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

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1 Electonic Jounal of Diffeential Equations, Vol. 2001(2001), No. 02, pp ISSN: URL: o ftp ejde.math.swt.edu (login: ftp) A STABILITY RESULT FOR p-harmonic SYSTEMS WITH DISCONTINUOUS COEFFICIENTS Bianca Stoffolini Abstact. The pesent pape is concened with p-hamonic systems div( A(x)Du(x),Du(x) p 2 2 A(x)Du(x)) = div( A(x)F (x)), whee A(x) is a positive definite matix whose enties have bounded mean oscillation (BMO), p is a eal numbe geate than 1 and F L is a given matix field. We find a-pioi estimates fo a vey weak solution of class W 1,,povidedis close to 2, depending on the BMO nom of A,andpclose to. This esult is achieved using the coesponding existence and uniqueness esult fo linea systems with BMO coefficients [St], combined with nonlinea commutatos. Conside the p-hamonic system 0. Intoduction div( Du(x) p 2 Du(x)) = 0 (0.1) in a egula domain R n. A vecto field u in the Sobolev space W 1, loc (, Rn ), >max{1,}, is a vey weak p-hamonic vecto [IS1],[L] if it satisfies Du p 2 Du, Dφ dx =0 φ Co (, Rn ). This definition was fist intoduced by Iwaniec and Sbodone in [IS1], they wee able to pove, using commutato esults, that thee exists a ange of exponents, close to p, 1< 1 <p< 2 <, such that if u W 1, 1 loc (, Rn ) is vey weak p-hamonic, then u belongs to W 1, 2 loc (, Rn ), so, in paticula, is p-hamonic. J. Lewis [L], using that the maximal functions aised to a small positive powe is an A p weight in the sense of Muckenhoupt, was able to obtain simila esults. Kinnunen and Zhou [KZ] gave a patial answe to a conjectue posed by Iwaniec and Sbodone; they poved that 1 can be chosen abitaly close to 1, if p is close to 2. Late Geco and Vede developed the same esult fo p-hamonic equations with VMO L coefficients, Mathematics Subject Classifications: 35J60, 47B47. Key wods: Bounded mean oscillation, Linea and Nonlinea Commutatos, Hodge Decomposition. c 2001 Southwest Texas State Univesity. Submitted Novembe 27, Published Januay 2,

2 2 Bianca Stoffolini EJDE 2001/02 using estimates fo linea elliptic equations with VMO coefficients [D],[IS2]. Ou esult is concened with p-hamonic systems with BMO coefficients: div( A(x)Du(x),Du(x) p 2 2 A(x)Du(x)) = div( A(x)F (x)) (0.2) whee A(x) = (A ij (x)) is a symmetic, positive definite matix with enties in BMO, F is a given matix field in L. Ou definition of vey weak p-hamonic vecto a pioi equies that the enegy functional is finite along a solution, that is: ADu dx < a closed subspace of W 1, 0 (, R n ), in addition fo evey φ C0 (, Rn ) ADu p 2 ADu, ADφ dx = F (x), ADφ dx (0.3) We will use the existence and uniqueness esult fo linea systems with bounded mean oscillation (BMO) coefficients to deive a new Hodge decomposition fo matix fields and, then, using commutatos, we will pove a continuity esult fo p close to 2, depending on the BMO-nom of A. The method of poof is diffeent fom the linea case; in fact, thee we have at ou disposal two commutato esults: one is a powetype petubation of the kenel of a linea bounded opeato, the othe is the Coifman-Rochbeg-Weiss esult about the linea commutato of a Caldeon-Zygmund opeato with a BMO matix. In the nonlinea case, we do not know of a esult fo nonlinea commutatos with a BMO function, so we can only use the commutato esult of powetype, applied to the natual Hodge decomposition coming fom the linea case. The statement is the following: Main Theoem. Fo given in such a way that 2 < ε, detemined by the BMO-nom of A, thee exists δ>0such that if p <δand u is a vey weak p-hamonic vecto, then ADu C F (0.4) Futhe developments ae pesented consideing some new spaces, the so-called gand L q spaces, in the spiit of [GIS]. 1. Definitions and peliminay esults Definition 1. Let be a cube o the entie space R n. The John-Nienbeg space BMO() [JN] consists of all functions b which ae integable on evey cube Q and satisfy: whee b Q = 1 Q Q b(y)dy. { 1 } b =sup b b Q dx : Q < Q Q

3 EJDE 2001/02 p-hamonic systems 3 Definition 2. Fo 1 <q< and 0 θ< the gand L q -space, denoted by L θ,q) (, R n n ), consists of matices F 0<ε q 1 Lq ε (, R n n ) such that F θ,q) = sup ε θ q F q ε < 0<ε q 1 These spaces ae Banach spaces, they wee intoduced fo θ = 1 in the study of integability popeties of the Jacobian [IS1] and wee used in [GISS] to establish a degee fomula fo maps with non-integable Jacobian. Definition 3. The gand Sobolev space W θ,p 0 (, R n ) consists of all vecto fields u belonging to 0<ε W 1,p ε 0 (, R n ) such that Du L θ,p (, R n n ); a nom on this space is Du θ,p). Next, we ecall a stability esult fo nonlinea petubation of a kenel of a bounded linea opeato; namely: T δ f = T ( f δ f), whee T : L p (,E) L p (,E) is a bounded linea opeato and E is a Hilbet space. Theoem 1. Let T : L p (,E) L p (,E) be a bounded linea opeato fo all p 1 p p 2 ; then fo 1 p 2 p δ 1 p 1 p thee is a constant C = C( T p1, T p2 ) such that if f belongs to the kenel of T, we get T ( f δ f) p C δ f p (1.1) A new Hogde decompostion. Conside a linea system with BMO coefficients: div(b(x)du(x)) = div F (x) whee B(x) is a symmetic, positive definite matix whose enties ae in BMO, F is a given matix field. We state the following existence and uniqueness esult fo the solution of the Diichlet poblem: Theoem 2. [St] Thee exists ε>0, depending on the BMO-nom of B, such that fo 2 < εthe Diichlet poblem: div(bdu)=divf F L (, R n n ), u Wo 1, (, R n ) admits a unique solution. In paticula the enegy functional Du ε B(x)Du, Du dx is finite and the following a-pioi estimate holds (1.2) Du C F (1.3) Remak. Note that, taking into account the unifom estimate (1.3) fo exponents in a ange detemined by the BMO-nom of B, we have actually existence and uniqueness in the gand Sobolev space W θ,2) 0 (, R n ). This Theoem can be ephased in tems of a new Hodge decomposition. Moe pecisely,

4 4 Bianca Stoffolini EJDE 2001/02 Theoem 2. Thee exists ε>0, depending on the BMO-nom of B, such that fo 2 <εamatixfieldf L (, R n n ) can be decomposed uniquely as it follows: F = BDφ + L with div L = 0 and φ W 1, o (, R n ). Theefoe, thee exists a bounded linea opeato S : L (, R n n ) L (, R n n ) given by S(F )=BDφ. It is sufficient to solve the linea system div(bdφ) =divf We will apply Theoem 1 to the opeato T = I S with B = A.Noticethat the squae oot opeato acting on matices with minimum eigenvalue fa fom zeo, fo example geate o equal than 1, is Lipschitz, theefoe the squae oot of A is still in BMO. The kenel of the opeato T consists of matix fields of the fom ADφ. 2. Poof of the Main Theoem Conside a vey weak p-hamonic vecto u W 1,,withdetemined by Theoem 2 and with finite enegy. Decompose ADu p ADu using the new Hodge decomposition: ADu p ADu = ADφ + L, div L =0 Let us obseve that T ( ADu) = 0; theefoe L is a nonlinea petubation of the kenel of a bounded linea opeato; we can apply Theoem 1 with δ = p to get the following estimate L Using the above equality we find Du dx ADu = C δ ADu (2.1) ADu p 2 ADu, L dx + F, ADφ dx. Using Hölde s inequality on the last two tems of the above expession and (2.1), ADu dx ADu L + F ADφ p+1 p+1 C p ADu + C F ADu p+1 Using Young s inequality and choosing such that C p < 1, we get the assetion. We will pove also the uniqueness of the vey weak p-hamonic vecto in a space lage than W 1,, efining estimate (0.4). We begin with establishing the following Theoem, that fo the p-hamonic case was established in [GIS].

5 EJDE 2001/02 p-hamonic systems 5 Theoem 3. Fo given in such a way that 2 <ε, detemined by Theoem 2, thee exists δ such that if p <δand u, v W 1, (, R n ) ae vey weak p-hamonic vectos espectively with data F, G L (, R n n ) with finite enegy, the following estimate holds: ADu ADv Cε p 2 ( F + G )+C { F G F G ( F + G )2 p (p 2) (1 <p<2) (2.2) Poof. Take u W 1, (, R n ) with finite enegy, a vey weak solution of the equation: div( ADu p 2 ADu) =div( AF ) (2.3) and v W 1, (, R n ) with finite enegy, a vey weak solution of Conside the Hodge decomposition of we have estimates: div( ADv p 2 ADv) =div( AG) (2.4) ADu ADv p ( ADu ADv) = ADφ + L ADφ L C ADu ADv C δ ADu ADv We can use ADφ as test function in (2.3) and (2.4) and subtact the two equations, to obtain: ADu p 2 ADu ADv p 2 ADv, ADu ADv p ( ADu ADv) = F G, ADφ + ADu p 2 ADu ADv p 2 ADv, L, and ( ADu + ADv ) p 2 ADu ADv 2 p+ C(p) F G ADφ + C(p) ( ADu + ADv ) p 2 ADu ADv L. Now, using Hölde s and Young s inequalities we get the assetion. This Theoem is the key to pove uniqueness of the solution of (0.2) in the gand Sobolev space W θ,p 0 when the ight-hand side is in a gand L θ,q) space. We state the following uniqueness Theoem.

6 6 Bianca Stoffolini EJDE 2001/02 Theoem 4. Fo each F L θ,q) (, R n n ) with q the Hölde conjugate of p, and pin the ange detemined by Theoem 2, the p-hamonic system (0.2) may have at most one solution in the closed subspace of W θ,p (, R n ): E θ,p = {u W θ,p (, R n ): ADu θ,p) < } and we get the unifom estimate fo the opeato H : L θ,q) (, R n n ) E θ,p that caies F into ADu: whee HF HG θ,p) C(n, p, A ) F G α θ,q) ( F θ,q) + G θ,q) ) 1 α (2.5) α = { p θ(p 2) p if p 2 p+θ(p 2) if p 2 q If, in addition, A is in L, we get existence. In fact, given F L θ,q) (, R n n ), we conside a convolution F k with a standad mollifie; the appoximations F k convege to F in L θ,q) (, R n n ) fo evey θ >θ. Next, solve the p-hamonic system: div( A(x)Du k (x),du k (x) p 2 2 A(x)Duk (x)) = div( A(x)F k (x)) fo u k W 1,p 0 (, R n ). We use estimate (2.5) with θ in place of θ to show that u k is a Cauchy sequence in W θ,p) 0 (, R n ): ADu k ADu j θ,p) C(n, p, A ) F k F j α θ,q) ( F k θ,q) + F j θ,q)) 1 α Passing to the limit in the integal identities: ADu k p 2 ADu k, ADφ dx = we then conclude that the limit u is in W θ,p 0 (, R n ). Refeences F k (x), ADφ dx [D] G. Di Fazio, L p estimates fo divegence fom elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. 7 (1996), [GIS] L. Geco, T. Iwaniec and C. Sbodone, Inveting the p-hamonic Opeato, Manuscipta Mathematica 92 (1997), [GISS] L. Geco, T.Iwaniec, C. Sbodone and B. Stoffolini, Degee fomulas fo maps with nonintegable Jacobian, Topological Methods in Nonlinea Analysis 6 (1995), [GV] L. Geco and A. Vede, A egulaity popety of p-hamonic functions, Annales Academiae Scientiaum Fennicae (to appea). [I] T.Iwaniec, Nonlinea Diffeential Foms, Lectues in Jyväskylä,, August [IS1] T. Iwaniec and C. Sbodone, Weak minima of vaiational integals, J.Reine Angew.Math. 454 (1994), [IS2] T. Iwaniec and C. Sbodone, Riesz tansfom and elliptic PDEs with VMO-coefficients, Jounal d Analyse Math. 74 (1998), [KZ] J. Kinnunen and S. Zhou, A note on vey weak P -hamonic mappings, Electonic Jounal of Diffeential Equations, 25 (1997), 1 4.

7 EJDE 2001/02 p-hamonic systems 7 [JN] F. John and L. Nienbeg, On functions of bounded mean oscillation, Comm. Pue Appl. Math. 14 (1961), [L] J. Lewis, On vey weak solutions of cetain elliptic systems, Comm. Pat. Diff. Equ. 18 (1993), [S] E. M. Stein, Hamonic Analysis, Pinceton Univesity Pess, [St] B.Stoffolini, Elliptic Systems of PDE with BMO coefficients, Potential Analysis (to appea). Bianca Stoffolini Dipatimento di Matematica e Applicazioni R. Caccioppoli, via Cintia, Napoli, Italy addess: stoffol@matna2.dma.unina.it

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

SARAJEVO JOURNAL OF MATHEMATICS Vol3 15 2007, 41 45 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