Estimates of Quasiconvex Polytopes in the Calculus of Variations

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1 Journal of Convex Analysis Volume 13 26), No. 1, 37 5 Estimates of Quasiconvex Polytopes in the Calculus of Variations Kewei Zhang epartment of Mathematics, University of Sussex, Falmer, Brighton, BN1 9RF, United Kingdom k.zhang@sussex.ac.uk Received: May 17, 23 Revised manuscript received: April 15, 25 We give direct estimates for the quasiconvex polytopes QK) generated by a finite set K M N n. More precisely, we bound the quasiconvex envelope Q dist, K) near a convex exposed face of CX) which does not have rank-one connections. Our estimates depend on the weak-1,1) bounds for certain singular integral operators and the geometric features of the convex polytope CK). We show by an example that our estimate is local and independent of the size of K, hence it is a better estimate than the polyconvex hull P K) which is size dependent. In the variational approach to material microstructure [2, 3, 12], the notion of quasiconvex hull QK) for a compact set K M N n was introduced in [17] to locate the average of microstructure. Although there are some examples of explicit calculation of quasiconvex hulls, in general, no methods have been developed to calculate QK). Let #K) be the number of elements of a finite set K. It is known that if #K) = 2 or 3, QK) = K if and only if K does not have rank-one connections [2, 18, 4]. When #K) 4, this is no longer the case see, e.g. [4, 2, 12]) and it is not known how one calculates QK) even for a general four-point set K. It is well-known [2] that if A, B K and ranka B) = 1, then the line segment [A, B] QK). There are two extreme cases which indicate that in general, it is very difficult to calculate QK) even for a finite set: i) if K is contained in a subspace without rank-one matrices [4], then QK) = K; ii) if all points of K are rank-one connected, e.g., when N = 1 or n = 1, QK) = CK). There are various notions of semiconvex hulls for K to bound QK) both from outside and inside. Let L c K), RK) and P K) be the closed lamination convex hull, the rank-one convex hull and the polyconvex hull of K respectively [17, 13, 22], then one has L c K) RK) QK) P K) CK). However, except CK) and L c K), other semiconvex hulls are also difficult to calculate even if K is a finite set. A qualitative result known as the equal hull property was obtained in [22] also see [7] for generalizations) which implies that QK) = CK) if and only if L c K) = CK). A key observation in [22] that motivates the present work is that if QK) is not convex, the non-convexity must occur near the boundary CK). The example of Tartar s four point set [2, 4, 12] see Example 8 below) suggests that to bound the quasiconvex hull QK) from outside with limited information that K is finite and QK) is not convex, one has to bound QK) near CK). ISSN / $ 2.5 c Heldermann Verlag

2 38 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations An interesting feature of quasiconvexity analysis for vector-valued mappings is the fact that the projection of the gradient on certain subspaces can control the whole gradient. More precisely, suppose E M N n is a subspace without rank-one matrices and let P E be the orthogonal projection to its orthogonal complement, then [4], Ω P E φx)) 2 dx c Ω φx) 2 dx, for φ W 1,2 Ω, R N ), where c > is a constant depending only on E. In this paper we use the weak-1,1) version of this inequality to study quasiconvex hull for a finite set. We call the quasiconvex hull QK) of a finite set K M N n a quasiconvex polytope. In this paper, we give a more quantitative estimate of QK) by establishing a lower bound of the quasiconvex envelope Q distx, K) of the distance function near an exposed face E of CK) that does not have rank-one connections. Intuitively, suppose dim CK) = 3 and the exposed face E is one dimensional, then along E we can chip a wedge like slice off CK) without touching QK) see Example 9). From the structure of convex polytopes [5, 14], we see that a k dimensional polytope P is the convex hull of all of its i-dimensional exposed) faces i =, 1,..., k 1). Therefore, if QK) CK) and K is finite, we may claim that there is at least one 1-face i.e., an exposed edge) which does not have rank-one connections [22]. Before we state our main results, let us first introduce some notation. We say that two matrices A, B M N n are rank-one connected if ranka B) = 1. A set K M N n has a rank-one connection there are two elements in K which are rank-one connected. From now on, we denote by K M N n a non-empty finite set with QK) CK). Let d = dim CK) be the affine dimension of the convex hull CK) and we assume that d > 1 as the case d = 1 is trivial. Let M M N n be the plane containing CK) with dim M = d. For a point X M N n and a set V M N n, let V X = {Y X, Y V }. Suppose L CK) is a non-trivial < dim L < d) exposed face of polytope CK) which does not have rank-one connections. Let E be the plane that contains K = L K with diml) = dime). Since it is well-known and easy to check by definition that quasiconvex hull is translation invariant in the sense QK X) = QK) X, we may assume that K hence E is a linear subspace of M N n. Let W be a supporting plane of CK) such that L = CK) W. Obviously, E W M, hence both W and M are subspaces and dim W = d 1. We denote by V be the orthogonal complement of E in W. enote by e the unit normal vector a matrix with norm 1) of W in M pointing to the half-space of M containing CK). Let F = span[e] V and define { } e X cos θ W = inf P F X), X K \ E, P F X), 1) where P F is the orthogonal projection from M N n to F. Since K is finite, we have cos θ W > and e X cos θ W P F X) for all X K. The above inequality still hold if X K and P F X) = because in this case X e.

3 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations 39 Next we may optimize the angle θ W by varying the possible supporting planes W satisfying W CK) = L. Let W be the collection of supporting planes of CK) such that L = W CK). Then the optimal angle θ, π/2) is defined as = sup{cos θ W, W W}. Since K is finite and M is a finite dimensional plane, it is easy to see that can be reached by some W with W = E V and F = span[e] V. From now on we drop the superscript and denote by W the optimal supporting plane and W, F, V the corresponding subspaces given above. Thus we have, for X K, e X P F X). 2) Note that E is the subspace generated by the exposed face L which contains. In general, E itself is not a supporting plane of CK). In case it is, that is, E = W, we have θ =, hence = 1. Also note that θ is the angle which makes cosθ ) the largest among all θ W. We also need some estimates for certain singular integral operators. Let E M N n be a subspace without rank-one matrices and let P E be the orthogonal projection to the orthogonal complement E of E. In a joint work with the author, Z. Iqbal [9] established both the strong-p, p) and the weak-1, 1) estimates u L p R n ) Cp, E) P E u) L p R n ), 1 < p <, u W 1,p, meas{x R n, ux) λ}) C E λ P E u) L 1 R n ), u W 1,1, where Cp, E) > and C E are constants. In this paper we give a more explicit description of the singular integral operator T E involved in these estimates. Since E does not have rank-one matrices, we have [4] If we let then 1 λ E = µ E >. inf{ P E a b) 2, a R N, b R n, a = b = 1} = µ E >. 3) λ E = sup{ P E a b) 2, a R N, b R n, a = b = 1}, Theorem 1. Let E M N n be a subspace without rank-one matrices and assume that dim E = k 1. Then for every φ C R n, R N ), meas{x R n, φx) λ}) C E P λ E φy)) dy, for all λ >, 4) R n where C E > is a constant in the form C E = Cn, N) µ [n/2]+1 E ), 4 ) with Cn, N) > a constant depending only on n, N while [n/2] being the integer part of n/2.

4 4 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations Our estimate for C E in 4 ) follows from [15, 16]. There might be sharper estimates than 4 ). However, since our approach in this paper uses only exposed faces, even if sharp estimate exists for C E, it is unlikely that 6) below gives sharp estimates for the quasiconvex hull. The weak-1,1) estimate for projection to a subspace was first used by Müller [11] to construct a non-convex quasiconvex homogeneous function of degree 1 in M 2 2, where the subspace E is the anti-conformal subspace [15, pp. 6]. For further applications in M 2 2, see [23]. The following is the main result of this paper. Theorem 2. Let K M N n be a finite set such that CK) QK). Assume that d = dimck)) with 1 < d N n and let L CK) be a non-trivial face without rank-one connections, K = K L, K and E = span[l]. Then ) Q distx, K) CE, θ, σ) distx, K) 1 + σ)c E P cos θ E X), 5) for any σ >, where CE, θ, σ) > is a constant given by 15) below, and P E is the orthogonal projection from M N n to the orthogonal complement of E. Furthermore, distx, K) > C E P E X) implies X / QK). 6) Remark 3. In case / ink, we may change the above estimates by taking X K, replacing E by E = E X and 5) and 6) being replaced respectively by ) Q distx, K) CE, θ, σ) distx, K) 1 + σ)c E P cos θ E X X ), 5 ) and distx, K) > C E P E X X ) implies X / QK). 6 ) due to the translation from QK X ) to QK). If we apply Theorem 2 to each non-trivial exposed faces of CK) without rank-one connections, we may define a non-convex set which stays between QK) and CK). An important feature of Theorem 2 is that our estimate 6) gives an alternative way of bounding QK) from outside that depends only on the local property of CK) near the exposed face L and the ellipticity property of E through P E. It is independent of the diameter size) of CK). More precisely, let the unbounded convex set M + M containing CK) be defined as M + = {X M, e X P F X) } with the relative boundary M + = {X M, e X = P F X) }, we see that our estimate 6) is not affected by the size of CK) for points near L as long as CK) stays in M +, that is, CK) is bounded by the wall M +.

5 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations 41 Inequality 6) sometimes gives better estimates of QK) than the widely used polyconvex hull P K) that may depend on the size of CK) see Example 1), even if the constant C E in 6) is not sharp. After notation and preliminaries, we establish Theorem 2 first, followed by that of Theorem 1. At the end of this paper, we give examples related to Theorem 1 and Theorem 2. We denote by M N n the space of all real N n matrices, with R Nn norm, measu) is the Lebesgue measure of a measurable subset U R n and let dista, K) = inf P K A P be the distance function from a point A M N n to a set K M N n. From now on let Ω be a nonempty, open and bounded subset of R n. We denote by u the gradient of a vector-valued) function u and we define the space C k Ω, R N ), Lebesgue spaces L p Ω, R N ) and Sobolev spaces W 1,p Ω, R N ) in the usual way. A continuous function f : M N n R is quasiconvex at A M N n if for any smooth function φ : Ω R N compactly supported in Ω, Ω fa + φx))dx Ω fa)dx. If f is quasiconvex at every A M N n, it is called a quasiconvex function [1, 1, 6]. The class of quasiconvex functions is independent of the choice of Ω. A continuous function f : M N n R is polyconvex if fa) = convex function of minors of the matrix A. It is well known that polyconvexity implies quasiconvexity [2] while the converse is not true. For a given continuous function f : M N n R, the quasiconvex envelope Qf) is defined by Qf) = sup{g f, g is quasiconvex}, and can be calculated by using the formula Qf)A) = inf fa + φx)) dx, φ C,RN ) where R n is the unit cube [6]. Similarly, the polyconvex envelope is defined as P f) = sup{g f, g is polyconvex}, In the study of material microstructure, the notion of quasiconvex hull QK) for a closed set K M N n was introduced in [17]. Roughly speaking, QK) contains all the weak limits that can be generated by a sequence of gradients u j approaching K: If distu j, K) in L 1 and u j converges weakly in L 1 to u, then ux) QK) a.e. The quasiconvex hull and polyconvex hull are defined as QK) = {X M N n, fx) sup fy ), for every quasiconvex f : M N n R}; Y K P K) = {X M N n, fx) sup fy ), for every polyconvex f : M N n R}. Y K

6 42 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations When K is compact, the quasiconvex hull QK) can be defined by a single quasiconvex function [21] as QK) = {A M N n, Q dist p A, K) = } for each 1 p <. This characterization of QK) will be used later in the proof of Theorem 2. For a finite set K = {A i } m i=1 M 2 2, the polyconvex hull of K is given by P K) = {X = m λ i A i, i=1 m m λ i det A i ) = det λ i A i ), λ i, i=1 i=1 m λ i = 1}. i=1 In particular [17, 18], if there is a point X M 2 2 and a real number α such that deta i X ) = α for 1 i m, then P K) = {X CK), detx X ) = α}. We need this characterization for P K) in this special case to calculate P K) in Example 1 below. We conclude our preparation by stating the following result [8] which is a consequence of the measurable selection lemma. Proposition 4. Let K R n be a compact subset and let u : Ω R n be a continuous mapping. Then there exists a measurable mapping ũ : Ω K such that for all x Ω ux) ũx) = distux), K). Now we establish our main results. We prove Theorem 2 first, accepting Theorem 1 for the moment. Proof of Theorem 2. If K is contained in a plane E without rank-one connections, it was established in [9] for any compact set K that Q dist, K) C E dist, K), where C E > is a constant depending on E. So we consider the general case that CK) may have rank-one connections. Let X M N n be fixed and let R n be the unit cube. Suppose φ j ) is a sequence in C, R N ) such that lim distx + φ j, K)dx = Q distx, K) := a. j We extend φ j to R n by zero outside so that φ j C R n, R N ). Now we apply the measurable selection lemma Proposition 1) to the function F x, Q) = X + φ j x) Q for x and Q K. There exists a measurable mapping X j : Ω K, such that X + φ j x) X j x) = distx + φ j x), K),

7 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations 43 almost everywhere in Ω. From 2), we see that e X j P F X j a.e. in Ω. Let distx + φ j, K)dx = a + δ j, where δ j and lim j δ j =. Since φ j is zero on the boundary of, φ j =. We have a + δ j = X + φ j x) X j x) dx e X + φ j x) X j x)) dx 7) e X + φ j x) X j x))dx e X j x)dx e X. From 2) we have, Combining 7) and 8) we have so that e X j x)dx a + δ j P F X j dx. 8) P F X j dx e X P F X j dx 1 a + δ j + e X ). 9) On the other hand, note that X j x) K M a.e., and E = F M, hence P M X j ) = a.e. and a + δ j = distx + φ j, K)dx = X + φ j X j dx 1) P E X + φ j X j ) dx P E φ j ) dx P E X) P F X j dx. Therefore, by 9) and 1), P E φ j ) dx a + δ j ) + P E X) + P F X j dx a + δ j ) + P E X) + 1 a + δ j + e X ) P E X) + a + δ j ). From Theorem 1 we have, meas {x R n, φ j x) > λ}) C E λ P E φ j dx C E λ P E X) + a + δ j ), for every λ >, where C E > is the constant in Theorem 1 given by 4 ). Since the distance function dist, K) satisfies dista, K) distb, K) A B 11)

8 44 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations for A, B M N n, we see that In other words, so that distx, K) > distx + φ j x), K) + λ implies φ j x) > λ. Choosing, for each fixed σ >, j) λ :={x, distx, K) > distx + φ jx), K) + λ} {x, φ j x) > λ}, meas j) λ ) C E P λ cos θ E X) + a + δ j ). λ = 1 + σ)c E P E X) + a), 12) we see that for sufficiently large j >, meas {x R n, φ j x) > λ}) < 1, so that a + δ j = distx + φ j x), K)dx distx + φ j x), K)dx [ [distx, K ) λ] \ j) λ 1 C E λ P E X) + a + δ j ) for sufficiently large j >. Passing to the limit in the above inequality, we obtain [ ] a [distx, K) λ] 1 C E P λ cos θ E X) + a). 13) Substituting 12) into 13), we have ) ) 1 σ a 1+cos θ 1 + σc distx, K) 1 + σ)c E P E 1 + σ cos θ E X). Thus ) Q distx, K) CE, θ, σ) distx, K) 1 + σ)c E P cos θ E X) for each fixed σ >, where Now if X M N n satisfies there is some σ > such that CE, θ, σ) = σc E 1+ ], 14) ) σ. 15) 1 + σ distx, K) > C E P E X), 16) distx, K) 1 + σ)c E P E X) >, hence from 14), Q distx, K) >, which implies X / QK). The proof is finished.

9 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations 45 Proof of Theorem 1. We first define a multiplier. Let E 1,..., E k be an orthonormal basis of E. We define a linear mapping for each R n with as L) : Mc N n Mc N n as [ k )) )) ] 1 T ) ) L)X) = I N N E j E j X, j=1 where Mc N n is the space of complex N n matrices. Clearly L is homogeneous of degree in and we if we write X as a vector in C Nn, there is an Nn Nn matrix M) such that L)X) = M)X, with the right hand side of the above equality the product of a matrix and a vector. We also have L)P E η )) [ k )) )) ] 1 T ) ) = I N N E j E j P E η )) j=1 [ k )) )) ] 1 T I N N E j E j =η. [ j=1 I N N k j=1 )) )) ] } T E j E j η Now if we define an operator T from L 2 M N n ) to L 2 M N n ) by its Fourier transform ) T f = L) f)), we see that [15] T is bounded with T P E u)) = u for any u W 1,2 R n, R N ) since the components of L) are all of C in R n \ {}. Let H) = I N N k j=1 )) )) T E j E j, we have H) 1 op 1 µ E, 17) where op is the operator norm of a matrix. Moreover, we have H 1 ) = H 1 ) H)H 1 ), ) Cn), H 1 ) Cn, N) µ 2 E, by noticing that E j = 1 and < µ E < 1. Thus we have, for each fixed m 1 m H 1 ) Cn, N, m) µ. k+1 E m

10 46 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations This implies that m M) Cn, N, m). 18) µ k+1 E m Therefore conditions for [16, pp , Prop. 2.b)] are satisfied, hence the weak-1,1) estimate for T holds [15, pp. 29, pp. 34]: meas{x R n, T f)x) λ}) C E λ R n f dx, where C E is in the form 4 ) due to the fact that we need estimate 18) up to the order m = [n/2] + 1 in [16, pp , Prop. 2.b)]. In particular if f = P E u) for u C R n, R N ), then T P E u)) = u, hence meas{x R n, ux) λ}) C E λ R n P E u) dx. Remark 5. The referee of this paper have made the following observations which make the definition of the multiplier faster and smoother than what I did above. efine H) : R N R N by H)η = η P E η ζ)ζ for each η R N, where ζ = / for R n \ {}. Similar to the proof above, we see that H) is invertible and satisfies the inequality in 17). Then the multiplier L) is defined by L)X) = H) 1 Xζ) ζ for each X M N n. This definition can then avoid some of the long formulas used in the original proof and no basis needs to be introduced. However, it would be nice to have more than one way to define the multiplier L). The original proof also gives explicit formulas in Example 7 below. Remark 6. By using the bound and the proofs of the relevant results in [15, 16] mentioned above, one can give explicit estimates of C E. A rough estimate is C E = CN, n, k) 1 + 1/µ n+1 E ). Since the calculation of the explicit bound is quite long by chasing each explicit constant in the pages cited above, we will not work this out here. Example 7. As we mentioned earlier, it is known [15, pp. 6] that for a complex valued smooth function defined on R 2 with compact support, one has meas{x R 2, fx) λ}) C f λ + i f x 1 x 2 If we write f = f 1 + if 2 and let u = f 1, f 2 ) as a R 2 -valued mapping, we have, after a slight change of the constant C >, meas{x R 2, ux) λ}) C λ L 1 PE u) L 1,.

11 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations 47 where E is the subspace of anti-conformal matrices with the basis E 1 = 2 2 ) 1, E 1 2 = 2 2 ) 1. 1 Example 8. We can give an explicit form of the multiplier corresponding to T for the one dimensional subspace E = span[a ] M N n with ranka ) > 1 and A = 1. The mapping L) associated with P E now is L)X) = I = I + )) )) ) T 1 ) ) A A X A )) A 1 A ) 2 )) T X ) ). Therefore, it might be possible to calculate an explicit bound C E in this case for Theorem 1. We conclude this paper by the following examples illustrating the effect of Theorem 2 near an exposed face of CK) without rank-one connections on QK). Example 9. We first consider Tartar s four-point configuration K = {A, B, C, } without rank-one connections [4] in the subspace of diagonal matrices Fig.1). The line segments are rank-one connections. By using a special quasiconvex function constructed in [19], one can show that QK) is given by the four legs and the square the figure on the left). Now suppose we only know that K M N n is a four-point set in a 2-dimensional plane without rank-one connections, we would like to bound QK) based on this very limited information. By using Theorem 2, we see that QK) is contained in the set given by the figure on the right. A A B B C C Figure 1: Left: Tartar s four point set and its quasiconvex hull. Right: The bound of QK) given by Theorem 2 Example 1. We illustrate the effect of Theorem 2 along an edge [A, B] with rankb A) > 1. We can chip off a slice from CK) without touching QK). Also note that if we translate the origin to the mid point of A and B, then F = span[e] V is the two dimensional subspace facing us.

12 48 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations B A O e V Example 11. This example shows that our estimate 6 ) in Theorem 2, Remark 3 is local as we mentioned earlier, hence may give a better estimate for the quasiconvex hull QK) for a finite set K than the polyconvex hull P K) which is widely used in bounding the quasiconvex hull. We show that P K) depends on the size of K, hence is not local. We consider a Tartar s construction in the subspace of diagonal matrices of M 2 2 as in Example 8. This time, we fix two neighbouring points A and B and allows the other two to move. Let A = diag2, ), B = diag, 2),, C h = diag h, h), h = diag1, h 1) with h >. Since A B = diag2, 2), ranka B) > 1, if we let E = span[a B] as in Remark 3, we have an estimate similar to 6 ) and it is easy to see that the bound near the line segment [A, B] is independent of large h >. Now we calculate P K). The idea is simple. We take any three point subset of K and calculate its polyconvex hull which is contained in P K). It is easy to see that the polyconvex hull is the lamination convex hull of the resulting union of polyconvex hulls for this three-point subsets. As described in [17, 18], we have, for the three point set K 1 = {A, B, C h } K M 2 2, there is a matrix X and a real number α such that deta X ) = detb X ) = detc h X ) = α. In our case, it is easy to see that X = h 2 /2h + 1))I and α = h 2 h + 2) 2 /4h + 1) 2 ), where I is the 2 2 identity matrix. Then P K 1 ) = {Y = diagx, y), dety X ) = α, x 2} {C h }, which is the union of the single point set {C h } and the part of a hyperbola connecting A and B which contains K 1 one the plane of diagonal 2 2 matrices. Note that the set {Y = diagx, y), dety X ) α} is a polyconvex set. If we repeat this calculation for the other three choices of three-point subsets of K, we see that there are four-pieces of hyperbola connecting neighbouring points which are in P K). We connect points in this set by laminates, we see that P K) is the region in the plane of 2 2 diagonal matrices bounded by these four-pieces of hyperbola. Now we show that P K) depends on h > and for large h, Theorem 2 gives a better estimate of QK) than P K). To see this, we only need to check the mid-point X h of the part of hyperbola between A and B and show that X h converges to the midpoint diag1, 1) of the line segment [A, B]. Let X h = diagx, x) be the mid-point of the hyperbola, we see that detx h C) = α, thus x + ) h 2 2 = h2 h + 2) 2 2h + 1) 4h + 1), hence x = h 2 h + 1, X h = h h + 1 I.

13 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations 49 B X O A C Clearly, X h diag1, 1), P K) is size dependent and our claim that 6) gives a better estimate then follows. Acknowledgements. I wish to thank the referee for several very helpful suggestions. References [1] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal ) [2] J. M. Ball, R.. James: Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal ) [3] J. M. Ball, R.. James: Proposed experimental tests of a theory of fine microstructure and the two-well problem, Philos. Trans. R. Soc. Lond., Ser. A ) [4] K. Bhattacharya, N. B. Firoozye, R.. James, R. V. Kohn: Restrictions on microstructure, Proc. R. Soc. Edinb., Sect. A ) [5] A. Brøndsted: An Introduction to Convex Polytopes, Springer, New York 1983). [6] B. acorogna: irect Methods in the Calculus of Variations, Springer, Berlin 1989). [7] G. olzmann, B. Kirchheim, J. Kristensen: Conditions for equility of hulls in the calculus of variations, Arch. Ration. Mech. Anal ) [8] I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam 1976). [9] Z. Iqbal: Variational Methods in Solid Mechanics, Ph Thesis, University of Oxford 1999). [1] C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations, Springer, Berlin 1966). [11] S. Müller: On quasiconvex functions which are homogeneous of degree one, Indiana Math. J ) [12] S. Müller: Variational models for microstructure and phase transitions, in: Calculus of Variations and Geometric Evolution Problems Cetraro, 1996), Lecture Notes in Math. 1713, Springer, Berlin 1999)

14 5 K. Zhang / Estimates of Quasiconvex Polytopes in the Calculus of Variations [13] S. Müller, V. Šverák: Attainment results for the two-well problem by convex integration, in: Geometric Analysis and the Calculus of Variations, Internat. Press, Cambridge 1996) [14] R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton 197). [15] E. M. Stein: Singular Integrals and ifferentiability Properties of Functions, Princeton University Press, Princeton 197). [16] E. M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton 1993). [17] V. Šverák: On the problem of two wells, in: Microstructure and Phase Transition,. Kinderlehrer et al. eds.), IMA Vol. Math. Appl. 54, Springer, New York 1993) [18] V. Šverák: On regularity of the Monge-Ampère equation without convexity assumptions, preprint 199). [19] V. Šverák: New examples of quasiconvex functions, Arch. Ration. Mech. Anal ) [2] L. Tartar: Some remarks on separately convex functions, in: Microstructure and Phase Transition,. Kinderlehrer et al. eds.), IMA Vol. Math. Appl. 54, Springer, New York 1993) [21] K.-W. Zhang: Quasiconvex functions, SOn) and two elastic wells, Ann. Inst. H. Poincaré, Anal. Non Linéaire ) [22] K.-W. Zhang: On various semiconvex hulls in the calculus of variations, Calc. Var. Partial iffer. Equ ) [23] K.-W. Zhang: On some quasiconvex functions with linear growth, J. Convex Analysis )