMO) [8, 9]; when 1 < p < 2 ≤ m or 2 ≤ p < 3 ≤ m with π1 (N ) ≃ {0} [14, Theorem 6.2]; when 3 ≤ p < m, π1 (N ) is finite, and π2 (N ) ≃ · · · ≃ π⌊p−1⌋ (N ) ≃ {0} [14, Theorem 6.2; 19]. The case when 4 ≤ p < m, π1 (N ), . . . , π⌊p−2⌋ (N ) are finite, and π⌊p−1⌋ (N ) ≃ {0} remains open. Ẇ 1−1/p,p (Bm−1 , N ) We are interested in the question of characterizing in general the range of the trace operator. T. Isobe [15] has provided characterization of the maps u : ∂M → N that are the traces of maps in Ẇ 1,p (M, N ) as the maps satisfying the two conditions: (oA ) The mapping u satisfies lim lim inf t→0 ε→0      ˆ |DU |p + [0,t)×∂M dist(U, N )p εp    : U ∈ Ẇ 1,p ([0, t) × ∂M, N ) and tr∂M U = u   < ∞. (oB ) The restriction of the mapping u to a generic triangulation M⌊p−1⌋ ∩ ∂M is homotopic in VMO(M⌊p−1⌋ ∩ ∂M, N ) to the restriction of a continuous function from M⌊p⌋ to TRACES OF SOBOLEV MAPS 3 N , where Mℓ with ℓ ∈ N stands for the ℓ–dimensional skeleton of a triangulation of M. Isobe’s first obstruction (oA ) is equivalent to u belonging to the image of the trace operator on Ẇ 1,p (∂M × [0, 1], N ); this condition (oA ) can be seen as an asymptotic condition on a family of Ginzburg–Landau functionals; according to the author the problem of characterizing in general maps satisfying (oA ) remains open [15, p. 367]. In (oB ), the notion of generic skeleton has to be understood in the sense of holding for almost every value of the parameter for a parametrized family of triangulations (see [13, Section 3; 23, Section 3; 24, Section 3]); the homotopy in VMO(M⌊p−1⌋ ∩ ∂M, N ) is understood in the sense of [8] (see also Section 2.4). When p 6∈ N, (oB ) can be simplified into requiring that restrictions of u to generic triangulations M⌊p−1⌋ ∩ ∂M are equal almost everywhere to restrictions of continuous function from M⌊p⌋ to N . Our goal in the present work is to characterize the image of the trace by the properties of mappings on lower-dimensional subsets. This approach is motivated by the fact that in the Gagliardo energy appearing in the definition (1.1) of the fractional Sobolev space 1− 1 ,p Ẇ p (∂M, N ), the quotient |u(x) − u(y)|p d∂M (x, y)p−1 can be interpreted as the minimal energy in Ẇ 1,p ([0, d∂M (x, y)]) to connect u(x) to u(y). Because of the quantitative nature of the phenomenon of analytical obstructions, we expect any characterization of the trace space to have some quantitative character. Finally, a workable characterization should be based on a robust definition of generic lower-dimensional set, as developed as topological screening by P. Bousquet, A. Ponce, and J. Van Schaftingen [4]. We first consider the case where the domain manifold M is the m–dimensional half-space m−1 × (0, ∞) with boundary ∂Rm = Rm−1 × {0}. We obtain the following. Rm + + := R Theorem 1.1. Assume that 2 ≤ m ∈ N, the map u : ∂Rm + → N is Borel–measurable, and λ > 1. If 1 < p < m, then the following statements are equivalent: (i) There exists U ∈ Ẇ 1,p (Rm U. + , N ) such that u = tr∂Rm + m (ii) There exists a Borel–measurable function w : ∂R+ → [0, ∞] such that ˆ w<∞ ∂Rm + and such that for every finite homogeneous simplicial complex Σ, every subcomplex Σ0 ⊂ Σ of codimension 1, and every Lipschitz–continuous map σ : Σ → Rm + which and satisfies σ(Σ0 ) ⊂ ∂Rm + ˆ w ◦ σ < ∞, Σ0 Ẇ 1,p (Σ, N ) there exists a mapping V ∈ such that trΣ0 (V ◦ σ) = u Σ0 and ˆ ˆ p−1 p λ |DV | ≤ γΣ0 ,Σ |σ|Lip w ◦ σ. Σ Σ0 (iii) There exist a constant θ > 0, a sequence (κi )i∈N in (0, ∞) converging to 0, and sets Hi ⊂ [0, κi ]m−1 , such that |Hi | lim inf m−1 > 0 i→∞ κ i 4 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN κ ,⌊p⌋ and if h ∈ Hi × {0}, then there exists a map V ∈ Ẇ 1,p (C+i tr V = u C κi ,⌊p−1⌋ +h and 0 ˆ m−⌊p⌋ |DV |p ≤ θ. κi κ ,⌊p⌋ C+i + h, N ) such that +h In order to understand the statement of Theorem 1.1, it is important to note that u denotes a mapping from ∂Rm + defined everywhere following [13, p. 66; 24, p. 5]. In other words, we do not consider equivalence classes of functions equal almost everywhere. (Given any σ or any h, there exists a map equal almost everywhere that satisfies (ii) or (iii) by being constant on the κ ,⌊p−1⌋ set σ(Σ0 ) or on C0 i + h.) In the paths condition (ii) and in the sequel, the simplexes of a simplicial complex inherit the metric and the measure from their canonical realization as an equilateral simplex of sidelength 1; on the full complex Σ, a measure is defined by additivity and a distance dΣ . Given simplicial complexes Σ and Σ0 , we have defined the quantity λ := sup γΣ 0 ,Σ (1.2) z∈Σ0 δ>0 Σ (z)| |Bλδ , δ|Σ0 ∩ BδΣ (z)| which is reminiscent of an Alhfors upper codimension-1 bound (see [16,17,22], where a doubling condition is made separately). The quantity |σ|Lip denotes the Lipschitz constant of the map σ |σ(x) − σ(y)| . |σ|Lip := sup |x − y| x,y∈Σ The function w appearing in (ii) can be interpreted as an extension energy density. In the paths condition (ii), we emphasize the facts that, as in singular homology, we do not assume anything about the local or global injectivity of σ — the map σ could even take a constant value where w is finite, in which case u ◦ σ is of course trivially extended by a Σ0´ constant — and that there is no Jacobian appearing in Σ0 w ◦ σ: we are integrating w ◦ σ on Σ0 rather than integrating w on the set σ(Σ0 ). In (iii), we define the canonical cubication of the half-space Rm + of size κ as follows. For ℓ ∈ {0, . . . , m} and κ > 0, we write κ κ (1.3) Q := Q ⊂ R : Q is a closed ℓ–dimensional face of − , 2 2 and we let then κ,ℓ (1.4)   m m κ,ℓ , Qκ,ℓ + := Q ∩ R+ : Q ∈ Q  m + κk with k ∈ Z m  m κ,ℓ+1 Qκ,ℓ 0 := Q ∩ ∂R+ : Q ∈ Q  κ,ℓ all intersect (with ℓ ∈ {0, . . . , m − 1} in the definition of Qκ,ℓ 0 ; since the faces of cubes of Q κ,ℓ m transversally ∂R+ , the cubes in Q0 are ℓ–dimensional) and (1.5) κ,ℓ = C+ [ Qκ,ℓ + , C0κ,ℓ = [ Qκ,ℓ 0 . In broad terms, the proof of Theorem 1.1 consists in deducing (ii) from (i) by a Fubini type argument (the proof is given in Section 2.1), (iii) from (ii) by the particularization to families of translations of canonical cubical complexes (the proof is given in Section 2.2), and (i) from TRACES OF SOBOLEV MAPS 5 (iii) by defining an extension by homogeneous extensions of extensions on cubical skeletons (the proof is given in Section 2.3). Assertion (iii) is very rigid because of the presence of a cubication, whereas assertion (ii) is very robust — it is invariant under diffeomorphisms whose derivative and its inverse are controlled uniformly — and is thus a natural candidate for a geometrical characterization of the image of the trace operator. This brings us thus to our geometric statement of Theorem 1.1 on manifolds. Theorem 1.2. Let M be an m–dimensional compact Riemannian manifold with boundary ∂M and 1 < p < m. There exists a δ > 0 small enough and a λ > 1 large enough with the following property. For any Borel–measurable map u : ∂M → N , there exists a map U ∈ Ẇ 1,p (M, N ) with tr∂M U = u if and only if there exists a Borel–measurable function w : ∂M → [0, ∞] such that ˆ w<∞ ∂M and such that for every homogeneous simplicial complex Σ, for every subcomplex Σ0 ⊂ Σ of codimension 1, and for every Lipschitz–continuous map σ : Σ → M satisfying ˆ w ◦ σ < ∞, σ(Σ0 ) ⊆ ∂M, |σ|Lip sup d0 ≤ δ(λ − 1), and Σ Σ0 there exists a mapping V ∈ Ẇ 1,p (Σ, N ) such that trΣ0 V = u ◦ σ ˆ ˆ p−1 p λ |DV | ≤ γΣ0 ,Σ |σ|Lip w ◦ σ. (1.6) Σ Σ0 and Σ0 Here we have defined d0 : Σ → R to be the distance to Σ0 in Σ by (1.7) d0 (y) := inf{dΣ (y, z) : z ∈ Σ0 }; the quantity supΣ d0 quantifies how far points in Σ can be from Σ0 . In comparison with Theorem 1.1, the map σ is assumed to satisfy the nonlinear conditions that σ(Σ) ⊆ M and σ(Σ0 ) ⊆ ∂M. The proof of Theorem 1.1 is based on the proof of Theorem 1.1 through suitable localization arguments. Finally, as in Isobe’s characterization by (oA ) and (oB ), the obstruction to the extension can be decoupled into a quantitative obstruction to the extension to a neighborhood of the boundary and a qualitative obstruction to the extension to the whole manifold. Theorem 1.3. Let M be an m–dimensional compact Riemannian manifold with boundary ∂M and let 1 < p < m. There exists a δ > 0 such that for each λ > 0 and u : ∂M → N the existence of an extension U ∈ Ẇ 1,p (M, N ) with tr∂M U = u is equivalent to: There exists a summable function w : ∂M → [0, ∞] such that for every finite homogeneous simplicial complex Σ, every subcomplex ´ Σ0 ⊂ Σ of codimension 1, and every Lipschitz– continuous mapping σ : Σ → M satisfying Σ0 w ◦ σ < ∞, one has: (a) If σ(Σ) ⊆ ∂M and |σ|Lip supΣ d0 ≤ δ(λ − 1), then there exists a mapping V ∈ Ẇ 1,p (Σ, N ) such that trΣ0 V = u ◦ σ Σ0 and ˆ ˆ p−1 λ |DV |p ≤ γΣ |σ| w ◦ σ. Lip 0 ,Σ Σ Σ0 6 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN (b) If dim Σ = ⌊p⌋ and if σ(Σ0 ) ⊆ ∂M, then u ◦ σ V Σ0 Σ0 is homotopic in VMO(Σ0 , N ) to for some V ∈ C(Σ, N ). The assertion (a) differs from the condition of Theorem 1.2, by the fact that in (a) we assume the stronger condition that σ ∈ ∂M in the whole of Σ instead of the weaker condition that σ ∈ ∂M in Σ0 and σ ∈ M in Σ, resulting in a weaker condition; in order to keep the equivalence we supplement (a) with (b), which is a reformulation of Isobe’s condition (oB ) as a condition on paths, as it appears in topological screening for the approximation of Sobolev mappings [4]. In the particular case where p 6∈ N, assertion (b) is equivalent to the fact that u◦σ is almost everywhere equal to the restriction to Σ0 of some V ∈ C(Σ, N ). In contrast Σ0 with Theorem 1.2, Theorem 1.3 does not give a quantitative estimate; such an estimate is precluded by the qualitative character of assertion (b). Acknowledgments. K.M. was supported by FSR Incoming postdoc. K.M. and J.V.S. were both supported by the Mandat d’Impulsion Scientifique F.4523.17, “Topological singularities of Sobolev maps” of the Fonds de la Recherche Scientifique–FNRS. 2. The case of half-spaces 2.1. From the extension to the paths condition. We prove that (i) implies (ii) in Theorem 1.1. Proposition 2.1. Let 2 ≤ m ∈ N, λ > 1, and p ∈ [1, ∞). There exists a constant C such that m given any U ∈ Ẇ 1,p (Rm + , N ) there exists a Borel–measurable function w : ∂R+ → [0, ∞] with ˆ ˆ |DU |p w≤C ∂Rm + Rm + with the following property: Suppose that Σ is a finite homogeneous simplicial complex, Σ0 ⊂ Σ is a subcomplex of m codimension 1, that the map σ : Σ → Rm + is Lipschitz–continuous and satisfies σ(Σ0 ) ⊂ ∂R+ , and that ˆ w ◦ σ < ∞. Σ0 Then there exists a map V ∈ Ẇ 1,p (Σ, N ) with trΣ0 V = U ◦ σ and ˆ ˆ p−1 p λ |DV | ≤ γΣ0 ,Σ |σ|Lip (2.1) w ◦ σ, Σ Σ0 almost everywhere on Σ0 Σ0 λ where γΣ is defined as in (1.2). 0 ,Σ Here and in the sequel, for 0 < η < 1 and ρ > 0 we define the solid spherical cap (2.2) Cρη := {x = (x′ , xm ) ∈ Rm : |x| < ρ and xm > ηρ} = Bρ ∩ (Rm−1 × (ηρ, ρ)) and note that |Cρη | = ρm |C1η |. The main ingredient of the proof is the following lemma. TRACES OF SOBOLEV MAPS 7 Lemma 2.2. Let 0 < η < 1, λ > η1 , and ρ > 0. Assume that F : Rm + → [0, ∞] is a m Borel–measurable function and that σ : Σ → R+ is a Lipschitz–continuous map defined on a homogeneous simplicial complex Σ. Let Σ0 ⊂ Σ be one-dimensional subcomplex, and assume that |σ|Lip < (ηλ − 1)ρ. Then we have ˆ ˆ  F ◦ (σ + d0 ξ) dξ (2.3) Cρη Σ (ηλ − 1)m (ρ + |σ|Lip )2m λ ≤ γΣ ,Σ η m ((ηλ − 1)ρ − |σ|Lip )m+1 0 ˆ ˆ Σ0 Rm + F (x)xm dx dz. |x − σ(z)|m λ We recall that the quantity γΣ was defined in (1.2); here and in the sequel we write 0 ,Σ m−1 m × (0, ∞). = x ∈ R+ = R (x′ , xm ) Proof of Lemma 2.2. By a change of variables x = σ(y) + d0 (y)ξ and by the non-negativity of the last component σm of σ : Σ → Rm + , we have ! ˆ ˆ ˆ ˆ  F (x) F ◦ (σ + d0 ξ) dξ = dx dy d0 (y)m Σ Cρη Σ ≤ (2.4) = |x−σ(y)|≤ρd0 (y) xm ≥ρηd0 (y)+σm (y) ! F (x) dx dy d0 (y)m ˆ ˆ Σ |x−σ(y)|≤ρd0 (y) xm ≥ηρd0 (y) ˆ ˆ Σ Σ0 ∩B Σ τd 0 (y) (y) ! F (x) dx dz dy, d0 (y)m |x−σ(y)|≤ρd0 (y) xm ≥ηρd0 (y) where τ > 1 is to be chosen later (see (2.9) below). Σ Noting that for every x ∈ Rm + satisfying |x − σ(y)| ≤ ρd0 (y) and z ∈ Σ0 ∩ Bτ d0 (y) (y) we have |x − σ(z)| ≤ |x − σ(y)| + |σ(y) − σ(z)| ≤ |x − σ(y)| + |σ|Lip dΣ (z, y) ≤ (ρ + |σ|Lip τ )d0 (y), where d0 was defined in (1.7) as the distance to Σ0 , and since, by assumption, σm (y) ≥ 0 we also have xm ≤ |xm − σm (y)| + σm (y) ≤ (ρ + |σ|Lip )d0 (y). Thus, we estimate by (2.4) ˆ ˆ  F ◦ (σ + d0 ξ) dξ Cρη (2.5) Σ ≤ (ρ + |σ|Lip )m ˆ Σ0 ˆ F (x) xm m  ˆ ρ+|σ|Lip τ y∈Σ xm Σ |x−σ(z)|≤ κ d (y,z)≤τ d0 (y) xm ≤d0 (y)≤ xκm K 1 Σ0 ∩ BτΣd0 (y) (y)  dy dx dz, 8 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN with K := ρ + |σ|Lip , κ := ηρ. We estimate now the innermost integral of the right-hand side of (2.5). Since Σ0 is compact, for every y ∈ Σ, there exists a point y0 ∈ Σ0 such that d0 (y) = dΣ (y, y0 ). Thus, if d0 (y) ≥ xm /K, we have Σ Σ Σ0 ∩ BτΣd0 (y) (y) ≥ Σ0 ∩ B(τ −1)d0 (y) (y0 ) ≥ Σ0 ∩ B(τ −1)xm /K (y0 ) . (2.6) Moreover, (2.7) n y ∈ Σ : dΣ (y, z) ≤ τ d0 (y) and xm K ≤ d0 (y) ≤ xm κ o Σ ⊆ B(τ xm )/κ (z). It follows thus from (2.6) and (2.7) that for every x ∈ Rm + and z ∈ Σ0 ˆ Σ |B(τ (τ − 1)xm dy xm )/κ (y0 )| ≤ sup (τ −1)xm K |Σ0 ∩ BτΣd0 (y) (y))| y0 ∈Σ0 |Σ0 ∩ B Σ K y∈Σ (2.8) dΣ (y,z)≤τ d0 (y) xm ≤d0 (y)≤ xκm K (τ −1)xm /K (y0 )| B Στ K δ (y0 ) (τ − 1)xm τ −1 κ ≤ . sup Σ K y0 ∈Σ0 δ|Σ0 ∩ Bδ (y0 )| δ>0 Recalling that by assumption (ηλ − 1)ρ − |σ|Lip > 0, we set τ := (2.9) ηλρ , (ηλ − 1)ρ − |σ|Lip so that one can directly check that τ −1 1 τ K = λ, = , τ −1 κ K (ηλ − 1)ρ − |σ|Lip Moreover, setting θ := (2.10) ρ+|σ|Lip τ κ ˆ = (ηλ−1)(ρ+|σ|Lip ) η((ηλ−1)ρ−|σ|Lip ) x∈Rm + |x−σ(z)|≤θxm F (x) m m−1 dx ≤ θ xm ρ + |σ|Lip τ (ρ + |σ|Lip )(ηλ − 1) = . κ η((ηλ − 1)ρ − |σ|Lip ) we get ˆ Rm + xm F (x) dx. |x − σ(z)|m Combining (2.5) with (2.8) and (2.10) we conclude. Proof of Proposition 2.1. Without loss of generality, we assume that ˆ |DU |p > 0. Rm + ν We consider a sequence (Uj )j∈N in C ∞ (Rm + , R ) such that for each j ∈ N ˆ 1 (2.11) |DUj − DU |p ≤ j , m 2 R ˆ+ 1 |Uj − U |p ≤ j , (2.12) m 2 B (0) ˆ j 1 (2.13) |Uj − u|p ≤ j . m−1 2 Bj (0)  TRACES OF SOBOLEV MAPS 9 Defining the function W : Rm + → [0, ∞] by p  W := |DU | +  (2.14) X p |DUj − DU | + X j∈N j∈N p m |Uj − U | χ Rm + ∩Bj (0) we have by (2.14), (2.11), and (2.12) ˆ (2.15) W = Rm + ˆ Rm + ≤ ˆ  |DU |p 1 + p |DU | Rm +  X ˆ m j∈N R+ 1+2 X 1 ˆ |DU |p , Rm +   m |Uj − U |p  |DUj − DU |p + χRm + ∩B (0)   j =5 j 2 j∈N  ˆ |DU |p . Rm + m We define the function w : ∂Rm + → [0, ∞] for each y ∈ ∂R+ by ˆ ˆ X W (x)xm p χ |U − u| |DU |p . dx + (2.16) w(y) := m−1 Bj (0) j m m m |x − y| R+ R+ j∈N We have by (2.15) and (2.16) ˆ ˆ ˆ ˆ W (x)xm xm W (x) dx dy = dy dx m |x − y| |x − y|m Rm ∂Rm ∂Rm Rm + + + + ˆ ˆ ˆ (2.17) 1 |DU |p . W ≤ C1 = m dy 2 + 1) 2 m m (|y| R R ∂Rm + + + We also have by (2.13) X (2.18) ˆ m−1 (0) j∈N Bj |Uj − u|p < ∞, so that in view of (2.16), (2.17), and (2.18) ˆ ˆ w ≤ C2 |DU |p < ∞. Rm + ∂Rm + m We assume now that σ : Σ → Rm + is Lipschitz–continuous, that σ(Σ0 ) ⊂ ∂R+ , and that ˆ w ◦ σ < ∞. Σ0 For each ξ ∈ Cρη , we define the map σξ := σ + d0 ξ : Σ → Rm +. (2.19) We claim that the map V := U ◦ σξ satisfies the conclusion for a suitable ξ. Indeed, by Lemma 2.2 with ρ = 2|σ|Lip /(ηλ − 1), (2.20) we first obtain ˆ Cρη (ηλ + 1)2m λ |σ|m−1 W ◦ σξ dµ dξ ≤ m Lip γΣ0 ,Σ m η (ηλ − 1) Σ ˆ ˆ Σ0 w ◦ σ. 10 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN Since |Cρη | = ρm |C1η |, there exists a ξ ∈ Cρη such that ˆ (2.21) λ (ηλ + 1)2m γΣ 0 ,Σ W ◦ σξ dµ ≤ m m η 2 η |C1 | |σ|Lip Σ Since we have assumed that (2.22) X ˆ ´ |DU |p Rm + ˆ w ◦ σ. Σ0 > 0, (2.21) implies in view of (2.14) that |DUj (σξ (y)) − DU (σξ (y))|p dy j∈N Σ + ˆ Σ m (σξ (y))|Uj (σξ (y)) − U (σξ (y))|p dy < ∞. χ Rm + ∩Bj (0) By Lipschitz–continuity of σξ and smoothness of Uj , we have D(Uj ◦ σξ ) = DUj (σξ ) · Dσξ almost everywhere (here, by · we mean the composition of differential as linear mappings, or equivalently, the multiplication of the Jacobian matrices) and thus by (2.22) ˆ X (2.23) |D(Uj ◦ σξ ) − DU (σξ ) · Dσξ |p dy j∈N Σ ≤ X ˆ |DUj (σξ (y)) − DU (σξ (y))|p |Dσξ |p dy < ∞. j∈N Σ Thus, by (2.22) and (2.23), we have ˆ ˆ p lim |Uj ◦ σξ − U ◦ σξ | + j→∞ B m (0)∩Rm + j Bjm (0)∩Rm + |D (Uj ◦ σξ ) − DU (σξ ) · Dσξ |p = 0, this implies that U ◦ σξ ∈ Ẇ 1,p (Σ, N ) and D(U ◦ σξ ) = DU (σξ ) · Dσξ . Finally, since by definition of w we have ˆ X (χB m−1 (0) ◦ σξ )|Uj ◦ σξ − u ◦ σξ |p < ∞, j∈N Σ0 j and hence Uj ◦ σ → u ◦ σ in Lploc (Σ0 ). By continuity of the trace, trΣ0 U ◦ σξ = u Σ0 . In order to conclude, we note that since ρ = 2|σ|Lip /(ηλ − 1) and ξ ∈ Cρη , we have |Dσξ | ≤ |Dσ| + |ξ| and 2|σ|Lip . (ηλ − 1) |ξ| ≤ Thus, ˆ Σ |D(U ◦ σξ )|p ≤ |σξ |pLip ˆ |DU ◦ σξ |p ≤ Σ |σ|pLip (ηλ + 1)p (ηλ − 1)p ˆ |DU ◦ σξ |p , Σ which gives, by (2.21), ˆ ˆ (ηλ + 1)2m−p (ηλ − 1)p p−1 λ p |DU ◦ σξ | ≤ |σ|Lip γΣ0 ,Σ w ◦ σ. 2m η m |C1η | Σ Σ0 We take η = λ+1 2λ and multiply w by a suitable constant, so that (2.1) holds.  TRACES OF SOBOLEV MAPS 11 2.2. From paths to cubical meshes. The implication (ii) =⇒ (iii) in Theorem 1.1 will follow from the next proposition. Proposition 2.3. Let 2 ≤ m ∈ N, λ > 1, p ∈ [1, ∞), and let u : ∂Rm + → N be a Borel– measurable map. Assume further that w : ∂Rm → [0, ∞] is a Borel–measurable function, such + that for every finite homogeneous simplicial complex Σ, any subcomplex Σ0 ⊂ Σ of codimension 1, and every Lipschitz–continuous map σ : Σ → ∂Rm + satisfying ˆ w ◦ σ < ∞, Σ0 there exists a map W ∈ Ẇ 1,p (Σ, N ) with trΣ0 W = u ◦ σ Σ0 almost everywhere on Σ0 with the estimate ˆ ˆ p−1 λ |DW |p ≤ γΣ |σ| w ◦ σ, (2.24) Lip 0 ,Σ Σ where the quantity λ γΣ 0 ,Σ Σ0 is defined in (1.2). Then, there exists a constant C > 0, such that for given h ∈ ∂Rm + and κ > 0 for which ˆ w < ∞, κ,⌊p−1⌋ C0 κ,⌊p⌋ there exists a map V ∈ Ẇ 1,p (C+ estimate + h, N ) satisfying trC κ,⌊p−1⌋ +h V = u 0 ˆ p |DV | ≤ C κ,⌊p⌋ C+ κ,⌊p⌋ The cubications C+ +h κ,⌊p−1⌋ and C0 ˆ +h with the w. κ,⌊p−1⌋ C0 +h κ,⌊p−1⌋ C0 +h where defined in (1.3)–(1.5). It follows from Proposition 2.3 that (ii) implies (iii) in Theorem 1.1, since by Fubini’s theorem ˆ ˆ ˆ w. w(x′ ) dx′ dh = κm−⌊p⌋ κ,⌊p−1⌋ [0,κ]m−1 C0 Rm−1 +h Proof of Proposition 2.3. We define for j ∈ N the sets 1,⌊p⌋ Σj := {Q ∈ Q+ : Q ⊂ [−j, j]m } 1,⌊p⌋ were Q+ 1,⌊p−1⌋ and Q0 and 1,⌊p−1⌋ : Q ⊂ [−j, j]m }, Σj0 := {Q ∈ Q0 are defined in (1.3)–(1.4). By a classical realization of cubes as simplicial complexes, we can assume that Σj is a simplicial complex and Σj0 is a simplicial subcomplex of Σj of codimension 1. Moreover, we observe that for every λ > 1, λ sup γΣ < ∞. j ,Σj j∈N 0 j m−1 by For κ > 0 and h ∈ we define the Lipschitz maps σκ,h : Σj → ∂Rm + ≃ R j (y) := κy ′ + h, where for y ∈ Σj we write y = (y ′ , ym ) ⊂ Rm−1 × R. By assumption, we σκ,h have ˆ ˆ 1 j w ◦ σκ,h ≤ ⌊p−1⌋ (2.25) w < ∞. κ,⌊p−1⌋ κ Σj0 C0 +h [0, κ]m−1 , 12 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN j By assumption, there exists a map W j ∈ Ẇ 1,p (Σj , N ) such that trΣj W j = u ◦ σκ,h 0 ˆ ˆ p−1 j j p |DW | ≤ C1 κLip . w ◦ σκ,h (2.26) and Σj0 Σj In view of (2.25) and (2.26), we have ˆ j )−1 |p = |DW j ◦ (σκ,h σj (Σj ) (2.27) Σj0 1 κp−⌊p⌋ ˆ |DW j |p Σj ≤ C1 κ⌊p−1⌋ ˆ Σj0 j ≤ C2 w ◦ σκ,h ˆ κ,⌊p−1⌋ C0 w. +h j In view of (2.27) and by weak compactness in Sobolev spaces, W j ◦ (σκ,h )−1 → V almost κ,⌊p⌋ everywhere on C+ κ,⌊p⌋ + h, where V ∈ Ẇ 1,p (C+ + h, N ), trC κ,⌊p−1⌋ +h V = u 0 ˆ ˆ |DV |p ≤ C3 w. κ,⌊p⌋ C+ κ,⌊p−1⌋ C0 +h κ,⌊p−1⌋ C0 and +h  +h 2.3. From cubical meshes to the half-space. We now prove the implication (iii) =⇒ (i) in Theorem 1.1. Proposition 2.4. Assume that 2 ≤ m ∈ N and 1 < p < m. There exists a constant C > 0 such that for every Borel–measurable map u : Rm−1 × {0} → N , every θ > 0, every sequence (κi )i∈N in (0, ∞) converging to 0, and every sequence of sets Hj ⊂ [0, κj ]m−1 , if (2.28) lim inf j→∞ |Hj | >0 κm−1 j κ ,⌊p⌋ and if for each h ∈ Hj × {0}, there exists V ∈ Ẇ 1,p (C+j u κ ,⌊p−1⌋ C0 j +h + h, N ) such that tr and m−⌊p⌋ κj (2.29) ˆ κ ,⌊p⌋ C+j κ ,⌊p−1⌋ C0 j +h V = |DV |p ≤ θ, +h then there exists a mapping U ∈ Ẇ 1,p (Rm + , N ) such that trRm−1 ×{0} U = u and ˆ |DU |p ≤ Cθ, (2.30) Rm + κ ,⌊p⌋ where the cubications C+j κ ,⌊p−1⌋ and C0 j are defined in (1.3)–(1.5). Proof. Step 1. Construction of Uhj by homogeneous extension. For each j ∈ N and every h ∈ Hj , we define the map Uhj : Rm + → N by a homogeneous extension. In order κ ,⌊p−1⌋ κj ,m−⌊p⌋ onto C j to define the extension we begin by introducing a retraction of Rm + \E + where the dual skeleton E κ,ℓ is defined for ℓ ∈ {0, . . . , m} and κ > 0 by E κ,ℓ := [ , {Q : Q is an ℓ–dimensional face of [0, κ]m + kκ, for k ∈ Zm }. κ,ℓ κ,ℓ−1 For ℓ ∈ {1, . . . , m}, we define the mapping Pκ,ℓ : C+ → C+ to be the homogeneous retraction defined on each cube Q ∈ Qκ,ℓ in the following way: Let xQ be the center of the cube Q, TRACES OF SOBOLEV MAPS 13 κj κj κj /2 ∂Rm + Figure 1. The mapping P κ,ℓ m so that Q ∩ E κ,m−ℓ = {xQ } (note that when Q ∩ ∂Rm + 6= ∅ then Q ∩ R+ is a half-cube and xQ ∈ ∂Rm + ). On this cube the map Pκ,ℓ : Q \ {xQ } → ∂Q (with the boundary taken in the ℓ–dimensional affine plane containing Q) is given by the formula x − xQ . (2.31) Pκ,ℓ (x) := xQ + κ |x − xQ |∞ κ,m−ℓ−1 → C κ,ℓ by We define now P κ,ℓ : Rm + + \E P κ,ℓ := Pκ,ℓ+1 ◦ · · · ◦ Pκ,m . (2.32) (The map P κ,ℓ is illustrated in Figure 1.) For any h ∈ Hj , we define U κj ,h for almost every x ∈ Rm + by U j,h (x) := V j,h(P κj ,⌊p⌋ (x − h) + h), (2.33) κ ,⌊p⌋ where V j,h ∈ Ẇ 1,p (C+j + h, N ) is a map given by assumptions, such that tr V j,h = u κj ,⌊p−1⌋ and (2.29) holds. C0 +h Step 2. Uniform boundedness in Lp of the gradients. We prove that when h ∈ Hj , the sequence (U j,h )j∈N remains bounded in Ẇ 1,p (Rm + ). We begin with a well-known lemma. κ,ℓ κ,ℓ−1 ,N) , N ), then we have V ◦ Pκ,ℓ ∈ Ẇ 1,p (C+ Lemma 2.5. If ℓ ∈ N, p < ℓ and V ∈ Ẇ 1,p (C+ with the estimate ˆ ˆ 2ℓp/2 p |DV |p . |DV ◦ Pκ,ℓ | ≤ κ (2.34) κ,ℓ−1 κ,ℓ p − ℓ C+ C+ Lemma 2.5 follows by the application of the next Lemma 2.6 on a suitable decomposition of κ,ℓ−1 ,N) cubes in pyramids (with a factor 2 coming from the fact that by definition of Ẇ 1,p (C+ traces coincide on common faces). Lemma 2.6. Let ℓ ∈ N and for κ > 0 we let Γκ := {x = (x′ , xℓ ) ∈ [−κ, κ]ℓ : 0 ≤ xℓ ≤ κ and xℓ = |x|∞ }. If ℓ > p, f ∈ Ẇ 1,p ([−κ, κ]ℓ−1 , N ), and if g : Γk → N is defined by g(x′ , xℓ ) = f (κx′ /xℓ ), 14 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN then g ∈ Ẇ 1,p (Γk , N ) and ˆ ℓp/2 κ |Dg| ≤ p−ℓ Γκ p Proof. We have ˆ  Df (κx′ /x ) ℓ Dg(x′ , xℓ ) = κ so that |Dg(x′ , xℓ )|2 = κ2 xℓ  |Df (κx′ /x )|2 ℓ x2ℓ + |Df |p . [−κ,κ]ℓ−1 ,− Df (κx′ /xℓ ) · x′  , x2ℓ |Df (κx′ /xℓ ) · x′ |2  κ2 ≤ ℓ |Df (κx′ /xℓ )|2 , x4ℓ x2ℓ and thus, by Fubini’s theorem and the change of variable y ′ = κx′ /xℓ , since p < ℓ, ˆ κˆ ˆ |Df (κx′ /xℓ )|p ′ p p/2 |Dg| ≤ ℓ κp dx dxℓ xpℓ Γκ 0 [−xℓ ,xℓ ]ℓ−1 ˆ κˆ ˆ ℓ−p−1 ℓp/2 p/2 ′ p xℓ ′ =ℓ |Df (y )| ℓ−p−1 dy dxℓ = κ |Df (y ′ )|p dy ′ . κ ℓ − p ℓ−1 ℓ−1 0 [−κ,κ] [0,κ] ✸ Iterating Lemma 2.5, in view of (2.33), (2.32), and (2.29), we obtain that for every h ∈ Hj , we have U j,h ∈ Ẇ 1,p (Rm + , N ) with the estimate (2.35) ˆ |DU j,h p | = ˆ |DV Rm + Rm + j,h ◦ (P κj ,⌊p⌋ p m−⌊p⌋ (· − h) + h)| ≤ C4 κ ˆ κ ,⌊p⌋ C+j |DV j,h |p - θ, +h the constants in the estimates depend only on m and p. Step 3. Convergence of the boundary data. We follow Brezis and Mironescu’s proof, see [7, Lemma 4.1 (Step 1)]. By the definition of the map U j,h in (2.33) and since U j,h ∈ Ẇ 1,p (Rm + , N ) (see (2.35)), for every h ∈ Hj we have trRm−1 U j,h = trRm−1 V j,h (P κj ,⌊p⌋ (· − h) + h) =: uj,h , thus, uj,h = u ◦ (P κj ,⌊p⌋ (· − h) + h) Rm−1 ×{0} . We are going to show that for a suitable choice of hj ∈ Hj , uj,hj → u in Lploc (Rm−1 ). We start with the observation that if k ∈ Zm−1 × {0}, then P κj ,⌊p⌋ (x− κk) = P κj ,⌊p⌋ (x)− κk, κj ,⌊p⌋ (x) ∈ ∂Rm is periodic. and thus the map ∂Rm + ∋ x 7→ x − P + Lemma 2.7. Let f : Rℓ → N be a Borel–measurable function and let Ψ ∈ L∞ (Rℓ ). Assume that for every k ∈ Zℓ , Ψ(x + κk) = Ψ(x). Then, we have for every Borel–measurable set A ⊂ Rℓ ˆ ˆ ˆ p ℓ |f (x) − f (x − Ψ(x − h))| dx dh ≤ κ sup |f − f (· − h)|p . [0,κ]ℓ A |h|≤kΨkL∞ (Rℓ ) A Lemma 2.7 is reminiscent to the opening of maps [6, Section 1.1] and the related estimates [5]. TRACES OF SOBOLEV MAPS 15 Proof. Since the function Ψ is periodic, we have by Fubini’s theorem and the change of variable z =x−h ˆ ˆ ˆ ˆ |f (x) − f (x − Ψ(z))|p dz dx |f (x) − f (x − Ψ(x − h))|p dx dh = ℓ ℓ A x−[0,κ] [0,κ] A ˆ ˆ |f (x) − f (x − Ψ(z))|p dz dx = ℓ A [0,κ] ˆ ℓ ≤κ sup |f − f (· − z)|p . ✸ |z|≤kΨkL∞ (Rℓ ) A κj ,⌊p⌋ (x−h)+h = x−Ψ(x−h). We set Ψ(x) := x−P κj ,⌊p⌋ (x), so that for every x, h ∈ ∂Rm +, P By Lemma 2.7 with ℓ = m − 1, f = u, and A = BR we obtain ! ˆ ˆ 1 |u − u(P κj ,⌊p⌋ (· − h) + h)|p dh ≤ sup ku − u(· − h)kpLp (BR ) . m−1 √ κj BR [0,κj ]m−1 |h|≤ ℓ−1 κj Since u ∈ Lploc (Rm−1 ), there exists a sequence (Rj )j∈N diverging to ∞ such that ˆ ˆ 1 lim |u(x′ ) − u(P κj ,⌊p⌋ (x′ − h) + h)|p dx′ dh = 0. j→∞ κm−1 [0,κ ]m−1 B j Rj j By our assumption (2.28), for j ∈ N large enough, we can choose an hj ∈ Hj ≃ Hj × {0} such that ˆ (2.36) lim |u − u(P κj ,⌊p⌋ (· − hj ) + hj )|p = 0. j→∞ B Rj We set (2.37) Uj := U j,hj . Conclusion. By (2.35), the sequence (Uj )j∈N that we defined in (2.37) is bounded in 1,p (Rm , Rν ) to a map Ẇ 1,p (Rm + + , N ). Therefore, up to a subsequence it converges weakly in Ẇ ν 1,p m U ∈ Ẇ (R+ , R ). Since N is compact, by Rellich–Kondrachov’s compactness theorem we have strong convergence in Lp (BR ) for every R > 0, which implies, up to a subsequence, convergence almost everywhere; hence U also takes values in the manifold N and thus U ∈ Ẇ 1,p (Rm + , N ). Finally, on the boundary we have trRm−1 Uj → trRm−1 U in Lploc (∂Rm + , N ) as j → ∞, and thus in view of (2.36), we conclude by continuity of the trace that trRm−1 U = u.  2.4. A qualitative necessary condition. Isobe’s characterization of the obstruction to the extension of Sobolev mappings [15] consisted of an analytical obstruction (oA ) and a topological obstruction (oB ). On the other hand, the characterization of Theorem 1.1 is essentially quantitative. As a complement to the proof of Theorem 1.1 and in preparation of the proof of Theorem 1.3, we state and prove the next qualitative necessary condition for the extension. Proposition 2.8. Let 2 ≤ m ∈ N, p ∈ (1, ∞). If u = tr∂Rm U for some U ∈ Ẇ 1,p (Rm + , N ), + ´ m then there exists a Borel–measurable function w : ∂R+ → [0, ∞] with ∂Rm w < ∞ such that if Σ + 16 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN is a finite simplicial complex of dimension at most ⌊p⌋, Σ0 ⊂ Σ is a subcomplex of codimension m 1, the map σ : Σ → Rm + is Lipschitz–continuous, σ(Σ0 ) ⊂ ∂R+ , and ˆ w ◦ σ < ∞, Σ0 then u ◦ σ Σ0 is homotopic in VMO(Σ0 , N ) to V dim Σ < p, then u ◦ σ Σ0 Σ0 for some V ∈ C(Σ, N ); if moreover is equal almost everywhere to V Σ0 for some V ∈ C(Σ, N ). We recall following [8] that a mapping V : Σ → Rν belongs to the space VMO(Σ, Rν ) whenever V is Borel–measurable and lim sup ρ→0 a∈Σ BρΣ (a) BρΣ (a) |V (x) − V (y)| dx dy = 0; the space VMO(Σ, Rν ) endowed with bounded mean oscillation semi-norm (2.38) kV kBMO = sup |V (x) − V (y)| dx dy < ∞ ρ>0 a∈Σ BρΣ (a) BρΣ (a) and the distance of convergence in measure is complete. We finally set VMO(Σ, N ) := {V ∈ VMO(Σ, Rν ) : V (x) ∈ N for almost every x ∈ Σ}. In particular, the maps V0 , V1 ∈ VMO(Σ, N ) are homotopic in VMO(Σ, N ) whenever there exists a map H ∈ C([0, 1], VMO(Σ, N )) such that V0 = H(0) and V1 = H(1); the continuity is understood with respect to convergence in measure and convergence with respect to the bounded mean oscillation semi-norm (2.38). We recall that Proposition 2.1 gave under the same assumptions the conclusion that u ◦ σ = trΣ0 W for some W ∈ Ẇ 1,p (Σ, N ). Proposition 2.8 would follow from ProposiΣ0 tion 2.1, embeddings of Ẇ 1,p (Σ, N ) into VMO(Σ, N ), an embedding of Ẇ 1−1/p,p (Σ0 , N ) into VMO(Σ0 , N ) together with a suitable approximation by continuous map. In order for this approach to work, one would need our assumption dim Σ ≤ p together with a regularity assumption on the simplicial complex: for instance the embedding theorem fails for simplicial complex composed of two simplices intersecting on a set of codimension at least p. In order to avoid these technical issues, we follow [4] and give a direct proof of Proposition 2.8. Proof of Proposition 2.8. As in the proof of Proposition 2.1, we define for y ∈ ∂Rm + ˆ γ W (x)xm    dx if lim |u(y) − U | = 0,  ρ→0 |x − y|m−1+γ (2.39) w(y) := Rm m R ∩Bρ (y) +    ∞ + otherwise, where 0 < γ ≤ 1 will be chosen later. The function W : Rm + → [0, ∞] is chosen as (2.40) W (x) :=  p   (M|DU |) (x)    ∞ if lim |U (x) − U | = 0, ρ→0 Rm + ∩Bρ (x) otherwise, TRACES OF SOBOLEV MAPS 17 with M|DU | : Rm + → [0, ∞] denoting the Hardy–Littlewood maximal function, given for x ∈ by Rm + ˆ 1 |DU |. M|DU |(x) = sup δ>0 |Bδ (x)| Bδ (x)∩Rm + By classical properties of Lebesgue points of functions and of traces (see for example [11, Sections 1.7 and 5.3]), the first case in the definitions (2.39) and (2.40) is taken almost everywhere. By the classical Hardy–Littlewood maximal function theorem, since p > 1, ˆ ˆ ˆ ˆ W (x)xγm w= dx dy = C W 1 m |x − y|m−1+γ ∂Rm R + + m ∂Rm + R+ (2.41) ˆ ˆ = C1 Rm + (M|DU |)p ≤ C2 |DU |p < ∞. Rm + Arguing as in the proof of Proposition 2.1, since γ ≤ 1, we obtain by an application of Lemma 2.2 that given an ℓ–dimensional simplicial complex Σ, a subcomplex ´Σ0 ⊂ Σ of codim mension 1, a Lipschitz–continuous σ : Σ → Rm + such that σ(Σ0 ) ⊂ ∂R+ and Σ0 w ◦ σ < ∞, η there exists a ξ ∈ Cρ such that ˆ W ◦ σξ < ∞, Σ η m where the map σξ : Σ → Rm + is defined in (2.19) and the solid cone Cρ ⊂ R is defined in (2.2). If W (x) < ∞ then by definition of W in (2.40), x is Lebesgue point of U and for each ρ > 0, by a suitable version of the Sobolev representation formula, we have in view of the definition of the maximal function ˆ |DU (z)| dz ≤ C4 ρM|DU |(x). |U (x) − U (z)| dz ≤ C3 (2.42) |z − x|m−1 Rm + ∩Bρ (x) Rm + ∩Bρ (x) If |x−y| ≤ ρ, then Bρ/2 ( x+y 2 ) ⊆ Bρ (x)∩Bρ (y). If x, y ∈ Σ and |y−x| ≤ δ, then |σξ (x)−σξ (y)| ≤ |σ|Lip δ. Thus, taking ρ := |σ|Lip δ, we obtain, in view of (2.42), the Lusin–Lipschitz inequality |U (σξ (x)) − U (σξ (y))| dz |U (σξ (x)) − U (σξ (y))| = B ρ ((σξ (x)+σξ (y))/2) 2 (2.43) ≤ C5  |U (σξ (x)) − U (z)| dz + Bρ (σξ (x)) |U (σξ (y)) − U (z)| dz Bρ (σξ (y)) 
≤ C6 ρ M|DU |(σξ (x)) + M|DU |(σξ (y)) . Let a ∈ Σ and δ > 0. Taking the mean value over x, y ∈ Σ ∩ Bδ (a) on both sides of (2.43) with ρ = |σ|Lip δ and applying Hölder’s inequality, we get, since dim Σ = ℓ, ˆ C7 |σ|Lip M|DU |(σξ (y)) dy (2.44) |U (σξ (x)) − U (σξ (y))| dx dy ≤ δℓ−1 BδΣ (a) BδΣ (a) BδΣ (a) ℓ ≤ C8 |σ|Lip δ1− p ˆ BδΣ (a) where the constants depend on Σ. (M|DU |)p ◦ σξ  p1 ℓ = C8 |σ|Lip δ1− p ˆ BδΣ (a) W ◦ σξ  p1 , 18 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN Similarly, if w(x) < ∞, then |u(y) − U (z)| dz ≤ C9 (2.45) ˆ |DU (z)| dz. |z − y|m−1 Rm + ∩Bρ (y) Rm + ∩Bρ (y) Hence, similarly as before, for x, y ∈ Σ0 we have ˆ  |DU (z)| dz + (2.46) |U (σξ (y)) − U (σξ (x))| ≤ C10 |z − σξ (x)|m−1 Rm + ∩Bρ (σξ (x)) ˆ |DU (z)| dz . |z − σξ (y)|m−1  Rm + ∩Bρ (σξ (y)) Hence, choosing ρ = δ|σ|Lip , taking the mean value over x, y ∈ BδΣ0 (a) on both sides, and applying Hölder’s inequality we obtain |U (σξ (x)) − U (σξ (y))| dx dy Σ Bδ 0 (a) Σ Bδ 0 (a) ≤ (2.47) Σ Bδ 0 (a) ≤ C11  ˆ Rm + ∩Bρ (σξ (x)) Σ Bδ 0 (a)  ˆ Rm + |DU (z)| dz dx |z − σξ (x)|m−1 γ |DU (z)|p zm dz dx |z − σξ (x)|m−1+γ ˆ γ/(p−1) Σ m Bδ 0 (a) R+ ∩Bρ (σξ (x)) = C12 δ 1− pℓ 1−1/p |σ|Lip ˆ Σ Bδ 0 (a) zm w◦σ  p1  p1 1 |z − σξ (x)|m−1−γ/(p−1) dz dx 1− p1 , provided γ < p − 1, with constants depending on Σ0 . Finally, if x ∈ Σ0 , y ∈ Σ and |x − y| ≤ δ so that |σξ (x) − σξ (y)| ≤ ρ = δ|σ|Lip , we have by (2.42), (2.45), and the triangle inequality, ! ˆ |DU (z)| dz + ρM|DU |(σξ (y)) . (2.48) |U (σξ (x)) − U (σξ (y))| ≤ C13 |z − σξ (x)|m−1 Rm + ∩Bρ (σξ (x)) Taking the average with respect to x ∈ BδΣ0 (a) and y ∈ BδΣ (a), we proceed as in (2.44) and (2.47), and we obtain (2.49) ! p1 ˆ ˆ 1− pℓ p−1 p |U (σξ (x)) − U (σξ (y))| dy dx ≤ C14 δ |σ|Lip w ◦ σ + |σ|Lip W ◦ σξ . Σ Σ Bδ 0 (a) Bδ (a) Σ Bδ 0 (a) BδΣ (a) By Lebesgue’s dominated convergence theorem, we have in view of (2.44), (2.47), and (2.49), 1 lim sup (2.50) |U (σξ (x)) − U (σξ (y))| dx dy = 0, ℓ δ→0 a∈Σ 1− p B Σ (a) B Σ (a) δ δ δ 1 lim sup (2.51) |U (σξ (x)) − U (σξ (y))| dx dy = 0, ℓ δ→0 a∈Σ0 1− p B Σ0 (a) B Σ0 (a) δ δ δ TRACES OF SOBOLEV MAPS 19 and 1 lim sup (2.52) δ→0 a∈Σ0 δ 1− pℓ Σ Bδ 0 (a) BδΣ (a) |U (σξ (x)) − U (σξ (y))| dx dy = 0. We define vδ (x) := Σ Bδ 0 (x) u◦σ and Vδ (x) := BδΣ (x) U ◦ σξ . By (2.50), we have limδ→0 dist (Vδ , N ) = 0, by (2.51) limδ→0 dist (vδ , N ) = 0, and by (2.52) limδ→0 kVδ − vδ kL∞ (Σ0 ) = 0. Hence for δ > 0 small enough, ΠN ◦ Vδ and ΠN ◦ vδ are welldefined and continuous respectively on Σ and on Σ0 and are close to each other on Σ0 . Hence for δ > 0 small enough, ΠN ◦ vδ is homotopic to the restriction of a continuous map. Since ΠN ◦ vδ → u ◦ σ in VMO(Σ0 , N ), we conclude that u ◦ σ is homotopic in VMO(Σ0 , N ) Σ0 Σ0 to the restriction to Σ0 of a continuous map on δ. The case p > ℓ follows from (2.45), (2.47), (2.49), and Campanato’s characterization of Hölder continuous functions by averages [10].  3. The global case In this Section we give the proof of Theorem 1.2 and Theorem 1.3. 3.1. Embedding into the half-space. In order to reduce the situation of manifolds to an open subset of a Euclidean half-space, we rely on the following isometrical embedding. Proposition 3.1 (Isometrical embedding into a half-space and retraction). If M is a compact Riemannian manifold with boundary ∂M, then there exists an isometric embedding i : M → Rµ such that i(M) ⊂ Rµ+ , i(∂M) ⊂ ∂Rµ+ , and there exists a smooth map ΠM : U → i(M) such that U ⊂ Rµ+ is relatively open, ΠM (U ∩ ∂Rµ+ ) ⊂ i(∂M), ΠM (U ∩ Rµ+ ) ⊂ i(M), and for every x ∈ i(M), x ∈ U, and ΠM (x) = x. Proof. By a collar neighborhood theorem, we can assume that M ⊂ M′ , where M′ is a compact Riemannian manifold without boundary and the inclusion is an isometry. We consider a function f : M′ → R such that f −1 (0) = ∂M, f −1 ((0, ∞)) = M and 0 < |Df | < 1 on M′ with respect to the metric g′ of M′ . In particular, g0 := g′ − Df ⊗ Df also defines a metric. By Nash’s embedding theorem, there exists a µ ∈ N and an embedding i0 : M′ → Rµ−1 which is isometric for the metric g0 . The mapping i′ : M′ → Rµ defined by i′ (x) = (i0 (x), f (x)) is : M → Rµ is then an isometric embedding for M′ endowed with the metric g′ and i := i′ M the required embedding. Since i′ (M′ ) is an embedded submanifold of the Euclidean space Rµ , there exists an open set U ′ ⊂ Rµ and Π′ ∈ C ∞ (U ′ , i′ (M′ )) such that i′ (M′ ) ⊂ U ′ and for every x ∈ i′ (M′ ), Π′ (x) = x. Moreover, since Df 6= 0 in the construction of the embedding i′ , the submanifolds i′ (M′ ) and ∂Rµ+ are transverse; there exists thus an open set U ∗ ⊂ Rµ and a δ > 0 such that i′ (M′ ) ∩ (Rµ−1 × (−δ, δ)) ⊂ U ∗ and a mapping Π∗ ∈ C ∞ (U ∗ , i(M)) such that for every (x′ , xµ ) ∈ U ∗ , Π∗ (x′ , xµ ) ∈ Rµ−1 × {xµ } and for every x ∈ i(M′ ) ∩ U ∗ , Π∗ (x) = x. By taking the set U ∗ smaller if necessary, we can also assume that for every x ∈ U ∗ and every t ∈ [0, 1], we have (1 − t)Π∗ (x) + tx ∈ U ′ . We conclude by defining the set U := U ∗ ∩ Rµ+ ∪ U ′ ∩ (Rµ−1 × (δ/2, ∞)) ,

20 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN and the mapping ΠM : U → i(M) for x = (x′ , xµ ) ∈ U by ΠM (x′ , xµ ) := Π′ ((1 − ψ(xµ ))Π∗ (x) + ψ(xµ )x), where the function ψ ∈ C ∞ (R, [0, 1]) is taken to satisfy ψ(t) = 1 when t ≤ δ/2 and ψ(t) = 0 when t ≥ δ.  3.2. Characterization of the trace space. We are now ready to prove Theorem 1.2 characterizing the traces of Sobolev maps between manifolds. The idea is to first use Proposition 3.1 to replace maps with a manifold in the domain to maps defined on a subset of the Euclidean half-space, by composing original maps with the retraction, and next to apply to those modified maps a localized version of Theorem 1.1. Proof of Theorem 1.2. Applying Proposition 3.1 we may assume, without loss of generality, that the manifold M is identified with its isometrical embedding into the half-space Rµ+ and that ΠM : U → M is the corresponding smooth retraction, where the set U ⊂ Rµ−1 × [0, ∞) is relatively open in Rµ+ . We define the sets U0 := U ∩ ∂Rµ+ (3.1) and U+ := U ∩ Rµ+ ; we choose the set V ⊂ Rµ+ relatively open in Rµ+ such that M ⊂ V and V ⊂ U, and define the sets V0 := V ∩ ∂Rµ+ (3.2) and V+ := V ∩ Rµ+ . Necessary condition. By assumption there exists a map U ∈ Ẇ 1,p (M, N ) such that tr∂M U = u. We define the maps Ū := U ◦ ΠM U+ and ū := u ◦ ΠM U0 , so that in particular Ū ∈ W 1,p (U+ , N ) and trU0 Ū = ū. We continue by observing that Lemma 2.2 and Proposition 2.1 with m = µ admit localized versions. First, in Lemma 2.2 under the additional assumption that for each y ∈ Σ we have σ(y) + d0 (y)Cρη ⊆ U+ , (3.3) the integral in (2.3) can be taken over U+ instead of Rµ+ , and we thus have ˆ ˆ ˆ ˆ  (ηλ − 1)µ (ρ + |σ|Lip )2µ λ F (x)xµ F ◦ (σ + d0 ξ) dξ ≤ dx dz. µ+1 γΣ0 ,Σ µ |x − σ(z)|µ η ((ηλ − 1)ρ − |σ|Lip ) η Cρ Σ0 U+ Σ Indeed, it suffices to observe that the dimension has changed from m to µ and that in view of the condition (3.3) and of the change of variable x = σ(y) + d0 (y)ξ, the integration domain of all the integrals with respect to x can be restricted to the set U+ . Next, for the localized version of Proposition 2.1, we define the function W̄ : U+ → [0, ∞] as in (2.14), with Rm + replaced by U+ and U by Ū , we define w̄ : V0 → [0, ∞] by (2.16), with the integrals restricted to U+ ; we have ˆ ˆ |DŪ |p . w̄ ≤ C (3.4) V0 V+ By Fubini’s theorem, there is an h ∈ ∂Rµ+ such that ∂M+ h ⊂ V and We define w := w̄ ◦ (ΠM M+h )−1 : ∂M → [0, ∞]. ´ ∂M+h w̄ ≤C ´ V0 w̄ < ∞. TRACES OF SOBOLEV MAPS 21 If the mapping σ : Σ → M is Lipschitz–continuous and if we set σ̄ = (ΠM then |σ̄|Lip ≤ C1 |σ|Lip . Thus, |σ|Lip supΣ d0 ≤ δ(λ − 1) implies for δ = δ̄/C̄ M+h )−1 ◦ σ, |σ̄|Lip sup d0 ≤ δ̄(λ − 1). (3.5) Σ Taking η = 1 2 + 1 2λ in (2.20), we get ρ = 4|σ̄|Lip /(λ − 1). (3.6) Thus, for any y ∈ Σ we have σ(y) + d0 (y)Cρη ⊂ V+ + sup d0 (y)Bρ ∩ Rµ+ y∈Σ and combining (3.6) with (3.5) we obtain sup d0 (y)ρ ≤ 4δ̄ ≤ 4C1 δ. y∈Σ This implies, from the choice of the set V, that for sufficiently small δ > 0 we have for all y ∈ Σ σ(y) + d0 (y)Cρη ⊂ U+ and thus condition (3.3) is satisfied. Moreover, since ΠM ◦ σ̄ = σ, we have ˆ ˆ w ◦ σ < ∞. w̄ ◦ σ̄ = Σ0 Σ0 We apply now localized Lemma 2.2 and proceed exactly as in the proof of Proposition 2.1: For σ̄(Σ0 ) ⊂ V0 , |σ̄|Lip ≤ δ̄(λ − 1), and ´ the Lipschitz–continuous function σ̄ : Σ → V+ with 1,p (Σ, N ) such that tr w̄ ◦ σ̄ < ∞ we obtain the existence of a map V ∈ Ẇ Σ0 V = ū ◦ σ̄ Σ0 = Σ0 u ◦ σ Σ0 and ˆ ˆ ˆ p−1 p−1 λ p−1 p λ |DV | ≤ γΣ0 ,Σ |σ̄|Lip w̄ ◦ σ̄ ≤ C1 γΣ0 ,Σ |σ|Lip (3.7) w ◦ σ. Σ Σ0 Σ0 Multiplying w by a suitable constant we obtain (1.6). This finishes the proof of the necessity part. ´ Sufficient condition: Let δ > 0 be fixed and let u : ∂M → N and w : ∂M → [0, ∞] with ∂M w < ∞ be Borel–measurable maps given by assumptions. Since ΠM (U0 ) ⊂ ∂M, the map w̄ := w ◦ ΠM : U0 → [0, ∞] is well-defined. If the mapping σ̄ : Σ → U is Lipschitz–continuous and if we set σ := ΠM ◦ σ̄ : Σ → M, then |σ|Lip ≤ C2 |σ̄|Lip and ˆ w ◦σ= Σ0 ˆ w̄ ◦ σ̄, Σ0 so that if σ̄(Σ0 ) ⊂ V0 , then for δ̄´ = δ/C2 the condition |σ̄|Lip supΣ d0 ≤ δ̄(λ − 1) implies |σ|Lip supΣ d0 ≤ δ(λ − 1), and if Σ0 w̄ ◦ σ < ∞, then by assumption there exists a map V ∈ Ẇ 1,p (Σ, N ) such that trΣ0 V = u ◦ σ Σ0 and ˆ ˆ p−1 λ |σ| |DV |p ≤ γΣ w ◦ σ. (3.8) Lip 0 ,Σ Σ Σ0 Thus by construction of σ and w̄, trΣ0 V = ū ◦ σ̄ Σ0 , where we have set again ū := u ◦ ΠM and ˆ ˆ p−1 λ |DV |p ≤ C2p−1 γΣ |σ̄| w̄ ◦ σ̄. Lip 0 ,Σ Σ Σ0 U0 , 22 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN For small enough κ0 > 0, we have by construction of U+ and V+ V+ ⊂ W+ := and V0 ⊂ W0 := {Q ∩ Rµ+ ∈ Qκ0 ,µ : Q ∩ Rµ+ ⊂ U+ } [ [ {Q ∩ ∂Rµ+ ∈ Qκ0 ,µ : Q ∩ ∂Rµ+ ⊂ U0 }, where the cubication Qκ0 ,µ is defined as in (1.3), with m replaced by µ. We define for ℓ ∈ {1, . . . , µ} and j ∈ N, the sets j,ℓ := W+ (3.9) W0j,ℓ := (3.10) and {Q ∩ Rµ+ : Q ∈ Qκj ,ℓ and Q ∩ Rµ+ ⊂ W+ }, [ {Q ∩ ∂Rµ+ : Q ∈ Qκj ,ℓ+1 and Q ∩ ∂Rµ+ ⊂ W0 }, [ j,ℓ ∪ W0j,ℓ−1 , W j,ℓ := W+ (3.11) where κj := 2−j κ0 . If j is large enough, then for every h ∈ [0, κj ]µ−1 × {0}, we have M ⊂ W j,µ + h ⊂ U+ . We let Σj be a sequence of homogeneous simplicial complexes, Σj0 ⊂ Σj be a sequence of subcomplexes of codimension 1 and σ j : Σj → W j,⌊p⌋ be a simplicial parametrization such that j,⌊p−1⌋ . We observe that for every λ > 0, σ j (Σj0 ) = W0 λ sup γΣ <∞ j ,Σj j∈N |σ j |Lip sup d0 ≤ C3 , and 0 Σj so that if λ is large enough, |σ j |Lip supΣj d0 ≤ δ̄(λ−1). Thus, as in (3.8), we obtain the existence of maps V j ∈ Ẇ 1,p (Σj , N ). We then may proceed as in the proofs of Propositions 2.3 and 2.4 to construct a map Ū ∈ Ẇ 1,p (V+ , N ) such that trV0 Ū = ū and ˆ ˆ ˆ w. w̄ ≤ C5 |D Ū |p ≤ C4 U0 V+ ∂M By Fubini’s theorem there is a set of positive measure of h ∈ ∂Rµ+ , such that we have M+h ⊂ V, Ū M+h ∈ Ẇ 1,p (M + h, N ), tr∂M+h Ū M+h = ū ∂M+h , and ˆ ˆ p |D Ū |p . |D Ū | ≤ C6 M+h For such h, we set U := Ū ◦ (ΠM and M+h (3.12) ˆ M )−1 V+ ∈ p Ẇ 1,p (M, N ) |DU | ≤ C7 ˆ and we have tr∂M U = u on ∂M w. ∂M This finishes the proof of the sufficiency part. Remark 3.2. In view of (3.4) and (3.12), the infima of  ´ p M |DU | and ´ ∂M w are comparable. 3.3. Combining a qualitative and quantitative condition. In this Section we focus on / N, then since dim Σ = ⌊p⌋ and V ∈ proving Theorem 1.3. Let us first remark that if p ∈ Ẇ 1,p (Σ, N ) we obtain by the Morrey–Sobolev embedding and the homotopy extension property that the condition (b) is equivalent to the existence of V ∈ C(Σ, N ) such that V |Σ0 = u ◦ σ almost everywhere on Σ0 . The first tool of the proof is the following proposition about the extension of boundary data already in W 1,p (∂M, N ). TRACES OF SOBOLEV MAPS 23 Proposition 3.3. Let M be a compact Riemannian manifold with boundary ∂M and let u ∈ Ẇ 1,p (∂M, N ) be a Borel–measurable map. Suppose that there exists a summable function w : ∂M → [0, ∞] with the following property: if Σ is a homogeneous simplicial complex of dimension ⌊p⌋, Σ0 ⊂ Σ is a subcomplex of Σ of dimension ⌊p−1⌋, σ : Σ → M is a Lipschitz–continuous map such that σ(Σ0 ) ⊆ ∂M satisfying ´ w ◦ σ < ∞, then u ◦ σ Σ0 is homotopic in VMO(Σ0 , N ) to the restriction of V Σ0 for Σ0 some V ∈ C(Σ, N ). Then there exists an extension U ∈ Ẇ 1,p (M, N ) with tr∂M U = u on ∂M. Proof. We use the same definitions as in the Proof of Theorem 1.2. In particular, for the summable function w : ∂M → [0, ∞] from´ the assumptions, we define w̄ := w ◦ ΠM . If σ̄ : Σ → N is Lipschitz–continuous and if Σ0 w̄ ◦ σ̄ < ∞, then, defining σ := ΠM ◦ σ̄, we ´ also have that σ is Lipschitz–continuous and Σ0 w ◦ σ < ∞. Thus, by assumption, the map ū ◦ σ̄ = u ◦ σ is homotopic in VMO(Σ0 , N ) to the restriction of V Σ0 for some V ∈ C(Σ, N ), where ū = u ◦ ΠM U0 . Again, proceeding as in the proof of the sufficient condition in Theorem 1.2, we obtain that for some fixed large enough j ∈ N and each h ∈ [0, κj ]µ−1 ≃ [0, κj ]µ−1 × {0}, where κj = 2−j κ0 > 0, we have M ⊂ W j,µ + h ⊂ U and for almost every h ∈ [0, κj ]µ−1 ≃ [0, κj ]µ−1 × {0}, and for each ℓ ∈ {0, . . . , µ − 1}, we have (3.13) ū W0j,ℓ +h ∈ Ẇ 1,p (W0j,ℓ + h, N ) j,⌊p−1⌋ and ū W j,⌊p−1⌋ +h is homotopic in VMO(W0 + h, N ) to the restriction V 0 j,⌊p−1⌋ W0 +h of a continuous map V ∈ C(W j,⌊p⌋ + h, N ). By a regularization argument and the homotopy extension property, there exists a Ū ⌊p⌋ ∈ C(W j,⌊p⌋ + h, N ) such that trW j,⌊p−1⌋ +h Ū ⌊p⌋ = ū. 0 j,ℓ−1 Now we define inductively maps Ū ℓ for ℓ ∈ {⌊p + 1⌋, . . . , µ}. Given Ū ℓ−1 ∈ Ẇ 1,p (W+ + j,ℓ + h, N ) on each Q ∈ Qκj ,ℓ by h, N ), we define Ū ℓ ∈ Ẇ 1,p (W+ ℓ Ū := ( ū Ū ℓ−1 (Pκj ,ℓ (· − h) + h) if Q ⊂ W0j,ℓ j,ℓ if Q ⊂ W+ , (but Q 6⊂ W0j,ℓ ), where the projection Pκj ,ℓ is defined in (2.31) and we take any h ∈ [0, κj ]µ−1 on which (3.13) j,ℓ + h, N ). We set Ū = Ū µ and in order holds. In view of Lemma 2.6, we have Ū ℓ ∈ Ẇ 1,p (W+ to obtain U ∈ Ẇ 1,p (M, N ) we use a Fubini argument exactly as at the end of the proof of the sufficient condition of Theorem 1.2.  The second tool is a localized version of Proposition 2.8. 1,p Proposition 3.4. Let 2 ≤ m ∈ N and p ∈ (1, ∞). If u = tr∂M U for ´ some U ∈ Ẇ (M, N ), then there exists a Borel–measurable function w : ∂M → [0, ∞] with ∂M w < ∞ such that if Σ is a finite simplicial complex of dimension at most ⌊p⌋, Σ0 ⊂ Σ is´ a subcomplex of codimension 1, σ : Σ → M is a Lipschitz–continuous with σ(Σ0 ) ⊂ ∂M, and Σ0 w ◦ σ < ∞, then u ◦ σ Σ0 is homotopic in VMO(Σ0 , N ) to V Σ0 for some V ∈ C(Σ, N ). 24 KATARZYNA MAZOWIECKA AND JEAN VAN SCHAFTINGEN Proof of Theorem 1.3. Necessary condition. The condition (a) follows immediately from Theorem 1.2 since the condition σ(Σ) ⊂ ∂M implies σ(Σ) ⊂ M and σ(Σ0 ) ⊂ ∂M; the condition (b) follows from Proposition 3.4. In view of the assumption in (a), by a variant of Theorem 1.2 applied to the manifold ∂M × (0, 1) with boundary ∂M × {0}1 there exists a map V ∈ Ẇ 1,p (∂M × [0, 1], N ) such that tr∂M×{0} V = u. By Fubini’s theorem, for almost every t ∈ (0, 1], V |∂M×{t} = tr∂M×{t} V ∈ Ẇ 1,p (∂M × {t}, N ). Through a suitable rescaling in the t variable we can assume without loss of generality that this is the case for t = 1. By Proposition 3.4 applied to the manifold ∂M × [0, 1] with boundary ∂M × {0, 1} and the boundary map V |∂M×{0,1} , there exists a summable function w̃ : ∂M × {0, 1} → [0, ∞] such that for every finite homogeneous simplicial complex Σ̃ of dimension ⌊p⌋, every subcomplex Σ̃0 ⊂ Σ̃ of codimension 1,´ every Lipschitz–continuous map σ̃ : Σ̃ → ∂M × [0, 1] satisfying σ̃(Σ̃0 ) ⊂ ∂M × {0, 1} and Σ̃0 w̃ ◦ σ̃ < ∞, the map V ◦ σ̃ Σ̃0 is homotopic in VMO(Σ̃0 , N ) to the restriction to Σ̃0 of a continuous map from Σ̃ to N . In order to apply Proposition 3.3, we define ŵ : ∂M → [0, ∞] by ŵ := w̃(·, 0) + w̃(·, 1) + w, where w is the summable map given by assumptions. If Σ is a finite ⌊p⌋–dimensional simplicial complex and Σ0 is a ⌊p − 1⌋–dimensional subcomplex, and if σ : Σ → M is Lipschitz– continuous such that σ(Σ0 ) ⊂ ∂M, we define Σ̃´ to be a simplicial realization of Σ0 × [0, 1] ´ := and set σ̃(y, t) (σ(y), t) ∈ ∂M × [0, 1]. If Σ0 ŵ ◦ σ < ∞, then Σ0 ×{0,1} w̃ ◦ σ̃ < ∞, and thus since dim(Σ) = ⌊p⌋ ≤ p, the maps V ◦ σ̃ Σ0 ×{0} and V ◦ σ̃ Σ0 ×{1} are homotopic in ´ VMO(Σ0 , N ). Moreover, in view of (b), since Σ0 w ◦ σ < ∞, the map u ◦ σ Σ0 = V Σ0 ×{0} is homotopic in VMO(Σ0 , N ) to the restriction to Σ0 of a map in C(Σ, N ). By transitivity of homotopies, V Σ0 ×{1} is homotopic in VMO(Σ0 , N ) to the restriction of a map in C(Σ, N ). By Proposition 3.3, V ∂M×{1} ∈ Ẇ 1,p (∂M × {1}, N ) is the trace of a map in Ẇ 1,p (M, N ) and the conclusion follows.  References [1] F. Bethuel, A new obstruction to the extension problem for Sobolev maps between manifolds, J. Fixed Point Theory Appl. 15 (2014), no. 1, 155–183. ↑2 [2] F. Bethuel and D. 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