ESAIM: Control, Optimisation Calculus of Variations UL: http://www.emath.fr/cocv/ June 1996, vol. 1, pp. 169-189 LOWE SEMICONTINUITY OF MULTIPLE INTEGALS AND CONVEGENT INTEGANDS HIPING LI Abstract. Lower semicontinuity ofmultiple integrals f(x u P) d f(x u P) d are studied. It is proved that the two can derive from each other under certain general hypotheses such as uniform lower compactness property locally uniform convergence of f to f. The result is applied to obtain some general lower semicontinuity theorems on multiple integrals with quasiconvex integr f, while f are not required to be quasiconvex. Keywords: Lower semicontinuity, convergent integrs, lower precompactness, weakly precompact, quasiconvex. 1. Introduction preliminaries Let be a measurable space with nite positive nonatomic complete measure, letf f : m N! [f+1g be extended real-valued functions satisfying certain hypotheses, let u :! m, P :! N be measurable functions in two linear topological spaces U V respectively. We consider integral functionals of the form I(u P )= I (u P )= f(x u(x) P(x)) d (1.1) f (x u(x) P(x)) d: (1.2) The main purpose of this paper is to study, under certain general hypotheses on U V f f,lower semicontinuity theorems of the form I(u P ) lim!1 I(u P ) (1.3) I(u P ) lim!1 I (u P ) (1.4) the relationship between them. The study of the relationship between (1.3) (1.4) was motivated mainly by the needs of convergence analysis for numerical solutions to some problems in calculus of variations. For example, in many cases, we have lower semicontinuity theorems of the form Department of Mathematics, Peking University, Beijing 100871, People's epublic of China. eceived by the journal August 7, 1995. Accepted for publication June 3, 1996. The research was supported by a grant of the National Natural Science Foundation of China.
170 HIPING LI (1.3), however, for one reason or another, the sequence (u P ) obtained by numerical methods is often a minimizing sequence with respect to I () rather than with respect to I(), a lower semicontinuity theorem of the form (1.4) is needed for convergence analysis (see Li [1]). In some applications it is equally important to know under what conditions (1.3) can be derived from (1.4) (see x3). Some useful results are established in this paper on the relationship between (1.3) (1.4). These results are then applied to establish some general lower semicontinuity theorems on multiple integrals with quasiconvex integrs. It is worth noticing that in these theorems f are not required to be quasiconvex which allows us to choose f more freely in numerical computations. For better understing of the background relevant known results, for the later use of this paper, we rst introduce some denitions hypotheses. Definition 1.1. A function f : m N! [f+1g is called L B; measurable, if it is measurable respect to the -algebra generated by products of measurable subsets of Borel subsets of m N. Definition 1.2. A function f : m N! is a Caratheodory function if (1): f( u P) is measurable for every u 2 m P 2 N, (2): f(x ) iscontinuous for almost every x 2. Definition 1.3. A function f : m N! [f+1g has the lower compactness property onu V if any sequence of f ; (x u (x) P (x)) is weakly precompact in L 1 () whenever (1): u converge in U, P converge in V (2): I(u P ) C<1 for all =1 2. f has the strong lower compactness property onu V if any sequence of f ; (x u (x) P (x)) is weakly precompact in L 1 () whenever (1) holds. Here f ; = min ff 0g. The assumption of lower compactness property provide us more freedom in applications than the nonnegative assumptions. For example, let U be L q ( m ), 1 q +1, with strong topology V be L p ( N ), 1 p +1, with weak topology, letg : m! N be such that jg(x u)j p0 cjuj q + b(x) for some constant c>0 function b() 2 L 1 (), where p 0 = f : m N! satisfy f(x u P ) hg(x u) Pi;c 1 juj q + b 1 (x) p, let p;1 for some constant c 1 > 0 function b 1 () 2 L 1 (), where hq P i = P N i=1 Q ip i, then it is easy to verify that f has the strong compactness property onu V. Definition 1.4. A sequence of functions f : m N! [f+1g is said to have the uniform lower compactness property, iff ; (x u (x) P (x)) are uniformly weakly precompact in L 1 (), in other words (see [2, 3]),
SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 171 f ; (x u (x) P (x)) are equi-uniformly integral continuous on, i:e: for any >0 there exists >0such that j 0 f ; (x u (x) P (x)) dj < for all, any measurable subset 0 satisfying ( 0 ) <, whenever u () converge in U, P () converge in V I (u P ) C<1 for all. Definition 1.5. A sequence of functions f : m N! [f+1g is said to converge to f : m N! [f+1g locally uniformly in m N, if there exists a sequence of measurable subsets l with ( n l )! 0asl!1such that for each l any compact subset G m N f (x u P ) ;! f(x u P ) unifomly on l G, as!1: Definition 1.6. A sequence of functions f : m N! [f+1g is said to converge to f : m N! [f+1g locally uniformly in the sense of integration on U V, if there exists a sequence of measurable subsets l with ( n l )! 0asl!1such that l ne(u P K) f (x u(x) P(x)) d! l ne(u P K) uniformly in U V for each l any xed K>0, where f(x u(x) P(x)) d E(u P K)=fx 2 :ju(x)j >K or jp (x)j >Kg: emark 1.7. In numerical computations of singular minimizers, truncation methods turned out to be successful (see [1]). In such applications, f can be as simple as f (x u P ) = minf; maxf f(x u P )gg which converge to f locally uniformly in m N if f is a Caratheodory function, which have the uniform lower compactness property on U V if f has the lower compactness property onu V. emark 1.8. When is a locally compact metric space, l in denition 1.5 denition 1.6 can be taken to be compact subsets of. In such a case, to say that f converge to f locally uniformly in m N is equivalent to saying that f converge to f uniformly on every compact subset in m N. Especially, iff converge to f uniformly on every bounded subset in m N then f converge to f locally uniformly in m N,iff fare Caratheodory functions f converge to f in measureonevery bounded subset in m N then f converge to f locally uniformly in the sense of integration on U V.
172 HIPING LI emark 1.9. To cover more applications, we may consider f fas mappings from U V to fmeasurale functions g :! [f+1gg. This allows us to consider, for example, interpolations of f(x u(x) P(x)) in nite element spaces. In fact, denitions results in this paper can be extended parallely to such mappings without diculty, as we will see that in the proofs of the theorems f are related to f only as such mappings. Definition 1.10. (See [4, 5, 6]) A function f : mn! is quasiconvex if 0 f(p + D(x)) dx f(p ) (0 ) for every P 2 mn, 2 C 1 0 (0 m ), every open bounded subset 0 n, where is the Lebesgue measure on n. Let n be open bounded. A Caratheodory function f : m mn! is quasiconvex in P if there exists a subset E with (E) = 0 such that f(x u ) is quasiconvex for all x 2 n E u 2 m. Example 1.11. Let 2, n = m = 2. Then the function f(du) =jduj 2 ; (tr Du) 2 is quasiconvex (in fact it is polyconvex, see [6, 9,10]) is unbounded below. In addition, f has the strong lower compactness propertyonl p ( 22 ) with weak topology for all p>2. If V is taken to be L 2 ( 22 ) with weak topology then f hasaslightly weaker lower compactness property which can be described by Chacon's biting lemma [5, 7, 8, 3] (see lemma 3.6) Let D( k )=fv :! k j v is measurable g: We assume that U D( m )V D( N ) are decomposable, i.e. if v() belongs to one of them, then T ()v() belongs to the same space whenever T is a measurable subset of, where T () is the characteristic function of T : ( 1 if x 2 T T (x) = 0 if x=2 T: We assume that U V satisfy the following hypotheses on their topologies: (H1): If v () =1 2 belong to one of the spaces converge there to zero if T! 0, then T ()v () alsoconverge to zero. (H2): The topology in U is not weaker than the topology of convergence in measure the topology of V is not weaker than the topology induced in V by the weak topology of L 1 ( N ). It is easy to see that if U is taken to be L q ( m ), 1 q +1, with strong topology V is taken to be L p ( m ) with weak topology for 1 p <+1 or weak topology for p = +1, then (H1) (H2) are satised. This covers most applications in Sobolev spaces. For applications concerning Orlicz spaces see for example [2].
SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 173 emark 1.12. Here throughout this paper, assumptions statements are referred to sets with measure-negligible projections on, i.e. they hold on a subset 0 with 0 =. In eshetnyak's result ( see theorem 1.2 in [11]), is taken to be a local compact metric space, f f : N! are nonnegative functions such that for any >0 there is a compact set A with ( n A) < f(x u) f (x u) being continuous on A N, f(x ) f (x ) are convex for almost all x 2, f! f locally uniformly in N as!1, V is taken to be L 1 ( N )withweak topology. In the case when f(x u )isconvex, (1.3) was proved (see Ioe [2]) under the hypotheses that f satises lower compactness property U, V satisfy (H1) (H2), (1.4) was proved (see Li [12]), by using (1.3), under the hypotheses that f has the lower compactness property, f have the uniform compactness property f! f locally uniformly in m N. The results of the type (1.3) concerning quasiconvexity off(x u ) P = Du can be found in [5, 6]. The following theorem, which will be used in x3, was established by Acerbi Fusco [5, 6]. Theorem 1.13. Let n bebounded open. Let F (u) = f(x u Du) dx u 2 W 1 p ( m ) where 1 p<1, where f : m mn! satises (1): f( ) is a Caratheodory function (2): f(x u ) is quasiconvex (3): 0 f(x u P ) a(x)+c (juj p + jp j p ) for every x 2 u 2 m P 2 mn, where C>0 a() 2 L 1 (). Then, the functional u! F (u) is sequentially weakly lower semicontinuous on W 1 p ( m ), i.e. (1.3) holds for P = Du P = Du with u *u in W 1 p ( m ),where in what follows "*" means "converges weakly to". Ball hang [6] generalized the above result to cover the case when jf(x u P )ja(x)+c (juj p + jp j p ) proved that (1.3) holds on each n E k, where fe k g is a nonincreasing sequence of measurable subsets of with lim k!1 E k =0. In x2, It is shown, under certain general hypotheses, such as uniform lower compactness property locally uniform convergence of f to f (see theorem 2.1 for details), that (1.3) (1.4) can somehow be derived from each other, the results are further developed into a theorem concerning ;-limit (see theorem 2.8, for ;-limit, ;-convergence their applications in calculus of variations see for example [13, 14]). These results generalized the result of Li [12] which was proved for the case when f(x u ) is convex. In x3, the results established in x2 are applied to prove some lower semicontinuity theorems of the form (1.3) (1.4) for quasiconvex integrs,
174 HIPING LI which generalize the results of Acerbi Fusco [5], Ball hang [6] Li [12]. 2. Lower semicontinuity convergent integrs The following theorem establishes the relationship between the lower semicontinuity theorems of the form (1.3) those of the form (1.4). Theorem 2.1. Let be a measurable space with nite positive non- atomic complete measure. Let U V satisfy (H1) (H2). Let fu g u2 U fp g P 2 V be suchthat u ;! u in U (2.1) P ;! P in V: (2.2) Let f ff g : m N! [f+1g satisfy (i): f ff g are L B- measurable (ii): f ; (x u(x) P(x)) f ; (x u (x) P (x)), f ; (x u (x) P (x)) are weakly precompact in L 1 () (iii): f! f locally uniformly in m N. Then, we have (a): provided that f(x u P ) d lim!1 f (x u P ) d (2.3) 0 f(x u P ) d lim!1 0 f(x u P ) d (2.4) for all measurable subset 0 (b): f(x u P ) d lim!1 f(x u P ) d (2.5) provided thatf ff g satisfy a further hypothesis (iv): f + (x u (x) P (x)) a(x) +f + (x u (x) P (x)), where a() 2 L 1 () is nonnegative f(x u P ) d lim!1 To prove the theorem, we need the following lemmas. f (x u P ) d: (2.6) Lemma 2.2. Let fu g u 2 U fp g P 2 V satisfy (2.1) (2.2) respectively. Let E 1 (K) E 2 (K) = fx 2 :ju (x)j >Kg = fx 2 :jp (x)j >Kg
SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 175 Then E (K) =E 1 (K) [ E 2 (K): (2.7) E (K) ;! 0 uniformly for as K!1: Proof. For any >0, since u 2 U, there exists K 1 () > 1such that fx 2 :ju(x)j >Kg <=2 8K >K 1 (): Thus, by (2.1) (H2), there exists () > 1such that E 1 (K) <=2 8 >() K>K 1()+1: (2.8) Since u 2 U for each, wehave lim K!1 E1 (K) =0 for each : Thus, for 2f1 2 ()g, there exists K 2 () > 1such that E 1 (K) <=2 8 2f1 2 ()g K>K 2 (): (2.9) Let K() = maxfk 1 ()+1 K 2 ()g, then (2.8) (2.9) give E 1 (K) <=2 8 1 K>K(): (2.10) On the other h, it follows from (2.2) (H2) that jp (x)j d C for some constant C>0. Thus, for any >0 there exists K() > 1 such that E 2 (K) <=2 8 1 K>K(): (2.11) Hence the lemma follows from (2.10) (2.11). 2 Lemma 2.3. Let f ff g satisfy the hypotheses in theorem 2.1. Let fa : m N! be dened by f A (x u P ) = minfa f(x u P )g: (2.12) Let fu g u 2 U fp g P 2 V satisfy (2.1) (2.2) respectively. Let f l g be asequence ofmeasurable subsets of, the existence of which is guaranteed bythehypothesis (iii) for f,suchthat ( n l ) ;! 0 as l!1 (2.13) f ;! f uniformly on l G (2.14) for each l any compact set G m N. Suppose for some constant C>0. f (x u P ) d C
176 HIPING LI Then, for any >0 A > 1, there existl() 1 ( A l) 1 such that Proof. l fa (x u P ) d f (x u P ) d + 8l l() ( A l): l fa (x u P ) d = f (x u P ) d + n l (;f (x u P )) d+ + l ( f A (x u P ) ; f (x u P )) d = f (x u P ) d + I 1 + I 2 : By (2.1), (2.2), (2.13) (ii), there exists l() > 0such that I 1 = n l (;f (x u P )) d n l (;f ; (x u P )) d < =2 if l l(): By (2.12), we have I 2 l ne (f(x u (K) P ) ; f (x u P )) d+ + E (K) (A ; f (x u P )) d = I 21 + I 22 (2.15) (2.16) where E (K) is dened by (2.7). By lemma 2.2, E (K)! 0 uniformly for as K!1. Thus it follows from (ii) that there exists K( A) > 1such that I 22 E (A ; f ; (K) (x u P )) d Let K = K( A), then < =4 if K K( A): G( K)=fu 2 m : juj KgfP 2 N : jp j Kg is a compact set in m N. ( A l) > 0such that Thus, we have ji 21 j <=4 It follows from (2.14) that there exists 8 ( A l): ji 2 j <=2 8 ( A l): (2.17) Thus (2.15) follows from (2.16) (2.17). 2 Lemma 2.4. Let f ff g satisfy the hypotheses in theorem 2.1. Let fu g u2 U fp g P 2 V satisfy (2.1) (2.2) respectively. Let Suppose F ( A) =fx 2 :f(x u (x) P (x)) >Ag: f (x u P ) d C (2.18)
SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 177 for some constant C>0. Then, for any >0 K 1, there exista() > 1 ( K) > 1 such that F( A) E (K)+ if A A() ( K) (2.19) where E (K) is dened by (2.7). Proof. By (iii), there is a sequence of measurable subsets f l g in such that ( n l ) ;! 0 as l!1 (2.20) f ;! f uniformly on l G (2.21) for each l any compact set G m N. For any >0, by (2.20), there is l 1 () 1 such that ( n l ) <=2 if l l 1 (): (2.22) By (2.18), l ne (K) f(x u P ) d = l ne (K) (f(x u P ) ; f (x u P )) d + n( l ne (K)) (;f (x u P )) d + C = I 1 + I 2 + C It follows from (ii) that I 2 n( l ne (K)) (;f ; (x u P )) d (;f ; (x u P )) d C 1 (2.23) for some constant C 1 > 0. It follows from (2.21) that there exists (l K) > 1 such that Thus we have ji 1 j l ne (K) jf(x u P ) ; f (x u P )j d 1 8 (l K): (2.24) f(x u P ) d C 2 8 (l K) (2.25) l ne (K) where C 2 = C + C 1 + 1 is a constant.
178 HIPING LI Denote ; = fx 2 :f(x u (x) P (x)) < 0g f + =maxff 0g then, by (2.25) lne(k) f + (x u P) d l ne (K) (;f ; (x u P )) d + C 2 = ( l ne (K))\ ; (;f(x u P )) d + C 2 = ( l ne (K))\ ; (f (x u P ) ; f(x u P )) d + ( l ne ; (K))\ (;f (x u P )) d + C 2 l ne jf(x u (K) P ) ; f (x u P )j d + (;f ; (x u P )) d + C 2 : It follows from this (2.23), (2.24) that l ne (K) f + (x u P ) d C 3 8 (l K) (2.26) where C 3 = C 1 + C 2 + 1 is a constant. Now (2.26) implies that there exists A() > 0 such that fx 2 l n E (K) : f(x u (x) P (x)) >Ag <=2 if A A() (l K): (2.27) Since where F ( A) E (K) [ ( n l ) [ F (l K A) F (l K A)=fx 2 l n E (K) :f(x u (x) P (x)) >Ag we have F( A) E (K)+ ( n l )+F(l K A): Taking l = l 1 () ( K) = (l 1 () K), by (2.22) (2.27), we conclude that (2.19) is true. 2 Proof of Theorem 2.1. Without loss of generality, we assume that f (x u P ) d C for some constant C>0. It follows from (iii) that there exists a sequence of measurable subsets f l g of such that l l+1 8l lim l!1 ( n l )=0 (2.28) f ;! f uniformly on l G (2.29) for each l any compact set G m N.
SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 179 Let E (K) be dened by (2.7). It follows from lemma 2.2 that there exists an increasing sequence fk i g such that 1X i=1 sup 1<1 fe (K i )g < 1: (2.30) Let i > 0 i =1 2 be a decreasing sequence of numbers satisfying lim i!1 i =0. LetA i = A( i =2 i ) l i = l( i ) i =maxf(l i K i ) ( i A i l i )g F i = F ( i A i )=fx 2 :f(x u i (x) P i (x)) >A i g where A() ( ) are dened by lemma 2.4 l() ( _ ) are dened by lemma 2.3. Then, by lemma 2.3, we have li fai (x u i P i ) d by lemma 2.4, we have 1X i=1 F i 1X i=1 f i (x u i P i ) d + i 8i (2.31) ( E i (K i )+ i =2 i ) < 1: (2.32) Let H j =((n lj ) [ ([ ij F i )) G j =n H j. It follows from (2.28) (2.32) that Thus, by the denition of f Ai G j G j+1 8j lim j!1 ( n G j )=0: (2.33) F i,wehave G j f(x u i P i ) d = li nf i f(x u i P i ) d + ( li nf i )ng j (;f(x u i P i )) d li nf i f(x u i P i ) d + ng j (;f ; (x u i P i )) d li nf i fai (x u i P i ) d + H j (;f ; (x u i P i )) d 8i j: It follows from this (2.31) that G j f(x u i P i ) d f i (x u i P i ) d + H j (;f ; (x u i P i )) d + i 8i j: (2.34) Let i!1in (2.34). By (2.4), we have G j f(x u P ) d By (ii) (2.33), we have lim i!1 f i (x u i P i ) d +lim i!1 H j (;f ; (x u i P i )) d: (2.35) lim (sup j!1 i1 H j (;f ; (x u i P i )) d) =0:
180 HIPING LI It follows from this (2.33), (2.35) that f(x u P ) d = lim j!1 G j f(x u P ) d lim i!1 f i (x u i P i ) d: This completes the proof of (a). Next, we prove (b). Without loss of generality, we assume that It follows from (ii), (iv) (2.36) that f(x u P ) d C<1: (2.36) f (x u P ) d C 1 < 1: By (2.6), for any >0, there exists ^() > 0 such that f(x u P ) d f (x u P ) d + 8 ^(): (2.37) By (ii), for any >0, there exists 1 () > 0such that j 0 f ; (x u P ) dj < 8 0 1 () (2.38) j 0 f ; (x u P ) dj < 8 0 1 (): (2.39) By (iv) (ii), for any >0, there exists 2 () > 0 such that 0 f + (x u P ) d Let 0 f + (x u P ) d + a(x) d 0 0 f(x u P ) d + 8 0 0 2 (): (2.40) E (K) =fx 2 :ju (x)j >K or jp (x)j >Kg: By lemma 2.2, for any >0 there exists K() > 0 such that E (K) < 8K K(): (2.41) By (iii), there exists a sequence of measurable subsets l such that ( n l )! 0asl!1 f! f uniformly on l G for each xedl compact set G m N. Thus, for any >0 there exists l() 1 such that ( n l ) < 8l l() (2.42) for any >0, l 1 compact set G m N there exists ( l G) > 0such that jf (x v Q) ; f(x v Q)j < (2.43) for all ( l G) (x v Q) 2 l G.
SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 181 Now, by taking =minf 1 () 2 ()g K = K() l = l() G = f(v Q) 2 m N : jvjk jp jkg we get from (2.37) { (2.43) that f(x u P ) d f (x u P ) d + l ne (K) l ne (K) () = maxf^() ( l G)g f (x u P ) d + f(x u P ) d + f(x u P ) d +4 n( l ne (K)) n( l ne (K)) 8 >(): f + (x u P ) d + f + (x u P ) d +3 This the arbitrariness of imply (2.5). 2 Corollary 2.5. If the hypothesis (iii) is replaced by the hypothesis (iii) 0 : f! f locally uniformly in the sense of integration on U V, in theorem 2.1, the conclusions of the theorem still hold. Proof. Since in the proof of theorem 2.1 the hypothesis (iii) was only used to show that there exists ( l K) > 0such that j l ne (K) f(x u P ) ; f (x u P ) dj <=4 for ( l K), the result follows. 2 Corollary 2.6. If the hypothesis (ii) is replaced by the hypothesis (ii) 0 : f has the strong lower compactness property ff g have the uniform lower compactness property, in theorem 2.1, the conclusions of the theorem still hold. Proof. The proof of the part (a) of theorem 2.1 remains valid, since (ii) 0 the assumption f (x u (x) P (x)) d C<1 give (ii). The proof of the part (b) of theorem 2.1 still holds, because (ii) 0, (iv) the assumption f(x u (x) P (x)) d C<1 imply f (x u (x) P (x)) d C 1 < 1 hence (ii). 2
182 HIPING LI Theorem 2.7. Let be ameasurable space with nite positive nonatomic complete measure. Let U V satisfy (H1) (H2). Let f ff g : m N! [f+1g satisfy (i): f ff g are L B- measurable (ii) 00 : f has the lower compactness property f have the uniform lower compactness property. (iii) 0 : f! f locally uniformly in the sense of integration on U V (iv) 0 : f + (x v Q) a(x)+f + (x v Q) for all (v Q) 2 m N, where a() 2 L 1 () is nonnegative. Let fu g u2 U fp g P 2 V be such that Suppose Then u ;! u in U P ;! P in V: f(x u P ) d lim!1 f(x u P ) d lim!1 Proof. Without loss of generality, we may assume f (x u P ) d: f(x u P ) d: f(x u (x) P (x)) d C 1 < 1: Thus (ii) 00, (iv) 0 give (ii). It is easily seen that (iv) 0 imply (iv). Hence the theorem follows from the same arguments as in the proof of the part (b) of theorem 2.1. 2 So far, only individual sequence f(u P )g satisfying (2.1) (2.2) is considered. If all sequences f(u P )g satisfying (2.1) (2.2) are considered at the same time, some ;-convergence results can be obtained from the above theorems. First, recall that the ;-limit of a sequence of integrals I ( ) can be dened by ;(U ; V ; ) lim!1 I (u P ) = infflim!1 I (u P ):(u P )! (u P )inu V g provided it exists the topologies of U V are metrizable [13, 14]. We have the following theorem as a consequence of theorem 2.1 theorem 2.7. Theorem 2.8. Let be ameasurable space with nite positive nonatomic complete measure. Let U V satisfy (H1), (H2) be metrizable. Let f ff g : m N! [f+1g satisfy (i): f ff g are L B- measurable (ii) 00 : f has the lower compactness property f have the uniform lower compactness property. (iii) 0 : f! f locally uniformly in the sense of integration on U V.
Then, (a): SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 183 f(x u P ) d ;(U ; V ; )lim!1 I (u P ) (2.44) provided thati(u P 0 )= 0 f(x u(x) P(x)) d is lower semicontinuous at (u P ) 2 U V for all measurable subsets 0 (b): f(x u P ) d lim!1 f(x u P ) d (2.45) for all (u P )! (u P ) in U V, i.e. f(x u P ) d is lower semicontinuous at (u P ) 2 U V,provided that f ff g satisfy a further hypothesis (iv) 0 : f + (x v Q) a(x)+f + (x v Q) for all (v Q) 2 m N, where a() 2 L 1 () is nonnegative f(x u P ) d ;(U ; V ; )lim!1 I (u P ): Corollary 2.9. Let be ameasurable space with nite positive nonatomic complete measure. Let U V satisfy (H1), (H2) be metrizable. Let f ff g : m N! [f+1g satisfy (i): f ff g are L B- measurable (ii) 00 : f has the lower compactness property f have the uniform lower compactness property. (iii) 0 : f! f locally uniformly in the sense of integration on U V. (iv) 0 : f + (x v Q) a(x)+f + (x v Q) for all (v Q) 2 m N, where a() 2 L 1 () is nonnegative. Then f(x u P ) d =;(U ; V ; )lim!1 I (u P ) (2.46) provided thati(u P 0 )= 0 f(x u(x) P(x)) d is lower semicontinuous at (u P ) 2 U V for all measurable subsets 0. In addition, we have in this case ;(U ; V ; )lim I (u P )=lim!1!1 f (x u(x) P(x)) d: (2.47) Proof. By theorem 2.8, we only need to show that f(x u P ) d lim!1 f (x u(x) P(x)) d: (2.48) This can be easily veried by using the inequality f(x u(x) P(X)) d f (x u(x) P(X)) d + l ne(k) (f(x u(x) P(X)) ; f (x u(x) P(X))) d ; n( l ne(k)) f + (x u(x) P(X)) d + n( l ne(k)) f ; (x u(x) P(X)) d
184 HIPING LI for all l 1 K>0, where E(K)=fx 2 :ju(x)j >K or jp (x)j >Kg by using lemma 2.2 the hypotheses (ii) 00, (iii) 0 (iv) 0. 2 3. Lower semicontinuity theorems for quasiconvex integrs In this section, some lower semicontinuity theorems in the form of (1.3) (1.4) are established for quasiconvex integr f(x u P ). In the following, U is taken to be L p ( m ) with strong topology V is taken to be L p ( mn ) with weak topology, where 1 p<1, is taken to be the Lebesgue measure on n. Theorem 3.1. Let n beopen bounded. Let f : m mn! satisfy (i): f( ) is a Caratheodory function (ii): f(x u P ) is quasiconvex in P (iii): f has the strong lower compactness property on U V (iv): f(x u P ) a(x)+b(x)(juj p + jp j p ) for every x 2, u 2 m P 2 mn,where a() b() 2 L 1 () are nonnegative functions. Let ff g : m mn! satisfy (a): f ( ) are L B- measurable (b): f have the uniform lower compactness property (c): f! f locally uniformly in m mn. Let u *u in W 1 p ( m ): (3.1) Then f(x u Du) d lim!1 f (x u Du ) d: (3.2) To prove the theorem, we begin with the following lemmas. Lemma 3.2. Let n beopen bounded. Let f : m mn! satisfy (i) {(iv) in theorem 3.1. For all integers 1, let functions ^f : m mn! be dened by ^f (x u P )=maxf; f(x u P )g: Let u *uin W 1 p ( m ). Then, for each xed 1, ^f (x u(x) Du(x)) d lim!1 ^f (x u (x) Du (x)) d: (3.3) Proof. Let E(b K)=fx 2 :jb(x)jkg:
SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 185 Denote K (x) the characteristic function of the set E(b K). It is easily veried that the functions ~f K = K ^f + 1 K 1 satisfy the hypotheses of theorem 1.13. Hence we have K(x) ^f (x u(x) Du(x)) d lim!1 K(x) ^f (x u (x) Du (x)) d: Since we have K(x) ^f (x u(x) Du(x)) d ^f ; (x u(x) Du(x)) d + K(x) ^f + (x u(x) Du(x)) d K(x) ^f (x u (x) Du (x)) d ^f (x u (x) Du (x)) d + lim!1 j (1 ; K(x)) ^f ; (x u (x) Du (x)) dj ^f (x u (x) Du (x)) d + I(K ) ^f ; (x u(x) Du(x)) d + K (x) ^f + (x u(x) Du(x)) d ^f (x u (x) Du (x)) d + lim!1 I(K ) for xed K : (3.4) By (iii) the denition of ^f,weknow that for any >0 there exists K() 1suchthat I(K ) < 8 K K() since lim K!1 ( n E(b K)) = 0. Thus (3.3) is obtained by letting k!1in (3.4) passing to the limit. 2 Lemma 3.3. Let n beopen bounded. Let f : m mn! satisfy (i) {(iv) in theorem 3.1. For all integers 1, let functions ^f : m mn! be dened by ^f (x u P )=maxf; f(x u P )g: Let u *uin W 1 p ( m ). Then f(x u(x) Du(x)) d lim!1 Proof. Denote F () = lim!1 ^f (x u (x) Du (x)) d: (3.5) ^f (x u (x) Du (x)) d: (3.6) Since f f +1 for all, F () is nonincreasing. On the other h, it follows from (iii) that F () is bounded from below. Hence lim!1 F () exists. Denote the limit by F.
186 HIPING LI Let Since f ^f for all, by lemma 3.2, we have f(x u(x) Du(x)) d F: (3.7) Given >0, by (3.6) F () F, there exists ( ) such that We have I( )= ^f (x u (x) Du (x)) d > F ; 8 ( ): (3.8) I( ) ( ^f (x u (x) Du (x)) ; ^f (x u (x) Du (x))) d: E( ) ^f (x u (x) Du (x)) d f ; (x u E( ) (x) Du (x)) d 8 where E( ) =fx 2 :f ; (x u (x) Du (x)) < ;g. Hence,by (iii), for any >0 there exists () > 0such that I( ) > ; 8 () : (3.9) It follows from (3.8) (3.9) that ^f (x u (x) Du (x)) d = ^f () (x u (x) Du (x)) d + I( ) > F ; 2 8 ( ()): Since >0 is arbitrary, this (3.7) imply (3.5). 2 Lemma 3.4. Let n beopen bounded. Let f : m mn! satisfy (i) {(iv) in theorem 3.1. Let u *uin W 1 p ( m ). Then 0 f(x u(x) Du(x)) d lim!1 for all measurable subset 0. 0 f(x u (x) Du (x)) d (3.10) Proof. Let 0 be the characteristic function of 0. Let f ~ = 0 f, f ~ = maxf; 0 fg. Itiseasilyveried that f ~ satises the hypotheses (i) { (iv) in theorem 3.1 as well. Hence, by lemma 3.3, we have 0 f(x u(x) Du(x)) d lim!1 0 ^f (x u (x) Du (x)) d (3.11) where ^f =maxf; fg. By (iii) the denition of ^f,we know that ^f have the uniform lower compactness property ^f! f locally uniformly in m mn. Thus (3.10) follows from (3.11) the part (a) of theorem 2.1. 2 Proof of theorem 3.1. The conclusion follows directly from lemma 3.4 the part (b) of theorem 2.1. 2 In theorem 3.1, the hypothesis that f has the strong compact property is not essential. Actually, we have the following stronger result.
SEMICONTINUITY OF INTEGALS AND CONVEGENT INTEGANDS 187 Theorem 3.5. Let n beopen bounded. Let u *uin W 1 p ( m ).Let f f : m mn! satisfy (i), (ii), (iv) (a), (c) in theorem 3.1 respectively, satisfy also the following hypothesis (h): f ; (x u (x) Du (x)) are weakly precompact in L 1 (). Then f(x u Du) d lim!1 f (x u Du ) d: Proof. Let f N =maxff ;Ng f N =maxff ;Ng. Then, for each xed N > 0, f N f N satisfy (i) { (iv) (a) { (c) in theorem 3.1 respectively. Thus, by theorem 3.1, we have, for each xed N>0, f N (x u Du) d lim!1 f N (x u Du ) d: (3.12) Let N!1,by(h)by passing to the limit in (3.12), we get the result. 2 The following lemma is a version of Chacon's biting lemma (see [5], see also [7, 8,3]). Lemma 3.6. Let n be bounded measurable, let g be a bounded sequence inl 1 (). Then there exist a subsequence g j of g a nonincreasing sequence ofmeasurable subsets E k with lim k!1 E k =0 such that g j are weakly precompact in L 1 ( n E k ) for each xed k, i.e. for any >0 xed k there exists ( k) > 0 such that 0 jg j (x)j d < provided that 0 n E k ( 0 ) <( k). Lemma 3.7. Let n beopen bounded. Let f ff g : m mn! satisfy (L): f(x u P ) f (x u P ) are bounded below by ;(a(x)+b(x)(juj p + jp j p )), where a() b() 2 L 1 () a(x) 0 b(x) 0 8x 2. Let u *uin W 1 p ( m ). Then there exist a subsequence u j of u a nonincreasing sequence ofmeasurable subsets E k with lim k!1 E k =0such that f ; (x u j (x) Du j (x)), f ; j (x u j (x) Du j (x)) are weakly precompact in L 1 ( n E k ) for each xed k. Proof. By lemma 3.6, there exist a subsequence u j of u a nonincreasing sequence of measurable subsets Ek ~ with lim k!1 Ek ~ = 0 such that h j (x) =(ju j (x)j p + jdu j (x)j p ) are weakly precompact in L 1 ( n E ~ k ) for each xedk. Let ^Ek = fx 2 :jb(x)j >kg E k = ^Ek [ Ek ~.Then E k are nonincreasing with lim k!1 E k = 0 g j (x) =a(x)+b(x)(ju j (x)j p + jdu j (x)j p ) are weakly precompact in L 1 ( n E k ) for each xed k. This the hypothesis (L) give the result. 2 8j
188 HIPING LI Theorem 3.8. Let n beopen bounded. Let f ff g : m mn! satisfy (i): f ff g are Caratheodory functions (ii): f(x u P ) is quasiconvex in P (iii): jf(x u P )j a(x) +b(x)(juj p + jp j p ) jf (x u P )j a(x) + b(x)(juj p + jp j p ) for every x 2, u 2 m P 2 mn, where a() b() 2 L 1 () are nonnegative functions (iv): f! f locally uniformly in m mn. Let u *u in W 1 p ( m ): (3.13) Then there exist a subsequence u j of u a nonincreasing sequence of measurable subsets E k with lim k!1 E k =0such that ne k f(x u Du) d lim j!1 ne k f j (x u j Du j ) d (3.14) for each k: Proof. By (iii), (3.13) lemma 3.7, there exist a subsequence u j of u a nonincreasing sequence of measurable subsets E k with lim k!1 E k = 0such that f ; j (x u j (x) Du j (x)) are uniformly weakly precompact in L 1 ( n E k ). Let f k = nek f f k j = nek f j. Applying theorem 3.5 to f k f k j u j,we obtain (3.14). 2 The author wishes to thank Professor J.M. Ball the referee for their valuable comments suggestions. eferences [1].-P. Li: Numerical methods for computing singular minimizers, Numer. Math., 71, 1995, 317{330. [2] A.D. Ioe: On lower semicontinuity of integral functionals, SIAM J. Control Optim., 15, 1977, 521{538. [3] J. Diestel J.J. Uhl, Jr: Vector Measures, Ameri. Math. Society, Providence, hode Isl, 1977. [4] C.B. Morrey: Multiple Integrals in the Calculus of Variations, Springer, New York, 1966. [5] E. Acerbi N. Fusco: Semicontinuity problems in the calculus of variations, Arch. at. Mech. Anal., 86, 1984, 125{145. [6] J.M. Ball K.-W. hang: Lower semicontinuity of multiple integrals the biting lemma, Proc.. Soc. Edinburgh, 114A, 1990, 367{379. [7] J.K. Brooks K.V. Chacon: Continuity compactness of measures, Adv. in Math., 37, 1980, 16{26. [8] J.M. Ball F. Murat: emarks on Chacon's biting lemma, Proc. Amer. Math. Soc., 107, 1989, 655{663. [9] F. Murat: A survey on compensated compactness, in Contributions to Modern Calculus of variations, L. Cesari. ed., Longman, Harlow, 1987.
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