Existence for some vectorial elliptic problems with measure data **)

Transcription

1 Riv. Mat. Univ. Parma 7) 5 006), -46 F RANCESCO L EONETTI and P IER INCENZO P ETRICCA *) Existence for some vectorial elliptic problems with measure data **) - Introduction The study of elliptic boundary value problems with measure data div a x; u x); Du x))) ˆ m in u ˆ 0 received great attention starting from[]. In such a paper the scalar case u R n! R has been considered and the model operator is the p-laplacian a i x; u; j) ˆjjj p j i. Some years later [], the anisotropic p i -laplacian has been dealt with a i x; u; j) ˆjj i j p i j i, that is, each component of the gradient D i u may have a different exponent p i ; this seems useful when dealing with some reinforced materials, []; see also [0], example.7., page 69. Let us point out that both [] and[] deal with the scalar case u! R; as far as the vectorial case u! R N is concerned, some contributions appeared in [7] and[4], where the vectorial p-laplacian is considered. In this paper we deal with the anisotropic *) F. Leonetti Dipartimento di Matematica Pura ed Applicata, UniversitaÁ di L'Aquila, 6700 L'Aquila, Italy; leonetti@univaq.it; P.. Petricca ia Sant'Amasio, 009 Sora, Italy. **) Received November 7 th 005 and in revised formmarch nd 006. AMS classification 5J60, 5D05, 5J55. We acknowledge the support of MIUR, GNAMPA, INdAM and CARISPAQ. [] EXISTENCE FOR SOME ECTORIAL ELLIPTIC PROBLEMS WITH MEASURE DATA

2 4 FRANCESCO LEONETTI and PIER INCENZO PETRICCA [] vector valued case < Pn D i jd i uj pi D i u) ˆ m u ˆ 0 in where u R n! R N with n, N, is a bounded open set and m ˆ m ;...; m N ) is a finite Radon measure on R n with values into R N. In this paper we prove existence of weak solutions for the Dirichlet problem.) in Section we give the precise result, whose proof appears in Section. We thank the referee for useful remarks and suggestions. - Notations and statements We study existence of weak solutions for the Dirichlet problem < Pn D i jd i uj pi D i u) ˆ m in u ˆ 0 where u R n! R N with n, N, is a bounded open set and m ˆ m ;...; m N ) is a finite Radon measure on R n with values into R N. We need some restrictions on the exponents p i ; we introduce the harmonic mean and we assume that p < n and p ˆ X n n p i! p n ) p )n < p p n ) i < n p i f;...; ng The right hand side of.) can be written as n p ) n p p i ˆ p i p n ) p n ) < thus p i < n p ) p n ) p i so we can take exponents q ;...; q n such that p i < q i < n p ) p n ) p i i f;...; ng;

3 [] EXISTENCE FOR SOME ECTORIAL ELLIPTIC PROBLEMS WITH MEASURE DATA 5 and we set q ˆ min q i ; q ˆ X n i n! ; q i q ˆ q n In this paper we will prove the following Theorem.. Under the previous assumptions there exists u W ;q 0 ; RN ) \ L q ; R N ) with D i u L q i ; R N ) i f;...; ng such that X n X N jd i uj pi D i u b D i W b dl n ˆ XN W b dm b W Cc ; RN ) bˆ bˆ In this theoremwe may take n ˆ, p ˆ p ˆ and p ; ); this example shows that our theoremcannot be derived from[5], []. The previous theoremimproves on [9] we remove the restriction on the support of the measure m and we allow a wider range for p i. - Proof of the Theorem Let fq k g kn be a decreasing sequence of positive numbers converging to zero. We mollify our measure m and we obtain smooth functions with f k ˆ m r Qk * m sup k f k k L ) jmj Rn ) < k We set p ˆ min i p i and we use direct methods in the calculus of variations in order to find n o u k ˆ u W ;p 0 ; R N ) D i u L p i ; R N ) i f;...; ng ; such that X n X N jd i u k j pi D i u b k D iw b dl n ˆ XN W b f b k dln W bˆ bˆ We bound u k we use a componentwise truncation argument [7], [] and estimates

4 6 FRANCESCO LEONETTI and PIER INCENZO PETRICCA [4] on level sets []; we set 0 0 t L >< t L L < t < L M t) ˆ t L > M t) t < 0 and we use the following test function W in.) we fix a f;...; Ng and we take W ˆ W ;...; W N ) with W b ˆ 0 for b 6ˆ a and W a ˆ M u a k ), where ua k is the a-th component of u k ˆ u k ;...; un k ). We define B L;k ˆ x L u a k x) < L ; and we insert the previous W into.) we get X n Di u k p i Di u a k dx ˆ fk a M ua k ) dx B L;k Since jm t)j, we can use.) so that the right hand side of.) is bounded by jmj R n ). Let us look at the left hand side since p i, we have Di u a p i k Di u k p i thus we get 4 Since we can select X n B L;k Di u a p i k dx c k ; L N q i < n p ) p n ) p i; n p ) u 0; 0; ) p n ) such that q i up i for every i. Now, without loss of generality, we can assume that q i ˆ up i for every i and u is close to n p ) p n )

5 [5] EXISTENCE FOR SOME ECTORIAL ELLIPTIC PROBLEMS WITH MEASURE DATA 7 Let us remark that < q i < p i for every i f;...; ng and q < p < n. Now we follow [] we set u)q l ˆ u so that l > ; we use Holder inequality Di u a q i k dx ˆ Di u a q i k ju a k j) lq i=p i ju a k j)lq i=p i dx 5 4 Di u a p i k ju a k j) l dx5 6 ˆ X 4 6 X 4 q i =p i Di u a p i k ju a 7 k j) l dx5 Lˆ0 B L;k L) l Di u a p i 7 k dx5 4 ju a k j)lq i= p i q i ) dx5 q i =p i q i =p i Lˆ0 B L;k q i =p i 4 ju a k j)lq i= p i q i ) dx5 4 ju a k dx j)q 5 q i=p i q i =p i We keep in mind.4) and l > we get 6 Di u a q i k dx c 4 ju a k dx j)q 5 q i=p i We use the anisotropic embedding theorem [] and we get u a L k q ) c Y n Di u a =n k L q i ) c Y n 4 4 ju a k j)q dx5 7 ˆ c 4 4 ju a k j)q dx5 [ =q i ) =p i )]=n u)= up) We recall that p < n, so u)= up) < =q thus.7) gives u a k L q ) c 5 k N; a ˆ ;...; N

6 FRANCESCO LEONETTI and PIER INCENZO PETRICCA [6] We insert this bound into.6) and we get 9 Di u a k L q i ) c 6 k N; i ˆ ;...n; a ˆ ;...; N Because of weak compactness, there exists u W ;q 0 ; RN ) \ L q ; R N ), with D i u L q i ; R N ) i f;...; ng, s i L q i p i ; R N ) and h i L q i ; R) for every i f;...; ng such that, up to a subsequence, 0 u k * u in W ;q 0 ; RN ); u k! u ae in ; D i u k * D i u in L q i ; R N ) i f;...; ng; jd i u k j p i D i u k * s i in L q i p i ; R N ) i f;...; ng; 4 We claimthat jd i u k D i uj * h i 0 in L q i ; R) i f;...; ng 5 h i ˆ 0 in i f;...; ng thus 6 D i u k! D i u in L ; R N ) i f;...; ng We prove claim.5) following [4]. We fix a smooth v R n! R N ; later, v will be chosen as a linear Taylor polynomial of u. We take cut off functions f R n! R and h R N! R such that f C c ; R); h C c RN ; R); with 0 f and 0 h. We take c R N! R N as follows c z) ˆ g jzj) z jzj where g C 0; ); R) with g and g 0 non-negative and bounded; moreover, we require that c j supp h) ˆ Id. Note s z) ˆ g0 jzj) zs z a jzj g jzj) d sa zs z a jzj jzj thus 7 X N aˆ j a XN s z)js 0 z; j R N

7 [7] EXISTENCE FOR SOME ECTORIAL ELLIPTIC PROBLEMS WITH MEASURE DATA 9 For every j f;...; ng it results that 0 jd j u k D j ujh u k v)fdx jd j u k D j vjh u k v)fdx jd j v D j ujh u k v)fdx jd j u k D j vj p j h u k v)fdxa jd j v D j ujh u k v)fdx =p j h u k v)fdxa =p j Let ~p ˆ max i p i. Then, using the monotonicity of z!jzj p z and the equality c ˆId on supp h), we get 0 T j k ˆ jd j u k D j vj p j h u k v)fdx 9 Xn Xn ~p jd i u k D i vj p i h u k v)fdx Xn ~p ˆ Xn ~p < ˆ ~p X n hjd i u k j pi D i u k jd i vj pi D i v; D i u k D i vih u k v)fdx hjd i u k j pi D i u k ; D i c u k v))ih u k v)fdx hjd i vj pi D i v; D i u k D i vih u k v)fdx hjd i u k j pi D i u k ; D i c u k v))ifdx hjd i vj pi D i v; D i u k D i vih u k v)fdx 9 = hjd i u k j pi D i u k ; D i c u k v))i[ h u k v)]fdx ;

8 40 FRANCESCO LEONETTI and PIER INCENZO PETRICCA [] We recall.) and we get 0 X n ˆ Xn hjd i u k j pi D i u k ; D i c u k v))ifdx hjd i u k j pi D i u k ; D i c u k v)f)idx hjd i u k j pi D i u k ; c u k v)d i fidx ˆ hm r Qk ; c u k v)fidx Moreover,.7) implies Xn hjd i u k j pi D i u k ; c u k v)d i fidx hjd i u k j pi D i u k ; D i c u k v))i[ h u k v)]fdx X N aˆ jd i u k j p i D i u a k X N sˆ We insert.0) and.) into.9) we get 0 T j k ˆ jd j u k D j vj p j h u k v)fdx < ~p hm r Qk ; c u k v)fidx Xn X N aˆ jd i u k j p i D i u a k X N s u k v)d i v s [ h u k v)]fdx hjd i u k j pi D i u k ; c u k v)d i s u k v)d i v s [ h u k v)]fdx 9 = hjd i vj pi D i v; D i u k D i vih u k v)fdx ; We now use.),.),.),.4) and we keep in mind that h; c; Dc are

9 [9] EXISTENCE FOR SOME ECTORIAL ELLIPTIC PROBLEMS WITH MEASURE DATA 4 continuous and bounded for every j ˆ ;...; n we get T j ˆ limsup T j k k! < ~p sup jc z)j fdjmj zr N Xn X N X N s a i aˆ sˆ 0 lim lim hs i ; c u v)d i s u v)d iv s [ h u v)]fdx 9 = hjd i vj pi D i v; D i u D i vih u v)fdx ; ; k! k! jd j u k D j ujh u k v)fdx ˆ h j h u v)fdx; jd j v D j ujh u k v)fdx ˆ jd j v D j ujh u v)fdx; lim k! h u k v)fdx ˆ h u v)fdx Take the limsup in.). We get, for every j ˆ ;...; n, 0 h j h u v)fdx 7 T j 0 h u v)fdx A =p j jd j v D j ujh u v)fdx where T j has been estimated in.). For a. e. a and every j ˆ ;...; n, we have see [6] and []) ju x) [u a) Du a) x a)]j lim dx ˆ 0; R!0 R 9 0 lim R!0 lim R!0 jdu x) Du a)jdx ˆ 0; js j x) s j a)jdx ˆ 0;

10 4 FRANCESCO LEONETTI and PIER INCENZO PETRICCA [0] lim R!0 jh j x) h j a)jdx ˆ 0; jmj B a; R)) limsup R!0 L n B a; R)) < where L n is the Lebesgue measure in R n and f x)dx ˆ L n f x)dx A) A We choose v to be the linear Taylor polynomial of u in a v x) ˆ u a) Du a) x a) Since we are still free to select f, h and c, let f R x) ˆ ~ x a f R n ; f ~ C R c B 0; ); R); 0 f ~ ; f x)dx ~ ˆ R n h R y) ˆ ~h y ; ~h Cc R B 0; ); R); 0 ~h ; ~h jb 0; ) ˆ ; c R y) ˆ R~c y ; R with ~c as in the above discussion. Then.7) can be written as follows 0 h j h R u v)f R dx A where 4 0 T j =pj B 0 T j ;R ˆ limsup k! ~p sup jc R z)j zr N Xn X N X N s a i aˆ sˆ C h R u v)f R dxa =p j jd j u k D j vj p j h R u v)f R dx f R djmj jd j v D j ujh R u v)f R dx hs i ; c R u v)d i f R a s u v)d iv s [ h R u v)]f R dx ) hjd i vj pi D i v; D i u D i vih R u v)f R dx

11 [] EXISTENCE FOR SOME ECTORIAL ELLIPTIC PROBLEMS WITH MEASURE DATA 4 Recalling.),...,.) and the definitions of h R, f R, we get 5 h j h R u v)f R dx ˆ h j a); lim R! lim R!0 Tj ;R ˆ 0; h R u v)f R dx ; lim R!0 jd j v D j ujh R u v)f R dx ˆ 0 As far as.6) is concerned, we give some details in the Appendix. Let us take R! 0 in.) we get, for every j ˆ ;...; n, 9 h j a) ˆ 0 for a.e a thus our claim.5) is proved. Now we have.6), thus, up to a subsequence, 40 Du k! Du ae in The previous pointwise convergence and the estimate jdi u k j pi D i u k q i ˆ Di u k p i p L i L q i ) c 7; k N; i f;...; ng ) allow us to get 4 jd i u k j p i D i u k * jd i uj p i D i u in L q i p i ; R N ) i f;...; ng Thus we can pass to the limit, as k!, in.) and it turns out that u solves.). This ends the proof. p 4 - Appendix For the convenience of the reader, we give some details about the proof of.6). We look at inequality.4) it suffices to prove that all the four integrals go to zero when R! 0. We keep in mind that 4 4 jc R z)j c R 0 f R x) R n

12 44 FRANCESCO LEONETTI and PIER INCENZO PETRICCA [] then 0 sup zr N jc R z)j jmj B a; R)) f R djmj c R R n and the right hand side goes to zero because of.). Now we deal with the second integral in the right hand side of.4) hs i x); c R u x) v x))d i f R x)idx hs i a); c R u x) v x))d i f R x)idx hs i x) s i a); c R u x) v x))d i f R x)idx ˆ I R II R Note that then We recall that D i f R x) ˆ R n h D ~ i f x a i R R ~c z) ˆ z for z B 0; supp ~h) c R u v) ˆ u v on ju vj R thus it is convenient to let A R ˆ x B a; R) ju x) v x)j R > We define v n to be the n dimensional Lebesgue measure of the unit ball in R n ; then L n B a; R)) ˆ v n R n and 0 Ln A R ) R n v n ju x) v x)j dx R

13 [] EXISTENCE FOR SOME ECTORIAL ELLIPTIC PROBLEMS WITH MEASURE DATA 45 We use.) and we get 4 L n A R ) lim R!0 R n ˆ 0 Now we are able to deal with I R as follows 0 I R ˆ h R n hs i a); c R u x) v x)) D i f ~ x a i R R idx A R 44 h R n hs i a); c R u x) v x)) D i f ~ x a i R R idx na R Ln A R ) R n js i a)jc Di f ~ L R n ) v njs i a)j Di f ~ L R n ) Using 4.) and.) we get ju x) v x)j dx R 45 lim R!0 I R ˆ 0 As far as II R is concerned, we argue as follows 0 II R c v Di n f ~ L R n ) js i x) s i a)jdx thus we can use.0) and we get 46 lim II R ˆ 0 R!0 Limits 4.5) and 4.6) guarantee that the second integral in.4) goes to zero when R! 0. In a similar manner we can show that the third integral goes to zero. The fourth integral is easier to be dealt with. Thus we have established.6). References [] L. AMBROSIO, N. FUSCO and D. PALLARA, Functions of bounded variation and free discontinuity problems, Mathematical Monographs, Oxford University Press, New York 000. [] L. BOCCARDO and T. GALLOUET, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 7 99),

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