A linear boundary value problem for weakly quasiregular mappings in

Calc. Var. 13, 295–310 (2001) Digital Object Identifier (DOI) 10.1007/s005260000074

A linear boundary value problem for weakly quasiregular mappings in space Baisheng Yan Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (e-mail: [email protected]) Received July 20, 2000 / Accepted September 22, 2000 / c Springer-Verlag 2000 Published online December 8, 2000 –


Abstract. Given a number L ≥ 1, a weakly L-quasiregular map on a do1,p main Ω in space Rn is a map u in a Sobolev space Wloc (Ω; Rn ) that satisfies |Du(x)|n ≤ L det Du(x) almost everywhere in Ω. In this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in space Rn with dimension n ≥ 3. It turns out this problem depends on the power p of the underlying Sobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in W 1,p (Ω; Rn ) can only assume a quasiregular linear boundary value; however, for all L ≥ 1 nL and 1 ≤ p < L+1 , we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in W 1,p (Ω; Rn ). Mathematics Subject Classification (1991):30C65, 30C70, 35F30, 49J30 1. Introduction In this paper, we use some techniques recently discovered in study of vectorial Hamilton-Jacobi equations in the calculus of variations to investigate the problem concerning the linear boundary values of weakly quasiregular mappings on a Lipschitz domain Ω in space Rn with dimension n ≥ 3. The main results of the paper have been recently announced in Yan [28]. We recall that (see, e.g., Astala [2], Iwaniec [11], Iwaniec and Martin [14]) a map u from a domain Ω in Rn to Rn is said to be weakly Lquasiregular, L ≥ 1 being a constant called the (outer) dilatation of u, if 1,p it belongs to a (local) Sobolev space Wloc (Ω; Rn ) for some p ≥ 1 and satisfies

296

(1.1)

B. Yan

|Du(x)|n ≤ L det Du(x)

for almost every x ∈ Ω, where Du = (∂ui /∂xj ) denotes the gradient matrix of u and |ξ| denotes the matrix norm defined by |ξ| = max|h|=1 |ξh|. Note that the class of weakly quasiregular mappings in the standard space 1,n Wloc (Ω; Rn ) becomes the class of usual quasiregular mappings (see e.g. Rickman [24]) and also coincides with the class of mappings with bounded distortion (see Reshetnyak [23]). Weakly 1-quasiregular maps will be called the weakly conformal maps. Clearly, a weakly conformal map is weakly Lquasiregular for all L > 1. In order to study the weakly L-quasiregular mappings in the framework of vectorial Hamilton-Jacobi equations in the calculus of variations, we consider the L-quasiregular sets defined by (for any L ≥ 1) (1.2)

KL = {ξ ∈ Mn×n | |ξ|n ≤ L det ξ},

where Mn×n denotes the real n × n matrices with norm |ξ| defined above. When L = 1, the set K1 will be called the conformal set. A map u ∈ 1,p Wloc (Ω; Rn ) is then weakly L-quasiregular if and only if it satisfies the special Hamilton-Jacobi equation Du(x) ∈ KL

a.e. x ∈ Ω.

Closely related to the set KL is an important function FL on Mn×n defined by (1.3) FL (ξ) = max{0, |ξ|n − L det ξ}. On one hand, this function characterizes completely the class of weakly 1,p L-quasiregular mappings as mappings in Wloc (Ω; Rn ) that satisfy FL (Du(x)) = 0 a.e. x ∈ Ω; on the other hand, function FL satisfies an important condition of quasiconvexity introduced in Morrey [16] in the calculus of variations (see also Ball [3], Ball and Murat [5], Dacorogna [7], Morrey [17]); namely, FL satisfies
1 FL (ξ) ≤ FL (ξ + Dφ(x)) dx |Ω| Ω ∀ ξ ∈ Mn×n , φ ∈ C0∞ (Ω; Rn ). (1.4) As an easy consequence of property (1.4), one easily shows that if a weakly L-quasiregular map u in W 1,n (Ω; Rn ) assumes an affine boundary value u|∂Ω = ξx + b then one must have ξ ∈ KL . One of the main results of this paper is to show this result is also valid for weakly L-quasiregular mappings in W 1,p (Ω; Rn ) if p is not too far below the dimension n. We prove the following theorem.

Linear boundary value problem for weakly quasiregular mappings

297

Theorem 1.1. For each L ≥ 1 there exists an index p = pn (L) < n such that if a weakly L-quasiregular map u in W 1,p (Ω; Rn ) assumes an affine boundary value u|∂Ω = ξx + b then ξ ∈ KL . Note that by our later existence theorem (see Theorem 1.3) and a recent significant conjecture of Iwaniec [12] the optimal number pn (L) would be nL given by pn (L) = L+1 for all n ≥ 3 and L ≥ 1. This can be shown to be the case if L = 1 and n is even; we have the following sharper result. Theorem 1.2. Let dimension n ≥ 4 be even. Suppose a weakly conformal map u in W 1,n/2 (Ω; Rn ) assumes an affine boundary value u|∂Ω = ξx + b. Then ξ ∈ K1 and u ≡ ξx + b a.e. in Ω. The proof of Theorem 1.1 and 1.2 relies on the existence of certain quasiconvex functions vanishing exactly on the set KL . The existence of ˇ ak and Yan such quasiconvex functions has been established in M¨uller, Sver´ [20], and Yan and Zhou [29], following important work of Iwaniec [11], Iwaniec and Sbordone [15] and Iwaniec and Martin [14]. The situation of linear boundary value problem may be completely different for weakly quasiregular mappings in Sobolev spaces W 1,p (Ω; Rn ) for power p not too close to the dimension n. We prove the following main existence result. nL Theorem 1.3. Let n ≥ 3, L ≥ 1 and 1 ≤ p < L+1 . Then given any affine map ξx + b there exists a weakly L-quasiregular map u in W 1,p (Ω; Rn ) such that u|∂Ω = ξx + b.

This result includes a completely new result even for weakly conformal mappings in Rn for all dimensions n ≥ 3. We have the following special case of Theorem 1.3, which implies that for even dimensions n the power n/2 is optimal for the conclusion of Theorem 1.2. Corollary 1.4. Let n ≥ 3. Then every affine map can be the boundary value of a weakly conformal map in W 1,p (Ω; Rn ) for any 1 ≤ p < n/2. More general boundary data can be considered for weakly quasiregular mappings. Throughout this paper, we say a map ϕ ∈ W 1,p (Ω; Rn ) is piecewise affine if there exist at most countably many disjoint open subsets Ωj of Ω whose union has full measure such that each ϕ|Ωj is affine. We shall prove the following stronger version of Theorem 1.3. nL . Then, for any pieceTheorem 1.5. Let n ≥ 3, L ≥ 1 and 1 ≤ p < L+1 1,p n wise affine map ϕ ∈ W (Ω; R ) and  > 0, there exists a weakly Lquasiregular map u ∈ ϕ + W01,p (Ω; Rn ) such that u − ϕLp (Ω) < .

298

B. Yan

The proof of Theorem 1.3 relies on some important ideas derived from new investigations of Gromov’s method of convex integration (see Gromov [10]). Such investigations have been initiated and successfully applied to the existence study of vectorial Hamilton-Jacobi equations of the form Du(x) ∈ ˇ ak [18], [19]; a similar method has been recently K by M¨uller and Sver´ applied to the vectorial Hamilton-Jacobi equations of more general form L(x, u(x), Du(x)) = 0 by M¨uller and Sychev [21]. A different approach to the existence study for vectorial Hamilton-Jacobi equations uses the Baire category method and has been pursued by Dacorogna and Marcellini [8], [9]. In this paper, we establish a general existence theorem (Theorem 3.2) on the Hamilton-Jacobi equation Du(x) ∈ K with affine boundary conditions for a given set K which may be unbounded, a case not covered by the general study in papers [8], [9], [18], [19], [21] mentioned above. Our existence theorem is given in a form that is sufficient for the proof of Theorems 1.3 and 1.5; some generalization can be established to cover more general Hamilton-Jacobi equations but will not be included in the present paper. Notation and Preliminaries Before we proceed to prove the main theorems in following sections, we explain some notation and preliminaries. Let m, n ≥ 1 be integers. We use Mm×n to denote the space of all real m × n matrices with the operator norm |ξ| = max|h|=1 |ξh|. We use rankξ to denote the rank of matrix ξ. Given a ∈ Rm , b ∈ Rn , let a ⊗ b be the rank-one matrix with elements (a⊗b)ij = ai bj . If m = n, we use det ξ to denote the determinant of square matrix ξ and also use D = diag(d1 , · · · , dn ) to denote the diagonal matrix D ∈ Mn×n with dii = di and dij = 0 for i = / j. We shall always assume Ω is a bounded open domain in Rn with Lipschitz boundary ∂Ω. For any measurable set E in Rn we use |E| to denote ¯ intE, convE and χE to denote, reits Lebesgue measure. We also use E, spectively, closure, interior, closed convex hull and characteristic function of any given set E. For 1 ≤ p ≤ ∞, let W 1,p (Ω; Rm ) be the usual Sobolev space of mappings from Ω to Rm with norm (see [1], [17]) uW 1,p = uLp + DuLp , where Du = (∂ui /∂xj ), i = 1, · · · , m, j = 1, · · · , n, is the gradient 1,p matrix of u. Let Wloc (Ω; Rm ) be the local Sobolev space. Note that for any Lipschitz domain Ω one can identify the space W 1,∞ (Ω; Rm ) with the space of all Lipschitz continuous mappings from Ω to Rm . We also denote

Linear boundary value problem for weakly quasiregular mappings

299

by W01,p (Ω; Rm ) the completion of C0∞ (Ω; Rm ), the set of all smooth maps with compact support in Ω, in W 1,p (Ω; Rm ) under the norm defined above. Two maps u, v in W 1,p (Ω; Rm ) are said to have the same boundary value and write u|∂Ω = v|∂Ω or u ∈ v + W01,p (Ω; Rm ) provided that u − v ∈ W01,p (Ω; Rm ). Finally, we state a simple result frequently used throughout the paper; the proof is elementary and is left to the interested reader. Lemma 1.6. Let {Ωj } be a set of at most countably many disjoint open 1,p (Ω ; Rm ) satisfy that subsets of Ω. Let u ∈ W 1,p (Ω; Rm ) and u j ∈ W j uj |∂Ωj = u|∂Ωj for each index j. Suppose j uj pW 1,p < ∞ if p < ∞ or  ˜ = χΩ\∪j Ωj u + j χΩj uj supj uj W 1,∞ < ∞ if p = ∞. Then the map u belongs to u + W01,p (Ω; Rm ). 2. Quasiconvex functions vanishing on quasiregular sets In this section, we aim to prove Theorems 1.1 and 1.2. Let F : Mm×n → R be a function. According to Morrey [16], F is said to be quasiconvex on Mm×n provided the property
1 F (ξ) ≤ (2.1) F (ξ + Dϕ(x)) dx |D| D holds for all ξ ∈ Mm×n , bounded domains D ⊂ Rn , and all smooth maps ϕ ∈ C0∞ (D; Rm ). Given any function f : Mm×n → R, we define the quasiconvexification of f , denoted by f qc , to be the function given by
1 qc f (ξ) = (2.2) inf f (ξ + Dϕ(x)) dx. ϕ∈C0∞ (D;Rm ) |D| D It is well-known that the function f qc is independent of the domain D and is always quasiconvex assuming f is continuous; see, e.g., [3], [7], [16], [17]. If F : Mm×n → R is quasiconvex function and satisfies (2.3)

|F (ξ)| ≤ C(|ξ|p + 1) ξ ∈ Mm×n ,

where C > 0, p ≥ 1 are some constants, we observe that, by a density argument (see e.g. [5], [7], [17]), property (2.1) also holds for all ϕ ∈ W01,p (D; Rm ). From this observation, we easily have the following result. Proposition 2.1. Let F : Mm×n → R be quasiconvex and satisfy (2.3). Suppose K ⊂ Mm×n is a set such that F |K = 0. If u ∈ W 1,p (Ω; Rm ) satisfies u|∂Ω = ξx+b and F (Du(x)) = 0 a.e. in Ω, then one has F (ξ) ≤ 0.

300

B. Yan

We now consider the L-quasiregular set KL ⊂ Mn×n defined above by (1.2). As defined by (1.3), let FL : Mn×n → R be the function given by FL (ξ) = max{0, |ξ|n − L det ξ}. Then, by (1.4), FL is quasiconvex and one easily sees that FL vanishes exactly on KL ; note also that the growth of FL is of n-th power: 0 ≤ FL (ξ) ≤ C |ξ|n . Another important property of this function FL is that it easily satisfies the so-called Ln -mean coercivity condition (see, e.g., [13], [30]):

(2.4) FL (Dϕ(x)) dx ≥ |Dϕ(x)|n dx ∀ ϕ ∈ C0∞ (D; Rm ). D

D

Existence of the quasiconvex functions with growth below the natural power n and vanishing exactly on the quasiregular set is the main content of our next theorem; we refer to [20], [30] for the proof and further discussions. Theorem 2.2 ([20], [30]). Let n ≥ 3. Then for every L ≥ 1 there exists a power pn (L) ∈ [n/2, n) such that for each p ≥ pn (L) one can always have a quasiconvex function g satisfying (2.5)

0 ≤ g(ξ) ≤ |ξ|p ,

g(ξ) = 0 ⇐⇒ ξ ∈ KL .

Furthermore, if n is even and L = 1, the optimal power pn (1) equals n/2. Remark. 1) Existence of quasiconvex functions satisfying (2.5) follows from property (2.4) and general results established in Yan and Zhou [30] using the important technique of nonlinear Hodge decomposition discovered in Iwaniec [11] and Iwaniec and Sbordone [15]. In fact, by [30], Theorem p/n 2.1, one can choose g to be the quasiconvexification function (FL )qc for p ≥ pn (L). 2) The special case when n is even and L = 1 has been considered ˇ ak and Yan in [20], using a special linear structure of the by M¨uller, Sver´ conformal set K1 and following the important work of Iwaniec and Martin [14]. 3) In view of some recent results in Iwaniec [12] (see also Astala [2]), we conjecture that the optimal power pn (L) in the theorem is given by nL pn (L) = L+1 for all n ≥ 3, L ≥ 1. Proof of Theorem 1.1 Let pn (L) < n be the number determined in Theorem 2.2 and let g be the function satisfying (2.5) with p = pn (L). Assume u is a weakly Lquasiregular map in ξx + b + W01,p (Ω; Rn ), where p = pn (L). Since g(Du(x)) = 0 a.e. in Ω, by Proposition 2.1, one has g(ξ) ≤ 0 and hence ξ ∈ KL , as claimed. This completes the proof.

Linear boundary value problem for weakly quasiregular mappings

301

Proof of Theorem 1.2 Let n ≥ 4 be even, and let u ∈ W 1,n/2 (Ω; Rn ) be a weakly conformal map and u|∂Ω = ξx + b. As in the proof of Theorem 1.1 it follows from Proposition 2.1 and Theorem 2.2 that ξ ∈ K1 . Therefore the map u ˜ = χΩ u+ 1,n/2 n n χRn \Ω (ξx + b) is a weakly conformal map in Wloc (R ; R ). By a well1,n known result of Iwaniec and Martin [14], u ˜ must belong to Wloc (Rn ; Rn ). Then a classical result of Liouville’s theorem (see Reshetnyak [23]) shows u ˜ is a M¨obius transformation; hence, u ˜ ≡ ξx + b. The proof of Theorem 1.2 is now completed. 3. A general existence theorem In this section we establish an existence theorem for vectorial HamiltonJacobi equation of the form Du(x) ∈ K,

a.e. x ∈ Ω,

where K is a given subset of Mm×n . In studying equations of this type, it is important to investigate certain special structures of the set K. Definition 3.1. Given any set K ⊂ Mm×n , define sets Lj (K) for j = 0, 1, 2, · · · inductively as follows: L0 (K) = K and, for j = 0, 1, · · · , Lj+1 (K) = {tξ + (1 − t)η | t ∈ [0, 1], ξ, η ∈ Lj (K), rank(ξ − η) ≤ 1}. Define the lamination hull of set K to be the set given by (3.1)

L(K) = ∪∞ j=0 Lj (K).

Remark. Clearly, from the definition, Lj+1 (K) = L1 (Lj (K)) ⊇ Lj (K) for all j = 0, 1, · · · . We refer to, e.g., [4], [7], [8], [9], [18], [19], [22], [29], [30], [31] for more properties of this and other generalized convex hulls in the calculus of variations. Definition 3.2. For each 1 ≤ p ≤ ∞, define βp (K) to be the set of all matrices ξ in Mm×n such that there exists a map u = uξ ∈ W 1,p (Ω; Rm ) satisfying Du(x) ∈ K a.e. x ∈ Ω, u|∂Ω = ξx. (3.2) Remark. 1) If ξ ∈ K, we can always choose uξ ≡ ξx; hence K ⊆ βp (K). Clearly, we always have βp (K) ⊆ βq (K) for any 1 ≤ q < p ≤ ∞. 2) The following result shows that the set βp (K) is independent of the domain Ω and that the map u ∈ ξx + W01,p (Ω; Rm ) can be chosen in such a way that u − ξxLp (Ω) is arbitrarily small.

302

B. Yan

Lemma 3.1. Let ξ ∈ βp (K) and let uξ be the map in the definition above. Then, for any bounded open set Σ in Rn and any  > 0, there exists a map v(y) ∈ ξy + W01,p (Σ; Rm ) such that v − ξyLp (Σ) <  and

1 1 g(Dv(y)) dy = g(Duξ (x)) dx |Σ| Σ |Ω| Ω for any continuous function g with |g(η)| ≤ C(|η|p + 1). Proof. Without loss of generality, assume 0 ∈ Ω. Fix a number δ > 0. Since Σ is open, for any y ∈ Σ, there exists y between 0 and δ such that the sets ¯ are contained in Σ for all 0 <  < y . Note that all such Ωy, ≡ y + Ω sets {Ωy, } form a Vitali cover of Σ; hence, by the Vitali covering lemma, ¯ and a null set N such that 0 < k < δ there exist disjoint sets {yk + k Ω} ¯ and Σ = ∪k (yk + k Ω) ∪ N. Define v = v δ : Σ → Rn by  k ξy + k uξ ( y−y δ k ) if y ∈ yk + k Ω for some k, v (y) = ξy otherwise in Σ. Then it is easy to see that v δ (y) ∈ ξy + W01,p (Σ; Rm ). Easy computation also shows that

1 1 δ g(Dv (y)) dy = g(Duξ (x)) dx |Σ| Σ |Ω| Ω for any continuous function g with |g(η)| ≤ C(|η|p +1). Finally, by choosing δ > 0 sufficiently small, one can make v δ − ξyLp (Σ) < . We now state our main existence theorem of this section. Theorem 3.2. Let K ⊂ Mm×n be a closed set and let A ⊂ βp (K) be a set satisfying
1 c0 = sup (3.3) |Duξ |p dx < ∞, ξ∈A |Ω| Ω where uξ ∈ W 1,p (Ω; Rm ) is some map satisfying (3.2). Suppose the lamination hull B = L(A) is open and bounded. Then B ⊂ βp (K). From this theorem, we easily have the following result; we refer to M¨uller ˇ ak [18] for similar results about open relations and to Gromov [10] and Sver´ and M¨uller and Sychev[21] for other related results. Corollary 3.3. Let A ⊂ Mm×n be a bounded set such that L(A) is open. ¯ Then L(A) ⊂ β∞ (A). ¯ Proof. The result follows easily from the theorem with K = A.

Linear boundary value problem for weakly quasiregular mappings

303

The following result is essential for proving our general existence theorem. Other versions of the result can be found in [8], [9], [18], [19] and [21]; a similar result in the case m = 1 has been used by Cellina in [6]. Lemma 3.4. Let U be an open set in Mm×n and let η ∈ U and η = tη1 + (1 − t)η2 with t ∈ (0, 1) and rank(η1 − η2 ) = 1. Then, for any  > 0, there exist piece-wise affine map u ∈ ηx+W01,∞ (Ω; Rm ) and finitely many points η3 , · · · , ηs in U such that Du(x) ∈ {η1 , η2 , η3 , · · · , ηs } a.e. in Ω and the measure of the set {x ∈ Ω | Du(x) = / η1 , Du(x) = / η2 } is less than . Proof. The proof follows closely some ideas in [6], [21]. We proceed in several steps. 1. Let η1 = η2 + a ⊗ b, where a ∈ Rm , b ∈ Rn and |b| = 1. Then η1 − η = a ⊗ b1 , η2 − η = a ⊗ b2 , where b1 = (1 − t)b, b2 = −tb. Since η ∈ U and U is open, we can choose b3 , · · · , bs in Rn such that ηj ≡ η + a ⊗ bj ∈ U for all j = 3, · · · , s, each bj is extreme point of the convex hull H ≡ conv{b1 , b2 , · · · , bs }, and 0 ∈ intH. Note that we can choose s = n + 2. 2. For each k = 1, 2, · · · , let {bk3 , · · · , bks } be a set of points contained in intH such that |bkj − b1 | < 1/k and 0 ∈ intconv{b1 , b2 , bk3 , · · · , bks } for all k. For simplicity, we also write bi = bki for i = 1, 2 and k = 1, 2, · · · . 3. For each given k = 1, 2, · · · , consider the function wk (x) = −1 + max bkj · x,

(3.4)

1≤j≤s

x ∈ Rn .

It is clear that wk is piece-wise affine, Lipschitz continuous on Rn and satisfies Dwk ∈ {b1 , b2 , bk3 , · · · , bks } a.e. in Rn . From the fact that 0 ∈ intconv{bk1 , · · · , bks }, the set Σ = {x ∈ Rn | wk (x) < 0} is a bounded open polyhedral convex set containing 0 and wk |∂Σ = 0 (see e.g. [25]). Therefore, we have proved that there exist a Lipschitz bounded domain Σ in Rn and a piece-wise affine function w = wk ∈ W01,∞ (Σ) such that Dw ∈ {b1 , b2 , bk3 , · · · , bks } a.e. in Σ. 4. A similar argument using the Vitali covering lemma as the one used in the proof of Lemma 3.1 shows that there exists a piece-wise affine function hk ∈ W01,∞ (Ω) with Dhk ∈ {b1 , b2 , bk3 , · · · , bks } a.e. in Ω. Note that it is important here that we allow our piece-wise affine maps to be affine on countably many sets. For each j = 1, 2, · · · , s, define Ωjk = {x ∈ Ω | Dhk (x) = bkj }. 5. We modify the values of hk on the set Ωjk for each j = 3, · · · , s so that the gradients belong to the  set {b1 , b2 , · · · , bs }. To do this, let j ∈ k k {3, · · · , s} be given, and let Ωj = ∞ ν=1 Σjν be such that hk |Σ k is affine jν

and Dhk |Σ k = bkj for all ν = 1, · · · . Since bkj ∈ intconv{b1 , b2 , · · · , bs }, jν

we have 0 ∈ intconv{b1 − bkj , · · · , bs − bkj }. As in Steps 3 and 4, we have a

304

B. Yan

˜ ν ∈ W 1,∞ (Σ k ) such that Dh ˜ ν (x) ∈ piece-wise affine Lipschitz function h jν 0  ∞ k . Define v k = h + ˜ {b1 − bkj , b2 − bkj , · · · , bs − bkj } a.e. in Σjν k j ν=1 χΣ k hν Ωjk .

on Let

Then

vjk

∈

hk + W01,∞ (Ωjk )

and

Dvjk

Gkij = {x ∈ Ωjk | Dvjk (x) = bi },

jν

∈ {b1 , b2 , · · · , bs } a.e. in Ωjk . k σij = |Gkij |/|Ωjk |.

Since vjk ∈ hk + W01,∞ (Ωjk ), we easily see that for all k = 1, 2, · · · bkj =

(3.5)

s 

k σij bi ,

i=1

s 

k σij = 1.

i=1

Note that, for any j ≥ 3, bkj → b1 as k → ∞ and, by the assumption of our selection, b1 is an extreme point of conv{b1 , b2 , · · · , bs }. We easily obtain from (3.5) that for each j = 3, · · · , s k lim σij = 0,

(3.6)

∀ i = 2, 3, · · · , s.

k→∞

6. Define fk = χΩ k ∪Ω k hk + 1

2

s

j=3 χΩjk

vjk on Ω. Then, fk is piece-wise

affine and belongs to W01,∞ (Ω), and one has Dfk (x) ∈ {b1 , b2 , · · · , bs } a.e. in Ω. Note that, for any i = 3, · · · , s, the set {x ∈ Ω | Dfk (x) = bi } equals (up to a set of measure zero) ∪sj=3 Gkij , where sets Gkij are defined in Step 5. Hence, by (3.6), we have lim |{x ∈ Ω | Dfk (x) = bi }| = 0 ∀ i ≥ 3.

k→∞

This proves the measure of the set {x ∈ Ω | Dfk (x) ∈ / {b1 , b2 }} tends to zero as k → ∞. We choose a large k¯ such that |{x ∈ Ω | Dfk¯ (x) ∈ / {b1 , b2 }}| < . 7. Finally, let u(x) ≡ ηx + fk¯ (x) a for x ∈ Ω. We easily see that u is a piece-wise affine map and satisfies all the requirements of the lemma. This completes the proof. Proposition 3.5. Let B = L(A) be an open set in Mm×n . Then for any ξ ∈ B and  > 0, there exist a piece-wise affine map u ∈ ξx + W01,∞ (Ω; Rm ) and two sets of finitely many points {α1 , α2 , · · · , αr } ⊂ A and {ξ1 , ξ2 , · · · , ξq } ⊂ B such that Du(x) ∈ {α1 , · · · , αr } ∪ {ξ1 , · · · , ξq },

a.e. x ∈ Ω

and the measure |{x ∈ Ω | Du(x) ∈ / {α1 , α2 , · · · , αr }}| < .

Linear boundary value problem for weakly quasiregular mappings

305

Proof. Given ξ ∈ B = L(A), we have ξ ∈ Lk (A) ⊂ B, where k is the minimal such index. Then ξ = tη1 + (1 − t)η2 for some t ∈ (0, 1) and η1 , η2 ∈ Lk−1 (A) with rank(η1 − η2 ) = 1. Since B is open and ξ ∈ B, by Lemma 3.1, we have a map u1 ∈ ξx + W01,∞ (Ω; Rm ) and a set {η3 , · · · , ηs } ⊂ B such that Du1 (x) ∈ {η1 , η2 } ∪ {η3 , · · · , ηs } a.e. x ∈ Ω; / {η1 , η2 }}| < 1 < . |{x ∈ Ω | Du1 (x) ∈ If k = 1, the result is proved since then η1 , η2 ∈ A. If k ≥ 2, we can apply similar arguments to each ηj ∈ Lk−1 (A) ⊂ B and to each component of the set Ωj = {x ∈ Ω | Du1 (x) = ηj } where u1 is affine to adjust the value Du1 (x) = ηj to values in the set Lk−2 (A) for most x ∈ Ωj , where j = 1, 2. Repeating this argument in a finite number of steps, one can eventually reach at the conclusion of the proposition. Assume now A ⊂ βp (K) satisfies (3.3) and B = L(A) is open. Let ξ ∈ B and let u be a map determined in the previous proposition. Let Σ = {x ∈ Ω | Du(x) ∈ A} = ∪∞ j=1 Σj , where u|Σj = αi(j) x + dj is affine for some 1 ≤ i(j) ≤ r. By Lemma 3.1 and condition (3.3), there 1,p m exists vj p∈ u + W0 (Σj ; R ) such that Dvj (x) ∈ K a.e. in Σj and Σj |Dvj | dx ≤ c0 |Σj |.  Let u ˜ = χΩ\Σ u + ∞ ˜ ∈ ξx + W01,p (Ω; Rm ) and j=1 χΣj vj . Then u satisfies (3.7) (3.8) (3.9) (3.10)

D˜ u(x) ∈ K ∪ {ξ1 , · · · , ξq }; u(x) ∈ {ξ1 , · · · , ξq }}; Ω = {x ∈ Ω | D˜ |Ω | < ;  u|p dx ≤ c0 |Ω \ Ω |. Ω\Ω |D˜

Proof of Theorem 3.2 The proof is based on the previous construction and the boundedness assumption on B = L(A). We assume, for a constant λ < ∞, |η| < λ for all η ∈ B. Let ξ ∈ B be given. We use the construction described above. Note that, u|p dx ≤ λp |Ω |. in addition to (3.7)–(3.10), it also follows that Ω |D˜ + In the following, let k → 0 be a decreasing sequence satisfying  1/p k k < ∞. ˜ defined above with Let u1 ∈ ξx + W01,p (Ω; Rm ) be the function u  = 1 . We modify the values of u1 on each open set where Du1 ∈ / K. Let Ω1 = Ω be the set defined in the construction. Write Ω1 = ∪∞ j=1 ∆j

306

B. Yan

such that u1 |∆j = ξj x + dj and ξj ∈ B. For each j, we use the previous construction for ξj ∈ B with domain ∆j and number  = 2 /2j to obtain a map v˜j ∈ u1 + W01,p (∆j ; Rm ) such that D˜ vj (x) ∈ K ∪ {ξ1 , · · · , ξq  };

(3.11) (3.12)

vj (x) ∈ {ξ1 , · · · , ξq  }}; ∆ j = {x ∈ ∆j | D˜

(3.13)

|∆ j | < 2 /2j ;



(3.14)

∆j \∆j

|D˜ vj |p dx ≤ c0 |∆j \ ∆ j |.

∞

˜j . Let Ω2 = ∪∞ j=1 ∆j . Then |Ω2 | < 2 . Define u2 = χΩ\Ω1 u1 + j=1 χ∆j v 1,p m Then u2 ∈ ξx + W0 (Ω; R ) satisfies that u2 = u1 on  Ω \ Ω1 , Du2 (x) ∈ B a.e. in Ω2 , Du2 (x) ∈ K for a.e. in Ω \ Ω2 , and Ω1 \Ω2 |Du2 |p dx ≤ c0 |Ω1 \ Ω2 |. We then modify the values of u2 on the set Ω2 as we did for u1 on Ω1 to obtain u3 and Ω3 . Continuing in this way, we obtain a sequence {uk } in ξx + W01,p (Ω; Rm ) and open sets Ωk ⊂ Ωk−1 ⊂ Ω such that |Ωk | < k ; Duk (x) ∈ B a.e. in Ωk ; Duk (x) ∈ K a.e. in Ω \ Ωk ; uk+1 = uk on Ω \ Ωk ;  p Ωk \Ωk+1 |Duk+1 | dx ≤ c0 |Ωk \ Ωk+1 |.

(3.15) (3.16) (3.17) (3.18) (3.19)

First of all, note that conditions (3.16), (3.19) yield
|Duk+1 |p dx ≤ c0 |Ωk \ Ωk+1 | + λp |Ωk+1 | ≤ C0 |Ωk |, Ωk

where C0 = max{c0 , λp }. Hence, by (3.18), Duk+1 − Duk Lp (Ω) = Duk+1 − Duk Lp (Ωk ) (3.20)

1/p

≤ 2 C0

|Ωk |1/p .

Furthermore, by (3.15)–(3.19), we easily obtain that  p (3.21) Ω |Duk+1 | dx ≤ C0 |Ω|;  p



(3.22) Ω dist (Duk (x); K) dx ≤ C |Ωk | < C k , where C is a constant and dist(η; K) is the distance function to the set K.  1/p Finally, conditions (3.20), (3.21) and the convergence of k k imply that the sequence {uk } is a Cauchy sequence in ξx + W01,p (Ω; Rm ). Let u ¯ ∈ ξx + W01,p (Ω; Rm ) be the limit of this sequence. Since K is closed,

Linear boundary value problem for weakly quasiregular mappings

307

condition (3.22) implies D¯ u(x) ∈ K for almost every x ∈ Ω. This proves ξ ∈ βp (K); hence, B ⊂ βp (K). The proof of Theorem 3.2 is now completed. Remark. From (3.21) in the proof above we easily see that the solution u ¯ ∈ ξx + W01,p (Ω; Rm ) obtained in the proof also satisfies
1 (3.23) |D¯ u|p dx ≤ C0 ≡ max{c0 , sup |η|p }. |Ω| Ω η∈B 4. Linear boundary values of weakly quasiregular mappings In this section, we prove main results Theorems 1.3 and 1.5. To apply the general existence theorem proved before, we introduce some notations. Given a number λ > 0, define R(n) = {ξ ∈ Mn×n | |ξ|n = | det ξ|}; Aλ = {ξ ∈ R(n) | |ξ| < λ}; Bλ = {ξ ∈ Mn×n | |ξ| < λ}; SO(n) = {ξ ∈ R(n) | det ξ = 1}.

(4.1) (4.2) (4.3) (4.4)

It is easy to see that the conformal set K1 = {ξ ∈ R(n) | det ξ ≥ 0}. We have the following result. Proposition 4.1. Ln−1 (Aλ ) = Bλ . Therefore Bλ = L(Aλ ) is an open set. Proof. The proof follows from a refinement of the argument in Yan [27]. Since Aλ ⊂ Bλ and Bλ is convex and hence L(Aλ ) ⊆ Bλ , it suffices to show Bλ ⊆ Ln−1 (Aλ ). Let ξ ∈ Bλ , ξ = / 0. By singular value decompositions of matrices, there exist Q1 , Q2 ∈ SO(n) such that ξ = Q1 diag(σ1 , · · · , σn ) Q2 , where |σ1 | ≤ · · · ≤ |σn |, σn = / 0 and |σn | < λ. Let R2 = σn Q2 ; then ξ = Q1 diag(1 , · · · , n−1 , 1) R2 ,

(4.5) where i =

σi σn

∈ [−1, 1] for i = 1, · · · , n − 1. Let, for j = 0, 1, · · · , n − 1,

Sj = {diag(1 , · · · , n ) | |i | ≤ 1, ∀ i ≥ 1; |k | = 1, ∀ k ≥ j + 1; n = 1}. (4.6) Note that any η = diag(1 , · · · , n ) ∈ Sj can be written as η = tη + + (1 − t)η − with t = (1 + j )/2 ∈ [0, 1] and η ± = diag(1 , · · · , j−1 , ±1, j+1 , · · · , n ) ∈ Sj−1 ;

308

B. Yan

clearly, rank(η + −η − ) = 1. Hence, Sj ⊆ L1 (Sj−1 ) for j = 1, 2, · · · , n−1. This proves Sn−1 ⊆ Ln−1 (S0 ). Finally, by (4.5), we have ξ ∈ Q1 Sn−1 R2 ⊆ Q1 Ln−1 (S0 ) R2 ⊆ Ln−1 (Q1 S0 R2 ) ⊆ Ln−1 (Aλ ). Therefore Bλ ⊆ Ln−1 (Aλ ); the proof is completed. nL , and let B be the open Proposition 4.2. Let n ≥ 3, L ≥ 1 and 1 ≤ p < L+1 n unit ball in R . Then for any ξ ∈ R(n) there exists a weakly L-quasiregular map uξ ∈ ξx + W01,p (B; Rn ) such that
1 |Duξ |p dx ≤ C |ξ|p , |B| B where C is a constant depending only on n, L and p.

Proof. Let ξ ∈ R(n) be given. If ξ ∈ K1 we choose uξ ≡ ξx. Assume ξ ∈ R(n) \ K1 . Define ξx L+1 uξ (x) = , δ= (4.7) , ∀x= / 0. L |x|δ nL , an elementary computation shows that uξ ∈ W 1,p (B; Since 1 ≤ p < L+1 n R ), Duξ (x) ∈ KL (in fact Duξ (x) ∈ ∂KL ) for all x = / 0 and uξ |∂B = ξx. Furthermore, using the formula |Duξ (x)| = |x|−δ |ξ|, one easily has
1 |Duξ |p dx ≤ cn,p,δ |ξ|p . |B| B This completes the proof. Finally let us notice that the maps uξ we have used in the proof also satisfy Duξ (x) ∈ K1 ∪ ∂KL for a.e. x ∈ B.

Proof of Theorem 1.3 / 0. Let λ = 2|ξ|. Then ξ ∈ Bλ . Let ξ ∈ Mn×n be given. We assume ξ = From Proposition 4.1, L(Aλ ) = Bλ is open and bounded; one also has supη∈Bλ |η|p ≤ C1 |ξ|p . Also, from Proposition 4.2, the set A = Aλ ⊂ βp (KL ) satisfies the condition (3.3) in Theorem 3.2 with constant c0 ≤ C2 |ξ|p . Therefore, Theorem 3.2 implies ξ ∈ βp (KL ). This proves the theorem. Using Lemma 3.1 and the remark following the proof of Theorem 3.2, we easily obtain the following result from Theorem 1.3. nL Corollary 4.3. Let n ≥ 3, L ≥ 1 and 1 ≤ p < L+1 . Then, for any ξ ∈ n×n n , b ∈ R and  > 0, there exists a weakly L-quasiregular map M u ∈ ξx + b + W01,p (Ω; Rn ) satisfying
(4.8) |Du |p dx ≤ C3 |ξ|p |Ω|, u − ξx − bLp (Ω) < , Ω

where C3 is a constant depending only on n and p.

Linear boundary value problem for weakly quasiregular mappings

309

Proof of Theorem 1.5 nL . Let ϕ ∈ W 1,p (Ω; Rn ) be a piece-wise Let n ≥ 3, L ≥ 1 and 1 ≤ p < L+1 ∞ affine map. Let Ω = ∪j=1 Ωj ∪ N , |N | = 0, be such that ϕ|Ωj = ξj x + bj is affine for all j ≥ 1. The condition ϕ ∈ W 1,p (Ω; Rn ) implies

(4.9)

∞ 

|ξj |p |Ωj | < ∞.

j=1

We apply Corollary 4.3 to each ξj , bj and Ωj and obtain weakly L-quasiregular maps uj ∈ ϕ + W01,p (Ωj ; Rn ) satisfying
 |Duj |p dx ≤ C3 |ξj |p |Ωj |, uj − ϕLp (Ωj ) < j/p . 2 Ωj  Then it is easily seen that the map u = j χΩj uj belongs to ϕ + W01,p (Ω; Rn ), is weakly L-quasiregular and satisfies u − ϕLp (Ω) < . The proof is completed. Finally, from the previous proofs and the note in the proof of Proposition 4.2, we easily obtain the following slightly stronger result. nL Theorem 4.4. Let n ≥ 3, L ≥ 1 and 1 ≤ p < L+1 . Then, for any piece1,p n wise affine map ϕ ∈ W (Ω; R ) and  > 0, there exists a map u ∈ ϕ + W01,p (Ω; Rn ) such that Du (x) ∈ K1 ∪ ∂KL a.e. in Ω and u − ϕLp (Ω) < .

Remark. We don’t know whether the set K1 ∪ ∂KL in this theorem can be replaced simply by ∂KL . ˇ ak Acknowledgement. The author would like to thank Tadeusz Iwaniec and Vladim´ır Sver´ for their interest in the result of this paper. He also thanks Dr. M. A. Sychev for a careful reading of the original manuscript.

References 1. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975 2. K. Astala, Analytic aspects of quasiconformality, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 617–626 3. J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337–403 4. J. M. Ball and R. D. James, Proposed experimental tests of a theory of fine microstructures and the two well problem, Phil. Trans. Roy. Soc. London 338A (1992), 389–450 5. J. M. Ball and F. Murat, W 1,p -Quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), 225–253

310

B. Yan

6. A. Cellina, On minima of a functional of the gradient: sufficient conditions, Nonlinear Anal. 20 (4) (1993), 343–347 7. B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, New York, 1989 8. B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1) (1997), 1–37 9. B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems, J. Funct. Anal. 152 (2) (1998), 404–446 10. M. Gromov, Partial Differential Relations, Springer-Verlag, New York, 1986 11. T. Iwaniec, p-Harmonic tensors and quasiregular mappings, Ann. of Math. (2) 136(3) (1992), 589–624 12. T. Iwaniec, Nonlinear Cauchy-Riemann operators in Rn , Trans. Amer. Math. Soc., to appear 13. T. Iwaniec and A. Lutoborski, Integral estimates for null Lagrangians, Arch. Rational Mech. Anal. 125 (1993), 25–79 14. T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29–81 15. T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. reine angew. Math. 454 (1994), 143–161 16. C. B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25–53 17. C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966 ˇ ak, Attainment results for the two-well problem by convex inte18. S. M¨uller and V. Sver´ gration, in Geometric Analysis and the Calculus of Variations, pp. 239–251, Internat. Press, Cambridge, MA, 1996 ˇ ak, Unexpected solutions of first and second order partial differ19. S. M¨uller and V. Sver´ ential equations, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 691–702 ˇ ak and B. Yan, Sharp stability results for almost conformal maps in 20. S. M¨uller, V. Sver´ even dimensions, J. Geom. Anal. 9(4) (1999), 671–681 21. S. M¨uller and M. A. Sychev, Optimal existence theorems for nonhomogeneous differential inclusions, Preprint 22. P. Pedregal, Laminates and microstructure, European J. Appl. Math. 4(2) (1993), 121– 149 23. Y. G. Reshetnyak, Space Mappings with Bounded Distortion, Transl. Math. Mono., Amer. Math. Soc., Vol. 73, 1989 24. S. Rickman, Quasiregular Mappings, Springer-Verlag, Berlin, 1993 25. R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1972 26. B. Yan, Remarks about W 1,p -stability of the conformal set in higher dimensions, Ann. Inst. H. Poincar´e, Analyse Non Lin´eaire 13 (6) (1996), 691–705 27. B. Yan, On rank-one convex and polyconvex conformal energy functions with slow growth, Proc. Roy. Soc. Edinburgh 127A (1997), 651–663 28. B. Yan, Linear boundary values of weakly quasiregular mappings, C. R. Acad. Sci. Paris Ser. I Math. 331 (2000), 379–384 29. B. Yan and Z. Zhou, Stability of weakly almost conformal mappings, Proc. Amer. Math. Soc. 126 (1998), 481–489 30. B. Yan and Z. Zhou, Lp -mean coercivity, regularity and relaxation in the calculus of variations, Nonlinear Anal., to appear 31. K. Zhang, On various semiconvex hulls in the calculus of variations, Calc. Var. Partial Differential Equations 6 (1998), 143–160