2 BAISHENG YAN When L =,it is easily seen that the set K = coincides with the set of conformal matrices, that is, K = = fr j 0 R 2 SO(n)g: Weakly L-qu

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1 RELAXATION AND ATTAINMENT RESULTS FOR AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL BAISHENG YAN Abstract. Consider functional I(u) = R jjdujn ; L det Duj dx whose energy-well consists of matrices satisfying jj n = L det : We show that the relaxations of this functional in various Sobolev spaces are signicantly dierent. We also make several remarks concerning various p-growth semiconvex hulls of the energy-well set and prove an attainment result for a special Hamilton-Jacobi equation, jduj n = L det Du, in the so-called grand Sobolev space W q) ( R n )withq = nl L+ :. Introduction and main results Given L, consider function f L () = jj n ; L det on the space M nn of n n matrices and integral functional (.) I(u) = jf L (Du(x))j dx = jjdu(x)j n ; L det Du(x)j dx where u is a mapping from domain R n to R n and Du(x) is the Jacobi matrix of u. In general, jj denotes the operator norm of m n matrix 2 M mn dened by jj =max h2r n jhj= jhj: The absolute energy minimizers of I(u) can be characterized as mappings satisfying the Hamilton-Jacobi equation: (.2) Du(x) 2 L = f 2 M nn jjj n = L det g a:e: x 2 : In the terminology used for phase transition problems (see, e.g., [3, 9, 0, ]), the set L is the energy-well of energy functional I(u): Note that L is also the boundary of the so-called L-quasiconformal set K L dened by K L = f 2 M nn jjj n L det g:

2 2 BAISHENG YAN When L =,it is easily seen that the set K = coincides with the set of conformal matrices, that is, K = = fr j 0 R 2 SO(n)g: Weakly L-quasiregular mappings are the mappings u 2 W p loc ( Rn ) satisfying Du(x) 2 K L almost everywhere in (see Iwaniec [6]). Many regularity and stability properties of weakly quasiregular mappings have been studied in connection with the certain semi-convex hulls and the attainment result of the quasiconformal sets K L in Yan [3, 4, 5]. In this paper we study similar problems related to the set L : The natural space for I(u) is the Sobolev space W n ( R n ): In this note we are interested in minimization of functional I(u) in dierent Sobolev spaces W p ( R n ). For this purpose, we dene the p-growth relaxation functions of I(u) as follows: (.3) g p () = inf u2w p ( R n ) =x jj jf L (Du(x))j dx: Clearly the function g p () is non-decreasing with respect to p : For any given function f : M mn! R, we dene the quasiconvexication of f to be (.4) f # () = inf 2C 0 ( Rm ) f( + D(x)) dx jj 8 2 M mn where C 0 ( R m ) stands for all smooth functions with compact support in : A density argument easily shows that g = jf L j # and hence the function g is also quasiconvex in the sense of Morrey. Recall that, in general, a function f : M mn! R is said to be quasiconvex (in the sense of Morrey) if it satises (.5) jj f( + D(x)) dx f() 8 2 M mn 2 C 0 ( R m ): It is well-known that any nite quasiconvex function f is always rankone convex that is, f( + t) is convex in t 2 R for any 2 M mn with rank = : A function f : M mn! R is called polyconvex if it can be written as a convex function of all subdeterminants therefore all convex functions are polyconvex. It can be shown that a polyconvex function is always quasiconvex. We refer to Acerbi & Fusco [], Ball [2], Ball & Murat [4], Dacorogna [5], Morrey [8], and Muller [9] for the proofs and more properties of these semiconvex functions.

3 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL 3 In what follows, we denote by f ; (0) the zero set of any scalar function f, that is, f ; (0) = f j f() =0g: We now state our main result. Theorem.. Let n 3 and L. Then (a) g p = jf L j # = maxff L 0g for all p n (b) for some >0, g p ; (0) = K L for all p n ; (c) g p 0 and the minimum is attained for all p< nl Part (c) of the theorem is equivalent to the following L+ : Theorem.2. Let n 3, L : Then, for any p< nl L+ and 2 M nn there exists a map u 2 W p ( R n ) with = x satisfying (.6) jdu(x)j n = L det Du(x) a:e: x 2 : Remark. This result sharpens an earlier result in Yan [4]. Note that, if = 0 the trivial map u 0 is a required solution of (.6) but, in this case, we can show existence of nontrivial solutions. In fact, we shall prove a sharper attainment result for a special class of nontrivial solutions of (.6) in a space strictly smaller than any W p ( R n ) for p< nl L+ : For a given q > we dene the grand Sobolev space W q) ( R m ) (see e.g. Iwaniec & Sbordone [7]) to be the space of all functions u 2\ p<q W p ( R m ) that satisfy [Du] q sup (q ; p) =p (.7) jdu(x)j dx p < : p<q jj Given any number 2 R, we dene the set (.8) S = fr(i +!!) j R 2 R 6= 0! 2 R n j!j =g where I is n n identity matrix and a b stands for the rank-one matrix (a i b j ). In the following, we shall use ( 2 n ) to denote the n n diagonal matrix with diagonal elements ::: n that is, we denote ( 2 n )= B A : 0 n Let J = I ; 2e e = (; ): Then J 2 = I and det J = ;: Dene JS = fjj 2 S g: Note that any matrix 2 S can be

4 4 BAISHENG YAN written as = Q(I +!!) for some >0, Q 2 SO(n) and! 2 R n with j!j = : Therefore, it follows easily that det = n ( + ) and jj = if j+j or jj = j+j if j+j : From this, one easily proves the following Proposition.3. For = L n; ; or L ; ;, S L for = ;L n; ; or ;L ; ;, JS L : In this paper, we shall also prove the following much stronger attainment result. Theorem.4. Let n 3 L be given. Then, for any 2 M nn, nl there exists a map u in the grand Sobolev space W L+ ) ( R n ) such that = x Du(x) 2 S [ JS 2 L a:e: x 2 for = L n; ; or L ; ; and 2 = ;L ; ; where the boundary condition = x is satised in the W p -sense for all p< nl L+ : Given any subset K of M mn and a number p, let C + p (K) be the set of continuous functions f : M mn! R satisfying 0 f() C(jj p +) fj K =0: Let QC + p (K), RC + p (K) andpc + p (K) be the class of quasiconvex, rankone convex, and polyconvex functions in C + p (K), respectively. Denition.. The p-growth quasiconvex, rank-one convex and polyconvex hulls of set K are dened, respectively, as follows: (.9) (.0) (.) Q p (K) =\ff ; (0) j f 2 QC + p (K)g R p (K) =\ff ; (0) j f 2 RC + p (K)g P p (K) =\ff ; (0) j f 2 PC + p (K)g: Remark. These p-growth semiconvex hulls have been referred to as the W p - or, simply, p-semiconvex hulls in Yan [2, 5]andYan & hou [6, 7], following the denition of usual semiconvex hulls without growth restriction given, for example, in Muller & Sverak [0] and Sverak []. We adopt the present denition here to stress the growth condition and to distinguish with the dierent and not equivalent denition of p-semiconvex hulls given in hang [8, 9]. The p-growth semiconvex hulls of the set K L have been studied in Yan [5]. Concerning the set L, we have the following

5 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL 5 Theorem.5. A p-growth semiconvex hull of L is the same as that of K L. For example, R p ( L ) = M nn for p < nl L+ and R p( L ) = K L for p nl L+ moreover, Q p( L )=K L for p n ; with some >0: 2. Relaxations and semiconvex hulls In this section, we prove Theorem., parts (a) and (b), and Theorem.5. Part (c) of Theorem. and Theorem.2 follow from Theorem.4, which will be proved later in the paper. First of all,we observe the following useful result. Lemma 2.. lim jtj! f L ( + t) = + for any given 2 M nn with rank =: Proof. It suces to note that det( + t) =det + ct for a constant c if rank =, and hence as jtj!: f L ( + t) (jtjjj;jj) n ;jcjjtj;jdet j! Proof of Theorem.(a). Using a well-known relaxation principle for integral functionals (see, e.g., [, 5]) one can easily show that g n = jf L j # and hence, by denition, (2.) 0 g p g q jf L j # jf L j 8 p q: Since f L = jj n ;L det is W n -quasiconvex [4], we easily have g n f L and hence g n maxff L 0g: (This can be also derived directly from property of the determinant.) Therefore, by (2.), to prove Theorem.(a), it is sucient to show jf L j # = maxff L 0g: But this follows from Lemma 2. and the following general result: Proposition 2.2. Let f be a quasiconvex function. Suppose that for each 2 M nn there exists a rank-one matrix such that Then jfj # =maxff 0g: lim f( + t) =+: jtj!

6 6 BAISHENG YAN Proof. Obviously, since f is quasiconvex, jfj # maxff 0g: We prove the equality. Suppose, for the contrary, jfj # ( 0 ) > maxff( 0 ) 0g for some 0 : Since jfj # ( 0 ) jf( 0 )j we must have f( 0 ) < 0: Now consider h(t) =f( 0 + t 0 ) where 0 is a rank-one matrix such thath(t)! + as jtj!: Then, since h(0) = f( 0 ) < 0, we would have t < 0 < t 2 such that h(t ) = h(t 2 ) = 0: This implies jfj # ( 0 + t i 0 ) = 0 for i = 2: From this, since jfj # is rank-one convex, we would arrive at the conclusion that jfj # ( 0 ) 0, a contradiction. Therefore, jfj # maxff 0g: Proof of Theorem.(b). Let f = (maxff L 0g) =n : Then jf L j jf L j # = f n. Using Holder's inequality in denition (.3), it follows that, for p<n (2.2) [(f p ) # ] n=p g p jf L j # = f n : Since f is a homogeneous function of degree which vanishes exactly on the set K L and is also L n -mean coercive in the sense dened in Yan & hou [7], it follows from [7, Theorem 2.] that (f p ) # () = 0 if and only if 2 K L for p n ; with some >0 and therefore, by (2.2), g p () = 0 if and only if 2 K L : This proves the result. Proposition 2.3. Let g 0 be a rank-one convex function. If g() = 0 for all 2 L then g() =0for all 2 K L : Proof. Let 2 K L n L : Then f L () < 0: In a similar way as above, we have f L ( + t ) = f L ( + t 2 ) = 0 for some t < 0 < t 2 with a given rank-one matrix : Thus g( + t i ) = 0 for i = 2, which, by the rank-one convexity of g, implies g() = 0 and proves the result. Proof of Theorem.5. From denition of semiconvex hulls, to prove the theorem, it is sucient to prove the equalities: QC + p ( L )=QC + p (K L ) RC + p ( L )=RC + p (K L ) PC + p ( L )=PC + p (K L ): Since the functions in QC + p ( L ), RC + p ( L ), and PC + p ( L ) are all rankone convex, these equalities follow easily from Proposition 2.3. The proof is completed. 3. The lamination hull and attainment results Given any set K M mn and number p 2 [ ], dene p (K) to be the set of matrices 2 M mn such that there exists a map

7 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL 7 u = u 2 W p ( R m ) satisfying (3.) Du(x) 2 K a:e: x 2 = x: Note that Lemma 3. in Yan [4] shows that the set p (K) is independent of the domain : From this denition, Theorem.2 is equivalent to the following Theorem 3.. p ( L )=M nn for all p< nl L+ : From this theorem, a similar argument as used for [4, Theorem 4.4] also derives the following shaper result we refer to [4] for details. Theorem 3.2. Let n 3, L and p < nl : Then, for any L+ piece-wise ane map ' 2 W p ( R n ) and > 0 there exists a map u 2 ' + W p 0 ( R n ) satisfying Du (x) 2 L L a.e. in and ku ; 'k L p () <: Let S be the set dened by (.8) in the introduction and dene a unbounded two-well set (3.2) W L = S [ JS 2 where 2 = ;L ; ; and is either L n; ; or L ; ; : Notice that Proposition.3 implies W L L therefore, Theorem 3. follows from an attainment result for set W L, which is a direct corollary of Theorem.4. Theorem 3.3. p (W L )=M nn for all p< nl L+ : To prove Theorem 3.3, or the more general Theorem.4, we need some techniques in convex integration theory we refer to [0,, 4] for more references on this theory. Denition 3.. Let L j (K) be dened for j = 0 2 inductively as follows: L 0 (K) =K and, for j =0 L j+ (K) =ft +(; t) t 2 [0 ] 2Lj (K) rank( ; ) g: The lamination hull of set K, L(K), is then dened to be (3.3) L(K) =[ j=0l j (K): The following useful attainment result has been given in Yan [4] see also Yan [3].

8 8 BAISHENG YAN Theorem 3.4 ([4, Theorem 3.2]). Let K M mn be a closed set and let A p (K) be a set satisfying (3.4) c 0 = sup jdu (x)j p dx < 2A jj where u 2 W p ( R n ) is some map satisfying (3:): Suppose the lamination hull B = L(A) is open and bounded. Then B p (K): Remarks. ) It follows from [4, Lemma 3.] and the remark following the proof of [4, Theorem 3.2] that, for any bounded domain R n and any 2 B = L(A), there exists a map u = u 2 W p ( R m ) such that (3.5) and (3.6) Du(x) 2 K a:e: x 2 = x jj jdu(x)j p dx maxfc 0 sup jj p g: 2B 2) A closer look at the proof of [4, Theorem 3.2] also shows that the map u = u satisfying (3.5), (3.6) above depends only on the family fu j 2 Ag in (3.4) and any xed number q p in particular, the solution u = u for any 2 B can be made independent of power p as long as p<qfor some q < : This can be seen from the choice of sequence f k g in the proof of [4, Theorem 3.2]. We notice that the construction of u = u depends on power p only through this sequence f k g that is required to satisfy k! 0 + and P k =p k < : However, if p<q, we can x such a sequence k satisfying k! 0 + and P k =q k < : Then the solution u = u constructed there is seen only depending on this sequence f k g and the family fu j 2 Ag given in (3.4). Note that the estimate (3.6) is independent of the sequence f k g. The main theorem of this section is the following Theorem 3.5. Let K M mn be a closed set and q > a number given. Suppose A M mn is a set such that for each 2 A there exists a map v = v 2 W q) (B R m ) satisfying Dv(x) 2 K a.e. x 2 B and = x where B is the unit ball in R n and suppose that (3.7) C q =sup[dv ] q B < : 2A If the lamination hull B = L(A) is open and bounded, then, for any bounded domain R n and any 2 B = L(A), there exists a map

9 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL 9 u = u 2 W q) ( R m ) satisfying (3:) and, therefore, B = L(A) p (K) for all p<q: Proof. Condition (3.7) implies that, for any p<q, (3.8) c p =sup 2A jbj B jdv (x)j p dx Cp q q ; p < : From this, using Theorem 3.4 quoted above and its remarks, we obtain, for each given 2 B = L(A), a map u = u depending only on the power q and the family fv j 2 Ag in (3.7) that belongs to W p ( R n ) for all p<qand satises (3.) and jj jduj p dx maxfc p sup jj p g: 2B Multiplying this estimate by q ; p, taking the =p power, and maximizing over p<qshow that [Du] q maxfc q q sup jjg < 2B where q = sup pq (q ; p) =p < is a constant. This shows u = u 2 W q) ( R n ) satisfying (3.) and completes the proof. 4. Proof of Theorem.4 In this nal section, we apply Theorem 3.5 to prove the main attainment theorem, Theorem.4. To do so, as in Yan [3, 4], we further introduce certain subsets in M nn : Let W L be the set dened by (3.2). Givenanumber >0 dene (4.) (4.2) (4.3) R(n) =f 2 M nn jjj n = j det jg A = f 2 R(n) jjj <g A ~ = A nf0g B = f 2 M nn jjj <g: Proposition 4.. L n ( ~A )=B : Therefore B = L( ~A ) is a bounded open set. Proof. Let 2 B : If 6= 0 then the proof of Proposition 4. in [4] has essentially shown that 2 L n; ( ~A ): Therefore one has only to show 0 2L n ( ~A ): To prove this, let 0 <t<be xed. Note that, since the diagonal matrices ( 2 n ) and (; 2 n ), where i =

10 0 BAISHENG YAN t, are in ~A and their dierence is rank-one, by Denition 3., we have (0 2 n )= ( 2 n )+(; 2 n ) 2 belongs to L ( ~A ) for all i = t: Now that matrices (0 2 n ) and (0 ; 2 n ) are in L ( ~A )anddier by rank-one, hence (0 0 3 n )= (0 2 3 n )+(0 ; 2 3 n ) 2 is in L 2 ( ~A ) for all i = t: Repeating this argument nitely many steps, we eventually arrive at 0 = (0 0 0) 2L n ( ~A ): Proposition 4.2. Let > 0, n 3, L and q = nl. Then, L+ for each 2 ~A, there exists a map v = v 2 W q) (B R n ) satisfying Dv(x) 2 W L a.e. x 2 B and = x such that (4.4) c 0 sup 2 ~ A [Dv ] q B C(n L) <: Proof. Let 2 ~A be given. We look for map v = v in the form of radial maps: v(x) = jxj x where is a number to be chosen later. Note that all such maps satisfy = x in the W p -sense if v 2 W p (B R n ) for some p : Let r = jxj,! = r ; x then a simple calculation shows that (4.5) (4.6) (4.7) Dv(x) =r (I +!!) det Dv(x) = ( + )r n det ( r jj j+j jdv(x)j = j+jr jj j+j > : If det > 0, we choose = to be one of the two numbers dened above, and then one easily sees that Dv(x) 2 S for all x 6= 0 in B and, a computation also shows that this function v = v belongs to W q) (B R n )andsatises (4.8) [Dv ] q B C (n L)jj: If det < 0 we let = 2 = ;L ; ; and in this case one easily sees that Dv (x) 2 JS 2 for all x 6= 0: Furthermore, we compute to get jbj B jdv (x)j p dx = jj p r 2p+n; dr = jjp 0 2 p + n = L jj p L +q ; p

11 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL which shows that v = v belongs to W q) (B R n ) and satises (4.9) [Dv ] q B C 2 (n L)jj: Combining (4.8), (4.9), one proves (4.4). The proofiscompleted. Proof of Theorem.4. Let 2 M nn be given. Let > 0 be a number such that jj < : Then 2 B : From Proposition 4., L( ~A ) = B is open and bounded. Also, from Proposition 4.2, the set A = ~A satises the condition (3.7) in Theorem 3.5 with K = W L and q = nl : Therefore, Theorem 3.5 implies that, there exists a map L+ u = u 2 W q) ( R n ) satisfying Du(x) 2 W L for a.e. x 2 and = x: Note also that [Du] q C 0 <: We complete the proof. References [] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (984), 25{45. [2] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (977), 337{403. [3] J. M. Ball and R. D. James, Proposed experimental tests of a theory of ne microstructures and the two well problem, Phil. Trans. Roy. Soc. London, 338A (992), 389{450. [4] J. M. Ball and F. Murat, W p -Quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58 (984), 225{253. [5] B. Dacorogna, \Direct Methods in the Calculus of Variations." Springer- Verlag Berlin, 989. [6] T. Iwaniec, An approach to Cauchy-Riemann operators in R n, preprint. [7] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal., 9 (992), 29{43. [8] C. B. Morrey, \Multiple Integrals in the Calculus of Variations." Springer- Verlag Berlin, 966. [9] S. Muller, Variational models for microstructure and phase transitions, in \Calculus of Variations and Geometric Evolution Problems," (Cetraro, 996), pp. 85{20, Lecture Notes in Math. 73, Springer, 999. [0] S. Muller and V. Sverak, Attainment results for the two-well problem by convex integration, Geometric analysis and the calculus of variations, 239{ 25, Internat. Press Cambridge, MA, 996. [] V. Sverak, On the problem of two wells, in \Microstructure and Phase Transition," pp. 83{90, (D. Kinderlehrer et al eds.), Springer-Verlag New York, 993. [2] B. Yan, On rank-one convex and polyconvex conformal energy functions with slow growth, Proc. Roy. Soc. Edinburgh, 27A (997), 65{663. [3] B. Yan, Linear boundary values of weakly quasiregular mappings, C. R. Acad. Sci. Paris Ser. I Math., 33 (2000), 379{384.

12 2 BAISHENG YAN [4] B. Yan, A linear boundary value problem for weakly quasiregular mappings, Cal. Var. Partial Dierential Equations, to appear. [5] B. Yan, Semiconvex hulls of quasiconformal sets, J. Convex Analysis, to appear. [6] B. Yan and. hou, Stability of weakly almost conformal mappings, Proc. Amer. Math. Soc., 26 (998), 48{489. [7] B. Yan and. hou, L p -mean coercivity, regularity and relaxation in the calculus of variations, Nonlinear Analysis, to appear. [8] K. hang, On various semiconvex hulls in the calculus of variations, Cal. Var. Partial Dierential Equations, 6 (998), 43{60. [9] K. hang, Rank-one connections at innity and quasiconvex hulls, J. Convex Analysis, 7 (2000), 9{45. Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA address: yan@math.msu.edu

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