INTERIOR REGULARITY FOR WEAK SOLUTIONS OF NONLINEAR SECOND ORDER ELLIPTIC SYSTEMS

Transcription

1 Proceedings of Equadiff- 2005, ISBN INTEIO EGULAITY FO WEAK SOLUTIONS OF NONLINEA SECOND ODE ELLIPTIC SYSTEMS JOSEF DANĚČEK, OLDŘICH JOHN, AND JANA STAÁ Abstract. Let divadu)) = 0 be a nonlinear ellitic system with C -matrix of coefficients. In our contribution we study the regularity of a weak solution belonging to W 2, Ω), where Ω is bounded domain in n, n 3. For d > 0 denote Ω d any subdomain of Ω with Lischitz boundary such that distx, Ω) > 2d for all x Ω d. We formulate the conditions connecting Du L 2 Ω), coefficient of elliticity ν, uer bound of derivatives of coefficients M and their modulus of continuity ω, guaranteeing Du C 0,α Ω d ). Key words. Nonlinear ellitic systems, weak solutions, regularity AMS subject classifications. 35J60, 35B65. Introduction. Let Ω n, n 3 be a bounded domain. Consider the system divadu)) = 0.) The detailed form of.) sounds like D α A α i Du)) = 0, i =, 2,..., N, where Einstein summation convention is used for α {, 2,..., n}. Suose that the matrix A = {A α i } belongs to C. egularity of the weak solution u W 2, Ω) of.) on Ω Ω is defined as Hölder continuity of Du on Ω. Denote A αβ ij ) = Aα i ), i, j =,..., N, α, β =,..., n.2) β j and write A) = {A αβ ij )} with A) for its Euclidean norm. We suose M > 0 nn : A) M,.3) ν > 0 nn ξ nn : A)ξ, ξ) νξ 2,.4) There is a function ω : 0, ) 0, ), ω0) = 0, ω nondecreasing, continuous, concave and bounded, such that, q nn : A) Aq) ω q )..5) Let further for d > 0 Ω d be any subdomain of Ω with Lischitz boundary such that distx, Ω) > 2d for all x Ω d. In what follows we establish for fixed d the conditions on ν, M, ω and Du L 2 Ω) guaranteeing that Du is Hölder continuous on Ω d. J. Daněček GAČ 20/05/2465, O. John GAČ 20/05/2465, MSMT , J, Stará GAČ 20/05/2465, GAČ 20/06/0352, MSM Technical University of Brno, Faculty of Civil Engineering, Deartment of Mathematics, Žižkova 7, Brno, Czech eublic, danecek.j@fce.vutbr.cz) Deartment of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, Praha 8, Czech eublic, John@karlin.mff.cuni.cz) Deartment of Mathematical Analysis, Charles University, Sokolovská 83, Prague 8, Czech eublic stara@karlin.mff.cuni.cz ) 65

2 66 J. Daněček, O. John and J. Stará 2. Basic estimate for weak solution. Algebraic lemma. In this section we reare needed estimates and formulate an algebraic lemma following the rocedure known from the deduction of artial regularity results. See e.g. Giaquinta []) Let u be a weak solution of.), x Ω d, 0, d. Denote A 0 = ADu) ), Ã = 0 [ADu) ) ADu) + tdu Du) )] dt, where Du) = = {y n ; y x < }. Using this notation we can rewrite.) as Slit u = v + w in a way that Dux) dx, diva 0 Du) = div[a 0 Ã)Du Du) )] on 2.) diva 0 Dv) = 0 on, v u W 2, 0 ), 2.2) diva 0 Dw) = div[a 0 ÃDu Du) )], w W 2, 0 ). 2.3) On the function v we can use Camanato s lemma saying that C > 0 ρ 0, ) : Dv Dv) ρ 2 C ρ )n+2 Dv Dv) ) As for w, we can use it in a weak formulation of 2.3) as a test function. By means of elliticity condition.4), Hölder inequality and the estimate.5) of A 0 Ã by the modulus of continuity ω we obtain ν B 2 Dw 2 ω 2 Du Du) ) Du Du) ) emark. In our notation we suress the deendence on x Ω d so that instead of writing x) we write etc. This simlification does not make any harm because 2.4), 2.5) work in Ω d uniformly. Using 2.4), 2.5) and taking account in the fact that u = v + w we obtain finally the estimate for u. With use of the notation Φρ) = Du Du) ρ 2 2.6) it reads as C, D deending on M/ν) x Ω d ρ : 0 < ρ < d 2.7) ρ ) n+2 D Φρ) C Φ) + ν 2 ω 2 Du Du) ) Du Du) ) For the function Φ denote U) = n Φ). Lemma 2. Algebraic lemma). Let A > 0, d > 0, β > 0 and δ n, n + 2) be given. There exist ε 0, C > 0 such that for each nonnegative nonincreasing function Φ defined on 0, 2d) satisfying the estimate ρ ) n+2 Φρ) A + B + B 2 U2)) Φ2), 0 < ρ < d 2.9)

3 with B < ε 0, B 2 U β 2) < ε 0 it holds Interior regularity 67 Φρ) C ρ δ, 0 < ρ < d. 2.0) If in accordance with this lemma we are able to estimate D ν 2 ω 2 Du Du) ) Du Du) 2 B + B 2 U β 2))Φ2) 2.) for 0 < < d with B, B 2 U β 2d) sufficiently small, we obtain 2.0) with the function Φ given in 2.6). It can be rewritten as ρ δ Du Du) ρ 2 C, x Ω d, ρ 0, d). 2.2) As this estimate is uniform with resect to x in Ω d, we conclude that Du belongs to the Camanato sace L 2,δ Ω d ). From the theorem of isomorhism between Camanato and Hölder saces see e.g. Kufner, John, Fučík [2]) we can conclude that Du C 0,δ n)/2 Ω d ). 3. Deduction of estimate 2.). Denote I = ω 2 Du Du) ) Du Du) 2 3.) With use of Young inequality for the coule of Young functions Θt) = t, sq Ψs) = q, where < n n 2, q = 3.2) we can write for any ositive ε I Θε Du Du) 2 ) + Ψ ε ω2 Du Du) ) = ε Du Du) 2 + ε ω 2 Du Du) ) = ε I + ε 2 I2. 3.3) We followed here the idea of the aer [3], where Young functions of different kind were used.) Estimate now the first integral. Using successively Hölder inequality, Sobolev embedding theorem and Cacciooli inequality we obtain I ) n 2) c Du Du) 2n n n 2 n )+2 ) c D 2 u n )+2 B ) c 2 Du Du) 2 2 n )+2 B 2 = cφ2) U 2). 3.4)

4 68 J. Daněček, O. John and J. Stará As for the second integral I 2, denoting we have E t = { x ; Dux) Du) > t } I 2 = ω 2 Du Du) ) dx = 0 d 2 su [ω t)] Du Du) t>0 dt In the last inequality denote + d 2 [ω t)] Et dt dt 3.5) d 2 ω = su [ω t)] 3.6) t>0 dt and use Hölder inequality once again. So we get in case of U2) 0) I 2 c ω Du Du) 2 ) n 2 2 c ω Φ2)U 2 2) 3.7) From 3.4), 3.7) and 3.3) we get I c ε U 2) + ) ε ω U 2 2) ) Φ2) 3.8) The otimal choice of ε which minimizes the exression on the right hand side of 3.8) is Using this, we have finaly ε = ω 2 I c ω U 2) ) 2 2 2). U 2 2) Φ2). 3.9) In case of U2) = 0 this estimate is trivial.) Coming back to 2.) we ut there B = 0 and take care of the smallness of the roduct ν 2 ω U 2 2). 3.0) From this and from the fact that the exression U2d) can be estimated by Du 2 L 2 Ω) /dn we conclude that the algebraic lemma can be alied if ν 2 ω Du 2 ) L 2 2 Ω) d n 3.) is sufficiently small. emark 2. Coming back to 2.8) we can see that in the case of ω = su t>0 ω 2 t) sufficiently small we can derive the estimate of the tye 2.)which does not deend on d so that we obtain the usual interior regularity in Ω. On the other hand, it is easy to construct an examle of modulus of continuity ω for which ω is big and ω for some, n n 2 ). is small Let = n n 2, T > 0, m > 0. Define ωt) = mt n for t 0, T and mt n for t > T. For this choice of ω we calculate ω = m 2 T 2 n, meanwhile ω n n = m 2. n 2

5 Interior regularity 69 EFEENCES [] M. Giaquinta, Multile integrals in the calculus of variations and nonlinear ellitic systems. Annals of Mathematics Studies 05, Princeton university ress, Princeton 983. [2] A. Kufner, O. John, S. Fučík, Function saces, Academia, Prague 977. [3] J. Daněček, O. John, J. Stará, Interior C,γ -regularity for weak solutions of nonlinear ellitic system. Math. Nachr., ),