J. Partial Diff. Eqs. 5(2002), 7 2 c International Academic Publishers Vol.5 No. ON THE W,q ESTIMATE FOR WEAK SOLUTIONS TO A CLASS OF DIVERGENCE ELLIPTIC EUATIONS Zhou Shuqing (Wuhan Inst. of Physics and Math., Chinese Acad. Sci., Wuhan 43007; Det. of Math of Hunan Norm. Univ., Changsha 4008, China) (E-mail: zhoushuqing97@263.net) Deng Songhai (Det. of Math of Xiangya Med. Inst. in Mid-east Univ., Changsha 40078, China) Li Xiaoyong (Det. of Math of Hunan Norm. Univ., Changsha 4008, China) (Received Mar. 9, 200) Local W,q estimates for weak solutions to a class of equations in di- Abstract vergence form D i (a ij (x) Du 2 D j u) = 0 are obtained, where q > is given. These estimates are very imortant in obtaining higher regularity for the weak solutions to ellitic equations. Key Words Divergence ellitic equation; local W,q estimate; reverse Hölder inequality. 2000 MR Subject Classification 35J65. Chinese Library Classification 075.25.. Introduction Using comactness method, Avellanda and Lin Fanghua in [] obtained L theory for ellitic systems of eriodic structure L ε = [ x α A αβ ij (x ε ) ] x β = f. Using the results in [], they in [2] also obtained C 0,α, C,α and C 0, regularity for homogenization roblem: n i,j= a ij ( x ε ) 2 u ε x i = f(x), x D, xj u ε (x) = g(x), x D, This roject is suorted by the National Natural Science Foundation of China.
8 Zhou Shuqing and Deng Songhai and Li Xiaoyong Vol.5 under certain conditions, where ε > 0, D is smooth domain in R n. Using Calderón- Zygmund decomositions theorem [3] and measure theory [4], Caffarelli and Petal in [5] established a determinant theorem for the weak solutions which have higher integrability to a class of homogenization roblems, and using this theorem, the authors obtained higher integrability for weak solutions to equations div(a(x, Du)) = 0, () then using this result, the authors obtained corresonding results for homogenization roblem with eriodic structure in [] and [2]. By the method different from that in [-2] and [5], Kileläinen and Koskela [6] obtained global integrability for the weak solutions to the equation (). Li Gongbao and Martio [7] obtained local and global integrability for the gradient of the weak solutions to the equation (). They also in [8] obtained that the weak solution to the equation () with very weak boundary value is exclusive. The L estimates established in [] layed crucial role in obtaining the results in [2]. But Caffarelli and Petal in [5] didn t obtain corresonding L estimates. In this aer, we discuss the weak solutions in W, to the following equation D i (a ij (x) Du 2 D j u) = 0. (2) Using the method in [5], we obtain L q integrability for the gradient of the weak solutions to the equation (2),where q is given to be bigger than, then establish the reverse Hölder inequality for the equation (2) by the method in [9] and [0], and obtain local W,q estimate for weak solutions to the equation (2). 2. W,q Estimate In this section, we discuss the weak solution in W, to the ellitic equation of divergence structure D i (a ij (x) Du 2 D j u) = 0, (3) where, a ij satisfies: λ ξ 2 a ij (x)ξ i ξ j Λ ξ 2, (4) where, λ, Λ > 0 are constants. We have the following theorem and corollary: Theorem 2. Suose q is bigger than ; if there exists ɛ > 0, a(x) I ɛ, (5) where a(x) = (a ij ), I is identical matrix and if u W, is a weak solution to the equation (3), then W,q loc (Ω), and for R, Ω, ( Du q + u q )dx 2 q [ ( Du + u )dx ], (6)
No. On the W,q estimte for weak solutions to... 9 where is a ball centered in x, with radius R, here, udx = udx. Corollary 2.2 If a ij is continuous and u W, is a weak solution to the equation (3), then q > 0, u W,q loc. 3. Some Preliminary Lemmas and Proof of Theorem 2. To rove Theorem 2., we first discuss the weak solution to -harmonic function, i.e. -Lalacian u div( Du 2 Du) = 0. (7) Lemma 3. [5] Suose u is a -harmonic function,, 2 are cubes with same center, while the length is different in Factor two. Then Du L () C(n, ) Du dx. (8) 2 We give a roof different from that in [5] and []. Proof Denote the length of by l. Let R = 5 2 2 l. Let denote the ball with the same center as the cube, and with radius R. We consider the following Dirichlet roblem: Du 2 Du Dϕdx = 0, x 2, ϕ W, 0 ( ), 2 u = 0, x \ 2. By [0], 0 < ρ < R, we have B ρ Du dx C( ρ R )n 2 Du dx. (9) Then by Theorem. in Chater 3 in [2], for 0 < ρ < R, u C 0, (B ρ ), furthermore, for all x, y, x y, u(x) u(y) C(n, )( Du dx). (0) x y Let y x in (0), we obtain Du L (Ω) B C(n, ) Du dx = C(n, ) R 2 2 Du dx. () Lemma 3.2 Suose u W, is a weak solution to the equation (3), and for some, Du dx λ. (2) Let u h be a solution to Dirichlet Problem { u h div( Du h 2 Du h ) = 0, x, (3) u h = u, x and suose (5) holds. Then
0 Zhou Shuqing and Deng Songhai and Li Xiaoyong Vol.5 where α = Du h dx Du dx, (4) D(u u h ) dx Cɛ α when 2 N; α = when < < 2. Du dx, (5) Proof Using ϕ = u u h as a testing function in the definition of weak solution, we immediately obtain (4). We now rove (5). When 2 N, by Proosition 5. in [3] and (5), we obtain D(u u h ) dx Du 2 Du Du h 2 Du h, Du Du h dx = Du 2 Du, Du Du h dx = C (I a(x)) Du 2 Du, Du Du h dx + C a(x) Du 2 Du, Du Du h dx = C (I a(x)) Du 2 Du, Du Du h dx Cɛ ( ) ( ) Du dx D(u u h ) dx, (6) from which we get (5). When < < 2, by Proosition 5.2 in [3] and (5) and (4), calculating as before, we obtain D(u u h ) dx C C C ( ( ( Du + Du h ) 2 2 ) 2 ( ( Du + Du h 2 )dx ( ) Du 2 Du Du h 2 Du h, Du Du h 2 dx ) Du 2 Du Du h 2 2 Du h, Du Du h dx ) 2 [ ( ) ( Du 2 dx ɛ Du dx D(u u h ) dx from which we obtain D(u u h ) dx Cɛ 2 ( ) ] 2, (7) ) ( ) Du 2 dx D(u u h ) 2 dx, (8)
No. On the W,q estimte for weak solutions to... therefore, (5) also holds when < < 2. We now rove Theorem 2.: By Lemma 3., Lemma 3.2 and Theorem A in [5], we obtain that u W,q loc. We now rove the estimate (6) holds. Choose a ball Ω, η a standard cut-off function, choose ϕ = η (u u R ), where u R = udx, as a testing function in (3). Using (4) and (5), we obtain λ η Du dx B R C η a ij Du 2 D i ud j udx B R = η a ij (u u R ) Du 2 D i ud j udx ( + ɛ) η Du dx + ( + ɛ)θ ( ) Dη u u h dx. (9) By Choosing θ sufficiently small, (9) imlies Du dx CR 2 Choosing such that max{, interolation theorem, we obtain u u R dx. (20) n n + } < <, from (20) and Hölder inequality and 2 Du dx CR [ ( u u R dx ) ] while 2 ( ) C Du dx, (2) ( ) u dx CR Du dx + C u dx. (22) Adding (22) to (2), we obtain ( ( Du + u )dx CR 2 ( Du + u )dx + C ( Du ) + u )dx. (23) Letting g = Du + u, and choosing R 0 sufficiently small such that θ = CR < CR 0 <, we get M ( ) d(x)(g) (x) C M d(x) (g)(x) + θm d(x) (g) (x), (24) 2
2 Zhou Shuqing and Deng Songhai and Li Xiaoyong Vol.5 where M d(x) (f)(x) is local maximum function of f(x), d(x) R 0, thus by Proosition. in Chater 5 in [2], there exists q such that, for t [, q ), u W,t loc and ( Du t + u t )dx 2 t [ C ( Du + u )dx ]. (25) The first art of the theorem shows that q >, thus the estimate (6) holds. Acknowledgement: The authors are grateful for their tutor Fang Ainong for his careful tuition. References [] Avellanda M. and Lin Fanghua, Comactness method in the theory of homogenization, Comm. Pure Al. Math., 40(987), 803-847. [2] Avellanda M. and Lin Fanghua, Comactness method in the theory of homogenization II, Comm. Pure Al. Math., 42(989), 39-72. [3] Stein E. M., Singular Integrals and Differentiability Proerties of Functions(in Chinese, translated by Minde C., etc.), Press of Beijing Univ., 986. [4] Halmos P. R., Measure Theory, Sringer-Verlarg, World Books Publishing House, 998. [5] Caffarelli L. A. and Peral I., On W, estimates for ellitic equatins in divergence form, Comm. Pure Al. Math., ()(998), -2. [6] Kileläinen T. and Koskela P., Global integrability of the gradients of solutions to artial differential equations, Nonlinear Analysis, TMA, 23(7)(994), 899-909. [7] Li Gongbao and Martio O., Local and global integrability of gradients in obstacle roblem, Acad. Sci. Fenn. Ser. A. I. Math., 9(994), 25-34. [8] Li Gongbao and Martio O., Uniqueness of solutions with very weak boundary values, Nonlinear Analysis, TMA,(to aear in). [9] Gehring F. W., The L -integrability of the artial derivatives of quasi-conformal maings, Acta Math., 30(973), 265-277. [0] Choe H. J., Regularity theory for a general class of quasi-linear ellitic differential equations and obstacle roblems, Arch. Rat. Mech. Anal., 4(99), 383-394. [] Dibenetto E., C +α local regularity of weak solutions of degenerate ellitic equations, Nonlinear analysis, TMA, 7(8)(983), 827-850. [2] Giaquinta, M., Multile Integrals in the Calculus of Variations and Nonlinear Ellitic Systems, Princeton Univ. Press, Princeton, New Jersey, 983. [3] Glowinski R. and Marroco A., Sur l aroximation, ar elements finis d ordre un, et la resolution, ar enalisation-dualite, d une classe de roblems Dirichlet non linearies, RAIRO, 975, R-2; 4-76.