Lp-integrability of the gradient of solutions to quasilinear systems with discontinuous coefficients
aa r X i v : . [ m a t h . A P ] O c t L p -INTEGRABILITY OF THE GRADIENT OF SOLUTIONS TOQUASILINEAR SYSTEMS WITH DISCONTINUOUSCOEFFICIENTS LUBOMIRA G. SOFTOVA
Abstract.
The Dirichlet problem for a class of quasilinear elliptic systems ofequations with small-BMO coefficients in Reifenberg-flat domain Ω is considered.The lower order terms supposed to satisfy controlled growth conditions in u and D u . It is obtained L p -integrability with p > D u where p depends explicitly onthe data. An analogous result is obtained also for the Cauchy-Dirichlet problemfor quasilinear parabolic systems. Introduction
In the present work we study the integrability properties of the weak solutions ofthe following Dirichlet problem D α (cid:16) A αβij ( x ) D β u j ( x ) + a αi ( x, u ) (cid:17) = b i ( x, u , D u ) a.a. x ∈ Ω u ( x ) = 0 on ∂ Ω(1)where Ω ⊂ R n , n ≥ Reifenberg flat domain (see Definition 2).The principal coefficients are discontinuous with ”small” discontinuity expressedin terms of their bounded mean oscillation (BMO) in Ω (cf. [20]). The matrix A ( x ) = { A αβij ( x ) } α,β ≤ ni,j ≤ N verifies A αβij ( x ) ξ iα ξ jβ ≥ λ | ξ | ∀ ξ ∈ M N × n , k A k ∞ , Ω ≤ M (2)with some positive constants λ and M. The non linear terms a ( x, u ) = { a αi ( x, u ) } α ≤ ni ≤ N and b ( x, u , z ) = { b i ( x, u , z ) } i ≤ N supposed to be Carath´eodory functions for x ∈ Ω , u ∈ R N , z ∈ M N × n and satisfy controlled growth conditions . Namely, for | u | , | z | → ∞ we have a αi ( x, u ) = O ( ϕ ( x ) + | u | nn +2 ) and b i ( x, u , z ) = O ( ϕ ( x ) + | u | n +2 n − )(3)with ϕ ∈ L p (Ω) , p > ϕ ∈ L q (Ω) , q > nn +2 . Mathematics Subject Classification.
Primary 35J57; Secondary 35K51; 35B40.
Key words and phrases.
Elliptic and parabolic divergence form systems, controlled growth con-ditions, BMO, Dirichlet data, Reifenberg flat domain.
Our aim is to show that the problem (1) satisfies the Calder´on–Zygmund prop-erty when Ω is ( δ, R )-Reifenberg flat and the coefficients are ( δ, R )-vanishing in Ω . Precisely, each bounded weak solution u ∈ W , ∩ L ∞ (Ω; R N ) of (1) gains betterregularity from the data ϕ and ϕ and belongs to W , min { p,q ∗ } ∩ L ∞ (Ω; R N ) where q ∗ is the Sobolev conjugate of q (see (4)).Similar result is obtained also for the Cauchy-Dirichlet problem for the parabolicquasilinear system u it − D α ( A αβij ( x, t ) D β u j + a αi ( x, t, u )) = b i ( x, t, u , D u )) a.a. ( x, t ) ∈ Q u ( x, t ) = 0 ( x, t ) ∈ ∂Q in a cylinder Q = Ω × (0 , T ) where Ω is ( δ, R )-Reifenberg flat and ∂Q = Ω ∪ { ∂ Ω × (0 , T ) } is the parabolic boundary.The problem of integrability and regularity of the solutions of linear and quasilin-ear elliptic/parabolic equations and systems is widely studied. Let us start with theclassical results concerning equations/systems with smooth coefficients presented inthe monographs [24, 25]. In the scalar case, N = 1 , the notorious results of DeGiorgi [10] and Nash [30] assert H¨older regularity of the solutions of linear diver-gence form equations with only L ∞ principal coefficients. One remarkable resultthat permits to obtain higher integrability of the weak solutions is due to Gehring[16]. He studied integrability properties of functions satisfying the reverse H¨olderinequality. It was noticed that some power of the gradient of the weak solutionssatisfies local reverse H¨older inequality. Modifying the Gehring lemma, Giaquintaand Modica [18] firstly obtain higher integrability of solutions of divergence formquasilinear elliptic equations. For the sake of completeness we give this result as itis presented in the monograph by Giaquinta [17, Theorem V.2.3]. Theorem 1.
Suppose that g ∈ L q (Ω) , F ∈ L q + δ (Ω) , g, F ≥ , q > , δ > and − Z B R ( x ) g q dx ≤ B − Z B R ( x ) gdx ! q + − Z B R ( x ) F q dx + θ − Z B R ( x ) g q dx for a.a. x ∈ Ω , R < min { d ( x, ∂ Ω) , R } where R > , B > , θ ∈ [0 , . Then g ∈ L p, loc (Ω) and − Z B R ( x ) g p dx ! /p ≤ C − Z B R ( x ) g q dx ! /q + − Z B R ( x ) F p dx ! /p for any ball B R ⊂ Ω , R < R , p ∈ [ q, p ) where C > , p > q depend only on B, θ, q, n.
UASILINEAR SYSTEMS IN DIVERGENCE FORM 3
There are various generalizations of the above theorem permitting to study ellipticand parabolic problems with Dirichlet and Neumann boundary conditions (see [2,3, 4, 5, 11, 15, 27]). Other results concerning higher integrability of divergenceform quasilinear equations and variational equations could be found in [9, 17, 19,21, 28]. The L p -estimates of derivatives obtained such way laid the foundation tothe so-called ”direct method” of proving partial regularity of solutions. Recently,the method of A-harmonic approximation permits to study the regularity of thesolutions without the use of the Gehring lemma. For more details we refer thereader to [1, 12, 13, 14], see also the references therein.The regularity theory for linear operators with smooth data was extended onoperators with discontinuous coefficients defined in rough domains. In [6, 7, 8] theauthors consider divergence form elliptic and parabolic equations and systems with BM O coefficients in Reifenberg flat domain with Dirichlet boundary conditionsextending such way the known results on operators with
V M O coefficients too (seealso [28, 34, 35, 22, 23] and the references therein). In [15] a reverse H¨older inequalityis established for quasilinear elliptic systems with principal coefficient being
V M O in x and under controlled growth conditions over the lower order terms. It permits theauthors to obtain interior H¨older continuity of solutions to scalar equations as well aspartial H¨older regularity of solutions to systems. In [31, 32] global H¨older regularityof solutions to elliptic quasilinear equations with V M O in x principal coefficientsis proved under strictly controlled growth conditions. Later this result is extendedfor quasilinear elliptic and parabolic equations in Reifenberg flat domains supposing controlled growth conditions and Dirichlet boundary data (see [11, 33, 36, 37]).In the present work we extend the results from [37] to elliptic and parabolicsystems with discontinuous data. Making use of the linear L p -theory for systems,developed in [7, 8] and the bootstrap method we prove D u ∈ L r with r dependingexplicitly on the data ϕ and ϕ in (3).2. Elliptic systems, definitions and main result
In the following we use the standard notations: • x = ( x , . . . , x n ) ∈ R n , ρ > B ρ ( x ) = { y ∈ R n : | x − y | < ρ } . • let Ω ⊂ R n be a bounded domain, x ∈ Ω and denote Ω ρ ( x ) = Ω ∩ B ρ ( x ) . • M N × n is the set of N × n -matrices. • For a vector function u = ( u , . . . , u N ) : Ω → R N we write | u | = X j ≤ N | u j | , D α u j = ∂∂x α u j , L.G. SOFTOVA D u = { D α u j } α ≤ nj ≤ N ∈ M N × n , | D u | = X α ≤ n j ≤ N | D α u j | . • Let f : Ω → R and | Ω | be the Lebesgue measure of Ω , then − Z Ω f ( y ) dy = 1 | Ω | Z Ω f ( y ) dy, k f k pp, Ω = k f k pL p (Ω) = Z Ω | f ( y ) | p dy. • For u ∈ L p (Ω; R N ) write k u k p, Ω instead of k u k L p (Ω; R N ) . • For each s ∈ (1 , ∞ ) recall that s ∗ means the Sobolev conjugate of ss ∗ = nsn − s if s < n arbitrary large number > s ≥ n. (4)For the function spaces we follow the notions of the monographs [24, 28]. Through allthe paper the standard summation convention on repeated upper and lower indexesis adopted. The letter C is used for various constants and may change from oneoccurrence to another.In [38] Reifenberg introduced a class of domains with rough boundary that canbe approximated by hyperplanes at every point and at every scale. Namely Definition 2.
The domain Ω is ( δ, R )-Reifenberg flat if there exist positive constants
R, δ < x ∈ ∂ Ω and each ρ ∈ (0 , R ) there is a local coordinatesystem { y , . . . , y n } with the property B ρ ( x ) ∩ { y n > δρ } ⊂ Ω ρ ( x ) ⊂ B ρ ( x ) ∩ { y n > − δρ } . (5)Reifenberg arrived at that concept of flatness in his studies on Plateau’s problemin higher dimensions and he proved that such a domain is locally a topological discwhen δ is small enough. It is easy to see that a C -domain is a Reifenberg flat with δ → R → . A domain with Lipschitz boundary with a Lipschitz constant lessthan δ also verifies the condition (5) if δ is small enough (say δ < / S β . It is a flat version ofthe Koch snowflake S π/ where the angle of the spike with respect to the horizontal is β. A domain Ω ⊂ R with S β ⊂ ∂ Ω is a Reifenberg flat if 0 < sin β < δ < / . Thiskind of flatness exhibits minimal geometrical conditions necessary for some naturalproperties in analysis and potential theory to hold. For more detailed overview ofthe properties of these domains we refer the reader to the papers [29, 39].From (5) it follows that ∂ Ω satisfies the ( A )-property (cf. [9, 17, 24]). Precisely,the measure | Ω ρ ( x ) | is δ -comparable to |B ρ ( x ) | , that is there exists a positive constant UASILINEAR SYSTEMS IN DIVERGENCE FORM 5 A ( δ ) < / A ( δ ) |B ρ ( x ) | ≤ | Ω ρ ( x ) | ≤ (1 − A ( δ )) |B ρ ( x ) | (A)for any fixed x ∈ ∂ Ω , ρ ∈ (0 , R ) and δ ∈ (0 , . This condition excludes that Ωmay have sharp outward and inward cusps. Moreover, for small δ they can beapproximated in a uniform way by Lipschitz domains with a Lipschitz constant lessthen δ (see [8, Lemma 5.1]). As consequence, they are W ,p -extension domains,1 ≤ p ≤ ∞ , hence the usual extension theorems, the Sobolev and Sobolev–Poincar´einequalities are valid in Ω . To describe the discontinuity of the principal coefficients we need of the following
Definition 3.
We say that a function a ( x ) is a ( δ, R ) -vanishing if there exist positiveconstants R and δ < such that sup <ρ ≤ R sup x ∈ Ω − Z Ω ρ ( x ) | a ( y ) − a Ω ρ ( x ) | dy ≤ δ , a Ω ρ ( x ) = − Z Ω ρ ( x ) a ( y ) dy. (6)We suppose that all A αβij ( x ) are ( δ, R )-vanishing. It implies that A ∈ BM O (Ω)with a small BMO norm k A k ∗ < δ. The nonlinear terms a ( x, u ) and b ( x, u , z ) are Carath´eodory functions for x ∈ Ω , u ∈ R N , z ∈ M N × n and satisfy the controlled growth conditions | a ( x, u ) | ≤ Λ( ϕ ( x ) + | u | nn − ) , ϕ ∈ L p (Ω) , p > | b ( x, u , z ) | ≤ Λ (cid:16) ϕ ( x ) + | u | n +2 n − + | z | n +2 n (cid:17) , ϕ ∈ L q (Ω) , q > nn + 2(8)with some positive constant Λ . In the particular case n = 2 the powers of | u | couldbe arbitrary positive numbers while the growth of | z | is quadratic (cf. [17, 24]).Under a weak solution to the problem (1) we mean a function u ∈ W ,p (Ω; R N ) , < p < ∞ satisfying Z Ω A αβij ( x ) D β u j ( x ) D α χ i ( x ) dx + X α ≤ n i ≤ N Z Ω a αi ( x, u ( x )) D α χ i ( x ) dx + Z Ω b i ( x, u ( x ) , D u ( x )) χ i ( x ) dx = 0 , j = 1 , . . . , N for all χ ∈ W ,p ′ (Ω; R N ) , p ′ = p/ ( p − . The conditions (7) and (8) are the naturalones that ensure convergence of the integrals above. Moreover, they are optimalsince a growth of the gradient greater than n +2 n leads to unbounded solutions as itis seen from the following example (cf. [25, 32]). The function u ( x ) ∈ W , ( B (0)) ,u ( x ) = | x | r − r − is a solution of the equation ∆ u = C | Du | r in B (0) . Note that u ( x ) L ∞ ( B (0)) for n +2 n < r < . L.G. SOFTOVA
In generally we cannot expect boundedness of each solution of (1) unless we addsome structural conditions. Consider, for instance, the system D α ( A αi ( x, u , D u )) = b i ( x, u , D u ) x ∈ Ωwhere A αi ( x, u , D u ) = X β ≤ n X j ≤ N ( A αβij ( x ) D β u j + a αi ( x, u ))are measurable in x ∈ Ω . Assume a pointwise coercive and sign conditions, bothof them for large values of the corresponding component of u , precisely: for every i ∈ { , . . . , N } there exist constants θ i , M i , ν ∈ (0 , + ∞ ) such that for u i ≥ θ i wehave ν | ξ i | − M i ≤ X α ≤ n A αi ( x, u , ξ ) ξ αi ≤ b i ( x, u , ξ ) for a.a. x ∈ Ω , ∀ ξ ∈ M N × n . (9)Suppose (7), (8) and (9) and let u ∈ W , ∩ L nn − (Ω; R N ) be a weak solution of (1)then for each i ∈ { , . . . , N } sup Ω u i ≤ θ i + K i where K i depend on M i , n, | Ω | and ν (see [26]). Theorem 4.
Let u ∈ W , ∩ L ∞ (Ω; R N ) be a weak solution of the problem (1) underthe conditions (2) , (7) and (8) . Then there exists a small number δ > such that if Ω is ( δ, R ) -Reifenberg flat domain and A αβij ( x ) are ( δ, R ) -vanishing with δ < δ < then u ∈ W ,r ∩ L ∞ (Ω; R N ) with r = min { p, q ∗ } . (10) Proof.
In [17, Chapter 5] Giaquinta considers quasilinear strongly elliptic systemswith L ∞ principal coefficients, under the conditions (7) and (8). Making use ofthe reverse H¨older’s inequality and the version of the Gehring lemma it is shownthat there exists an exponent r > u ∈ W ,r loc (Ω , R N ) (cf. [17, The-orem V.2.3],[9, Chapter III] or [28, Lemma 3.2.23] ). Since, roughly speaking,Caccioppoli-type inequalities hold up to the boundary, the method for obtaininghigher integrability can be carried over up to the boundary. In [17, Chapter 5] itis done for the Dirichlet problem in Lipschitz domain. Since the Reifenberg flatdomain can be uniformly approximated by Lipschitz domains the same result stillholds true. Precisely, there is r > k D u k r, Ω ≤ N ∀ r ∈ [2 , r )(11)where N and r depend on n, Λ , λ, k ϕ k p, Ω , k ϕ k q, Ω , | Ω | , k D u k , Ω . UASILINEAR SYSTEMS IN DIVERGENCE FORM 7
Let n > u ∈ W , ∩ L ∞ (Ω; R N ) be a solution of (1). Fixing that solutionin the nonlinear terms we get the linearized problem D α ( A αβij ( x ) D β u j ) = f i ( x ) − div( A i ( x )) a.a. x ∈ Ω u ( x ) = 0 on ∂ Ω(12)where f i ( x ) = b i ( x, u , D u ) , f ( x ) = b ( x, u , D u ) , A i ( x ) =( a i ( x, u ) , . . . , a ni ( x, u )) , A ( x ) = ( A ( x ) , . . . , A N ( x ))and by (7), (8) and (11) we get k A k p, Ω ≤ C (cid:18) k ϕ k p, Ω + k u k nn +2 ∞ , Ω (cid:19) k f k q , Ω ≤ C (cid:18) k ϕ k q , Ω + k u k n +2 n − ∞ , Ω + k D u k n +2 nq n +2) n , Ω (cid:19) (13)with p > q = min n q, r nn +2 o . Further, for all f i ∈ L q (Ω) , i = 1 , . . . , N there exists a vector field F i ( x ) ∈ L q ∗ (Ω , R n ) such that f i ( x ) = div F i ( x ) . Denote F ( x ) = ( F ( x ) , . . . , F N ( x )) , then by [32, Lemma 3.1]) we have k F k q ∗ , Ω ≤ C k f k q , Ω , q = min n q, r nn + 2 o . (14)Thus the problem (12) becomes D α ( A αβij ( x ) D β u j ( x )) = div( F i ( x ) − A i ( x )) a.a. x ∈ Ω u ( x ) = 0 , x ∈ ∂ Ω . (15)For linear systems as above we dispose with the regularity result of Byun and Wang[8, Theorem 1.7] that asserts there exists a small positive constant δ = δ ( λ, p, n, N )such that for each ( δ, R )-vanishing A αβij , for each ( δ, R )-Reifenberg flat Ω , and foreach matrix function F − A ∈ L r (Ω; M N × n ) , with r = min { p, q ∗ } , the solution u ∈ W , ∩ L ∞ (Ω; R N ) of (15) belongs to W ,r ∩ L ∞ (Ω; R N ) and the followingestimate holds k D u k r , Ω ≤ C k F − A k r , Ω , r = min { p, q ∗ } (16)with C = C ( λ, p, n, N, | Ω | ) . Our goal is to show the inclusion D u ∈ L r (Ω; M N × n ) with r = min { p, q ∗ } . Forthis we study the following cases:1) If q ≤ r nn +2 then q = q in (14) and r ≡ r = min { p, q ∗ } . L.G. SOFTOVA
2) If q > r nn +2 , then q = r nn +2 and q ∗ = r nn + 2 − r if r nn + 2 < n arbitrary large number > r nn + 2 ≥ n. Consider again two sub-cases:2 a ) If n > r nn +2 then r = min { p, r nn +2 − r } . If r = p then the theorem holdstrue otherwise D u ∈ L r (Ω : M n × N ) with r = r nn +2 − r . b ) If n ≤ r nn +2 then q ∗ is arbitrary large number, that implies r = p andthe theorem holds true once again.It is easy to see that r ≡ r unless r nn + 2 < q and r nn + 2 < n when u ∈ W ,r ∩ L ∞ (Ω; R N ) with r = r nn +2 − r . It holds for any solution of thelinearized problem (15) including the one fixed in the coefficients in (1).Consider once again (13) with D u ∈ L r (Ω; M N × n ) . Hence f i ∈ L q (Ω) with q = min { q, r n ( n +2) − r ( n +2) } and the associated vector-field F i belongs to L q ∗ (Ω; R n ) . Than F i − A i ∈ L r (Ω; R n ) with r = min { p, q ∗ } . Applying [8, Theorem 1.7] tosystem (15) and repeating the same procedure as above we get that the theoremholds with r ≡ r if i ) r n ( n + 2) − r ( n + 2) ≥ n or ii ) r n ( n + 2) − r ( n + 2) ≥ q. (17)Otherwise r = r n ( n +2) − r ( n +2) − r n if r n ( n + 2) − r ( n + 2) < q and r n ( n + 2) − r ( n + 2) < n (18)Repeating the same procedure k -times we get that the assertion holds if r n k ( n + 2) k − r P k − s =0 n s ( n + 2) k − − s ≥ min { n, q } . (19)Direct calculations give that (19) is equivalent to k > min (h log r r − . log n + 2 n i , h log r (2 q + qn + 2) q ( n + 2)( r − . log n + 2 n i + 1 ) where [ x ] means the integer part of x. The case n = 2 is simpler and is left to the reader. (cid:3) UASILINEAR SYSTEMS IN DIVERGENCE FORM 9 Quasilinear parabolic systems
Let Q = Ω × (0 , T ) be a cylinder in R n +1 with Ω being ( δ, R )-Reifenberg flat.Denote by C ρ the parabolic cylinder C ρ ( x, t ) = B ρ ( x ) × ( t − ρ , t ) , Q ρ ( x, t ) = Q ∩ C ρ ( x, t ) for ( x, t ) ∈ Q,a Q ρ ( x,t ) = − Z Q ρ ( x,t ) a ( y, τ ) dydτ = 1 | Q ρ ( x, t ) | Z Q ρ ( x,t ) a ( y, τ ) dydτ. Let 1 < r < ∞ and u : Q → R N .
1. The space W , r ( Q ; R N ) consists of all functions u ∈ L r ( Q ; R N ) having a finitenorm k u k rW , r ( Q ; R N ) = k u k rr,Q + k D u k rr,Q .
2. The space W ,r ∗ ( Q ; R N ) = L r (0 , T ; W ,r (Ω; R N )) ∩ W ,r (0 , T ; W − ,r ′ (Ω; R N )) ,r ′ = r/ ( r −
1) consists of the functions u ∈ W , r ( Q ; R N ) for which there exist vectorfunctions g ∈ L r ( Q ; R N ) and F ∈ L r ( Q ; M N × n ) such that u t = div F − g a.e. in Q in the sense of distributions, that is, for each vector function χ ∈ C ∞ ( Q ) with χ ( x, T ) = 0 holds Z Q u · χ t dxdt = Z Q ( F · Dχ + g · χ ) dxdt . (20)The space W ,r ∗ ( Q ; R N ) is endowed by the norm k u k W ,r ∗ ( Q ) = k u k W , r ( Q ) + inf ((cid:18)Z Q | F | r + | g | r (cid:19) /r ) where the infimum is taken over all F and g satisfying (20). The closure of C ∞ ( Q )with respect to this norm is denoted by ◦ W ,r ∗ ( Q ; R N ) . V ( Q ; R N ) stands for the Banach space of all functions u ∈ W , ( Q ; R N ) for which k u k V ( Q ; R N ) = ess sup t ∈ [0 ,T ] k u ( · , t ) k , Ω + k D u k ,Q < ∞ . V , ( Q ; R N ) consists of all u ∈ V ( Q ; R N ) that are continuous in t with respectto the norm of L (Ω; R N )lim ∆ t → k u ( · , t + ∆ t ) − u ( · , t ) k , Ω = 0 . The norm in V , ( Q ; R N ) is given by k u k V , ( Q ; R N ) = max t ∈ [0 ,T ] k u ( · , t ) k , Ω + k D u k ,Q . We consider the Cauchy-Dirichlet problem for the strongly parabolic quasilinearsystem u it − D α ( A αβij D β u j + a αi ( x, t, u )) = b i ( x, t, u , D u ) a.a. ( x, t ) ∈ Q u ( x, t ) = 0 ( x, t ) ∈ ∂Q. (21)The principal coefficients satisfy A αβij ∈ L ∞ ( Q ) and A αβij ξ iα ξ jβ ≥ ν | ξ | ∀ ξ ∈ M N × n , ν = const > . (22)In addition we suppose that A αβij are ( δ, R )-vanishing, that issup <ρ ≤ R sup ( x,t ) ∈ Q − Z Q ρ ( x,t ) | A αβij ( y, τ ) − A αβij Q ρ ( x,t ) | dydτ ≤ δ (23)which implies small BM O norm of each A αβij . The functions a αi ( x, t, u ) , b i ( x, t, u , z ) are Carath´eodory ones and verify the con-trolled growth conditions (see [2, 25]) X α ≤ n i ≤ N | a αi ( x, t, u ) | ≤ Λ( ψ ( x, t ) + | u | n +2 n )(24) X i ≤ N | b i ( x, t, u , z ) | ≤ Λ( ψ ( x, t ) + | u | n +4 n + | z | n +4 n +2 ) . (25)with ψ ∈ L p ( Q ) , p > ψ ∈ L q ( Q ) , q > n +2) n +4 . A vector function u ∈ ◦ W , ∗ ∩ L ∞ ( Q ; R N ) is a weak solution to (21) if for anyfunction χ ∈ ◦ W , ∗ ( Q ; R N ) , χ ( x, T ) = 0 we have Z Q u i ( x, t ) χ t ( x, t ) dxdt − Z Q ( A αβij ( x, t ) D β u j ( x, t ) + a αi ( x, t, u )) D α χ i ( x, t ) dxdt + Z Q b i ( x, t, u , D u ) χ i ( x, t ) dxdt = 0 . Theorem 5.
Let u ∈ ◦ W , ∗ ∩ L ∞ ( Q ; R N ) be a weak solution to (21) under theconditions (22) - (25) . Then there exists a small positive constant δ < such that if Ω is ( δ, R ) -Reifenberg flat and A αβij are ( δ, R ) -vanishing with < δ < δ then u ∈ ◦ W ,r ∗ ∩ L ∞ ( Q ; R N ) with r = min { p, q ∗∗ } where q ∗∗ = q ( n +2) n +2 − q if q < n + 2 arbitrary large number > if q ≥ n + 2 . UASILINEAR SYSTEMS IN DIVERGENCE FORM 11
Proof.
The higher integrability of the gradient of the solution follows by the modifi-cation of the Gehring lemma due to Arkhipova [4, Theorem 1] which is very efficientfor the study of parabolic systems with controlled growth conditions in domainswith boundary ∂ Ω satisfying a kind of (A)-property. Recently, similar result is ob-tained in [11] for domains having strongly Lipschitz boundary. Since the Reifenbergflat domain can be approximated uniformly with Lipschitz domains with a smallLipschitz constant [8, Lemma 5.1]) we have that there exists r > u ∈ ◦ W , ∗ ( Q ; R N ) holds k D u k r,Q ≤ N ∀ r ∈ [2 , r )where r and N depend on the data of the problem and k D u k ,Q . Take a solution u ∈ ◦ W , ∗ ∩ L ∞ ( Q ; R N ) of (21) and fix it in the lower order terms. Thus we get thelinearized problem u it − D α ( A αβij D β u j ) = f i ( x, t ) + div A i ( x, t ) a.a. ( x, t ) ∈ Q u ( x, t ) = 0 on ∂Q (26)where A i ( x, t ) = ( a i ( x, t, u ) , . . . , a ni ( x, t, u )) , A ( x, t ) = { a αi ( x, t, u ( x, t )) } α ≤ ni ≤ N f i ( x, t ) = b i ( x, t, u , D u ) , f = ( f ( x, t ) , . . . , f N ( x, t )) . Making use of the conditions (24) and (25) we get X i ≤ N k A i k p,Q ≤ C (cid:16) k ψ k p,Q + k u k n +2 n ∞ ,Q (cid:17)X i ≤ N k f i k q ,Q ≤ C (cid:16) k ψ k q ,Q + k u k n +4 n ∞ ,Q + k D u k n +4 n +2 q n +4) n +2 (cid:17) with p > , q = min n q, r ( n +2) n +4 o . Let Ω ′ ⊂ R n be C -domain such that Ω ⋐ Ω ′ andconsider the cylinder Q ′ = Ω ′ × (0 , T ) . Suppose that f i ( x, t ) is extended as zero outof Q. It is well known (cf. [25]) that for each f i ∈ L q ( Q ′ ) the linear problem F it − ∆ F i = F it − D α ( δ αβ D β F i ) = f i ( x, t ) a.a. ( x, t ) ∈ Q ′ F i ( x, t ) = 0 on ∂Q ′ has a unique solution F i ∈ ◦ W , q ( Q ′ ) and the estimate holds k F i k q ,Q ≤ k F i k ◦ W , q ( Q ′ ) ≤ C k f i k q ,Q . Denote by F = ( F , . . . , F N ) , F αi = ( A αβij − δ αβ δ ij ) D β F j and F i = ( F i , . . . , F ni ) . Since the Sobolev trace theorem holds for domain with Reifenberg flat boundary we get that F i | S ∈ W − q , − q ( S ) where S = ∂ Ω × (0 , T ) . Consider the linear system ( u i − F i ) t − D α ( A αβij D β ( u j − F j ))= D α (( A αβij − δ αβ δ ij ) D β F j ) + div ( A i )= div ( F i + A i ) a.a. ( x, t ) ∈ Q u − F = − F on ∂Q. (27)By the imbedding theorems, DF i ∈ L q ∗∗ (cf. [25, Ch.II,Lemma 3.3]) and k DF i k q ∗∗ ,Q ≤ C k f i k q ,Q Hence F i + A i ∈ L r ( Q ) with r = min { p, q ∗∗ } . Applying [7, Corollary 2.10] on thelinear problem (27) we get k D ( u − F ) k r ,Q ≤ C (cid:16) k D F k q ∗∗ ,Q + k A k p,Q (cid:17) ≤ C (cid:16) k f k q ,Q + k A k p,Q (cid:17) . Hence k D u k r ,Q ≤ C (1 + k f k q ,Q + k A k p,Q ) . Applying the bootstrapping arguments , we obtain as in the elliptic case that thetheorem holds after k iterations with k ≥ min (h log( r / ( r − n + 4) / ( n + 2)) i , h log r [ q ( n + 4) − n + 2)] q ( r − n + 4) . log n + 4 n + 2 i + 1 ) . (cid:3) Acknowledgments.
The author is indebted to the referee for the valuable remarkswhich have lead to improvement of the article.
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