A NOTE ON VERY WEAK SOLUTIONS FOR A CLASS OF NONLINEAR ELLIPTIC EQUATIONS

Size: px

Start display at page:

Download "A NOTE ON VERY WEAK SOLUTIONS FOR A CLASS OF NONLINEAR ELLIPTIC EQUATIONS"

Giles Fleming
5 years ago
Views:

1 SARAJEVO JOURNAL OF MATHEMATICS Vol , A NOTE ON VERY WEAK SOLUTIONS FOR A CLASS OF NONLINEAR ELLIPTIC EQUATIONS LI JULING AND GAO HONGYA Abstact We pove a new a pioi estimate fo vey weak solutions of a class of nonlinea elliptic equations 1 Intoduction Let be a bounded egula domain in R n We conside vey weak solutions w W 1, 0, Rm with > max{1, p 1} of the nonlinea system div g + w p 2 g + w + bx = 0 1 whee g L, R m n, bx L p 1, R m, the natual exponent p > 1 thoughout in this pape Equation 1 is undestood in the weak sense, that is g + w p 2 g + w ϕdx + bxϕxdx = 0 2 fo evey ϕ W 1, p+1 0, R n Iwaniec Sbodone studied the p hamonic system in [2] div u p 2 u = 0 3 They have shown that thee exist 1 = 1 p, m, 2 = 2 p, m,, satisfying 1 < 1 < p < 2 < such that evey vey weak p hamonic mapping u W 1, 1 loc, Rm belongs to W 1, 2 loc, Rm They conjectued Conjectue 1 in [2] that evey 1 > max{1, p 1} would do fo the egulaity esult fo the p hamonic system, but thei estimate fo 1 was vey close to p 2000 Mathematics Subject Classification 30C62, 35J60 Key wods phases Vey weak solution, Hodge decomposition, egula domain The fist autho is suppoted by youngteache s foundation of noth China electic powe univesity

2 42 LI JULING AND GAO HONGYA Juha Kinnunen Shulin Zhou studied the following nonhomogeneous system in [3] div g + w p 2 g + w = div h 4 whee g L, R m n h L p 1, R m n ae matix fields They have shown that if wx W 1, 0, Rm with > max{1, p 1} is the vey weak solution of 4, can be chosen abitaily close to one when p is close to two The objective of ou note is to study equation 1 we get a esult simila to the one of [3] 2 Main esults Fo the convenience of the eade we ecall the fomulation of the Hodge decomposition Theoem 3 in [2] Lemma 1 Let be a egula domain in R n w W 1, > 1 let 1 < ε < 1 Then thee exist φ W 1, 0, Rm with 1+ε 0, R m a divegence fee matix field H L 1+ε, R m n such that w ε w = φ + H 5 H 1+ε, C, m ε w 1+ε, 6 Remak Fot the definition of a egula domain see [2], thee we notice that balls cubes ae egula domains in R n The following Lemma comes fom [4] Theoem 23 Lemma 2 Poincae theoem Let be a bounded domain in R n Then thee exists a constant C = Cp, n, < such that fo evey u W 1,p 0 u p, C u p, 7 Theoem 1 Let > max{1, p 1} suppose that w W 1, 0, Rm, with satisfying 1 Then thee exists δ = δm, > 0 such that if max{ p 2, 1 } < δ, then w dx Cp, m,, g + bx p 1 dx 8 Poof Using Lemma 1 with ε = p, we obtain functions φ 1 W 1, 0, R m H 1 L p+1, R m n such that p+1 w p w = φ 1 + H 1 9

3 A NOTE ON VERY WEAK SOLUTIONS FOR A CLASS OF 43 H 1 ϕdx = 0, fo evey ϕ W 1, p 1 0, R m 10 H 1 p+1, C 1 p w p+1,, C 1 = C, m 11 In paticula, we have φ 1 p+1, C p w p+1, 12 Since φ 1 can be used as a test function in 2, we obtain g + w p 2 g + w φ 1 dx + bxφ 1 xdx = 0 Inseting 9 we aive at w dx = w p 2 w H 1 dx + bxφ 1 xdx w p 2 w g + w p 2 g + w φ 1 dx = I 1 + I 2 + I 3 13 We begin with estimating I 1 By using Lemma 1 again with ε = p 2, we obtain φ 2 W 1, p 1 0, R m H 2 L p 1, R m n such that w p 2 w = φ 2 + H 2 14 H 2 ϕdx = 0, fo evey ϕ W 1, p+1 0, R m 15 H 2 p 1, C 1 p 2 w p 1,, C 1 = C, m 16 Using 14, 15, 9, 10 16, we have I 1 = φ 2 + H 2 H 1 dx = H 1 H 2 = w p w φ 1 H 2 dx = w p w H 2 dx C 1 p 2 w,

4 44 LI JULING AND GAO HONGYA The same easoning shows that hence I 1 C 1 p w, I 1 C 1 min{ p 2, p } w, 17 Then we estimate I 2, by Hölde inequality, we have p 1 I 2 = bx φ 1 dx bx p 1 dx φ 1 Since φ 1 W 1, p+1 0, R m, using Lemma 2 hence φ 1 p+1, C φ 1 p+1, I 2 C bx p 1, φ 1 p+1, C bx p+1 dx p 1, w p+1 p+1, 18 By vitue of 12, we may estimate I 3 in the same way as in [2], by using the Lipschitz popety of w p 2 w we have I 3 C g w + g p 2 φ 1 dx C g, g + w p 2, φ 1 C g, g + w p 2 C g p 1, w p+1, p+1,, w p+1, + g, w 1, 19 Using 13, 17, we get 1 C 1 min{ p 2, p } w, C bx p 1, w p+1, + g p 1, w p+1, + g, w 1, The only point emaining is to sepaate w fom the tems in the ight h side This can be done outinely with the aid of Young s inequality We continue in this fashion obtaining the estimate, fo evey θ > 0 1 C 1 min{ p 2, p } θ w, C θ g, + b p 1 p 1, In paticula, if C 1 p 2 < 1, then 2 holds Estimates fo the constant C 1 = C, m can be found in [1] fomula 11 in [2] Using these estimates it is easy to see that we may choose C 1 = Cm, p,, This completes the poof of the theoem

5 A NOTE ON VERY WEAK SOLUTIONS FOR A CLASS OF 45 Refeences [1] T Iwaniec, p hamonic tensos quasiegula mapping, Anal Math, , [2] T Iwaniec C Sbodone, Weak minima of vaiational integals, J Reine Angew Math, , [3] Juha Kinnunen Shulin Zhou, A note on vey weak p-hamonic mapping, Electon J Diffe Equ, No , 1 4 [4] Yu G Reshetnyak, Space Mapping with Bounded Distotion, Tanslation of Mathematical Monogaphs, 73, Ameican Mathematical Society, Povidence, Rhode Isl, 1989 [5] D Giachetti, F Leonetti R Schianchi, On the egulaity of vey weak minima, Poc R Soc Edinb, Sect A, , Received: Septembe 28, 2005 Revised: Septembe 23, 2006 Li Juling Depatment of Applied Mathematics Noth China Electic Powe Univesity Baoding , PRChina E mail: ljul@sohucom Gao Hongya Depatment of Mathematics Hebei Univesity Baoding , PRChina E mail: hongya-gao@sohucom

A STABILITY RESULT FOR p-harmonic SYSTEMS WITH DISCONTINUOUS COEFFICIENTS. Bianca Stroffolini. 0. Introduction

A STABILITY RESULT FOR p-harmonic SYSTEMS WITH DISCONTINUOUS COEFFICIENTS. Bianca Stroffolini. 0. Introduction Electonic Jounal of Diffeential Equations, Vol. 2001(2001), No. 02, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.swt.edu o http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) A STABILITY RESULT