CLASSES OF ELLIPTIC MATRICES

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1 CLASSES OF ELLIPTIC MATRICES ANTONIO TARSIA Received 12 December 2005; Revised 20 February 2006; Acceted 21 February 2006 The equivalence between some conditions concerning ellitic matrices is shown, namely, the Cordes condition, a generalized form of Camanato s condition, and a generalized form of a condition of Buică. Coyright 2006 Antonio Tarsia. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. 1. Introduction Let Ω be an oen bounded set in R n, n>2, with a sufficiently regular boundary, and let A(x ={a ij (x} i,j=1,...,n be a real matrix, with coefficients a ij L (Ω. We consider the following roblem: u H 2,2 H0 1,2 (Ω, n a ij (xd ij u(x = f (x, a.e. x Ω. i,j=1 (1.1 If f L 2 (Ω, it is known (see the counterexamles in 6] that roblem (1.1isnotwell osed with the only hyothesis of uniform elliticity on the matrix A(x: there exists a ositive constant ν such that n a ij (xη i η j ν η 2 n, a.e.inω, η = ( η 1,...,η n R n. (1.2 i,j=1 It is therefore essential, in order to be able to solve Problem (1.1, to assume some hyotheses on A(x strongerthan(1.2. In this aer we consider some of these ones and comare them. More recisely, we will consider the following conditions and show that they are equivalent. Hindawi Publishing Cororation Journal of Inequalities and Alications Volume 2006, Article ID 74171, Pages 1 8 DOI /JIA/2006/74171

2 2 Classes of ellitic matrices Condition 1.1 (the Cordes condition, see 5, 8]. A(x R n2 0, a.e. in Ω, and there exists ε (0,1 such that ( ni,j=1 a ii (x 2 ni,j=1 a 2 ij(x n 1+ε, a.e.inω. (1.3 Condition 1.2 (Condition A x. There exist four real constants σ, γ, δ, with σ>0, γ>0, δ 0, γ + δ<1, 1, and a function a(x L (Ω, with a(x σ a.e. in Ω,suchthat n ξ ii a(x i=1 n a ij (xξ ij γ ξ n n + δ ξ 2 ii i,j=1 i=1 (1.4 for all ξ ={ξ ij } i,j=1,...,n R n2,a.e.inω. When = 1, the above condition will be simly denoted by Condition A x ; it was defined in 10], where it has also been shown to be equivalent to the Cordes condition. If a(x is constant on Ω, Conditon A x is the formulation for linear oerators of Camanato s condition A, (see 4], which was defined for nonlinear oerators. A articular version of Condition A x, that is, with = 2and(x constant, is stated in 7] for nonlinear oerators. Condition 1.3 (Condition B x. There exist four real ositive real constants σ, c 1, c 2, c 3 and a function β L (Ωsuchthat (i 0 <c 1 c 2 c 3 < 1, (ii β(x σ a.e. in Ω, and moreover n n ( n 2 n β(x a ij (xξ ij ξ ii c 1 ξ ii c 2 ξ ii ξ n 2 c 3 ξ 2 n (1.5 2 i,j=1 i=1 i=1 i=1 for all ξ ={ξ ij } i,j=1,,n R n2,a.e.inω. If β(x isconstantonω, we will denote this condition as Condition B; it has been defined by Buică in2]. The imortance of Conditions A x or B x is in the fact that they allow to show in a relatively simle manner, by means of near oerators theory (see 4, 9] or weakly near oerators theory (see 1 3], that roblem (1.1 is well osed. The usefulnessof showing the equivalence among these conditions is due to the fact that to verify whether a matrix satisfies Condition A x or B x is very comlicated, even if n = 2, while to verify whether it satisfiesthecordes condition is much simler.

3 2. A rocedure of decomosition for matrices Antonio Tarsia 3 In this section we consider a short rocedure of decomosition of the matrices A and I which has been develoed in 10]. We set Ω 0 = { x Ω : there exists b(x R such that b(xa(x = I } ; Ω 1 = Ω\Ω 0. (2.1 Remark 2.1. Set M = su Ω A(x, ν = inf Ω A(x, accordinglyn ν (A(x I nm. Then, for each x Ω 0,weobtain1/M b(x 1/ ν. We can assume measω 1 > 0, since otherwise as we will see in the following it is easy to show the equivalence between the above conditions. We set for all x Ω 1 : W(x ={B(x: B(x = si + ra(x, s,r R}; Σ x = W(x S(I,1 (where S(I,1={B : B I R n 2 < 1}. Let v 1,w 2 W(x betherojectionsofi on the lines through the zero vector of R n2 and tangent to Σ x.moreoverletv 2 be the rojection of I on the line through the zero vector of R n2 and erendicular to v 1,andletw 1 betherojectionofi onthelinethrough the zero vector of R n2 and erendicular to w 2. In this manner we find two systems of orthogonal vectors {v 1,v 2 }, {w 1,w 2 },withv i = v i (x,w i = w i (x, i = 1,2. Each of them is a basis in the lane W(x. Then I = v 1 + v 2 = w 1 + w 2, and there are L functions a i = a i (xandb i = b i (x, i = 1,2, such that A(x = a 1 (xv 1 (x+a 2 (xv 2 (x = b 1 (xw 1 (x+b 2 (xw 2 (x. ( As v 1 = w 2 = n 1 and v 2 = w 1 =1, then for i = 1,2, a 2 i a 2 1(n 1 + a 2 2 = (a 1 v 1 + a 2 v 2 a 1 v 1 + a 2 v 2 = (A(x A(x = A(x 2 ;hereifb ={b ij } i,j=1,,n and C ={c ij } i,j=1,,n,weset(b C = ni,j=1 b ij c ij. Set Q v (x,ν,τ = { ξ R n2 : ξ = sv 1 + tv 2,0< ν s, t τ }, Q w (x,ν,τ = { ξ R n2 : ξ = sw 1 + tw 2,0< ν s, t τ }, R ( { } x,ν 0,τ 0 = ξ R n 2 : ξ = sw 2 + tv 1,0< ν 0 s, t τ 0, C ( { Σ x = v : v W(xsuchthat z Σx, t>0for which v = tz }, C ρ (x = { v : v C ( } Σ x : t>0suchthat I tv <ρ, 0<ρ<1. (2.2 The following roositions are roved in 10]. Proosition 2.2. For all τ,ν > 0 with ν τ, τ 0, ν 0, 0 <τ 0 < ν 0, such that for all x Ω 1, Q v (x,ν,τ Q w (x,ν,τ R ( x,ν 0,τ 0. (2.3 Proosition 2.3. For all τ 0,ν 0, 0 <τ 0 < ν 0,thereexistsρ (0,1 such that for all x Ω 1, R ( x,ν 0,τ 0 Cρ (x. ( Condition B x Proosition 3.1. Condition A x and Condition B x are equivalent.

4 4 Classes of ellitic matrices Proof. We assume that A satisfies Condition A x.itfollows(from(1.4 with = 1 by squaring both members 2 2a(x ( A ξ γ 2 ξ 2 +2γδ ξ + δ 2 2 (3.1 then 2a(x ( A ξ (1 δ 2 2 2γδ ξ γ 2 ξ 2. (3.2 This is Condition B x with b(x = 2a(x, c 1 = 1 δ 2, c 2 = 2γδ, c 3 = γ 2. Conversely, we set A(x = β(xa(x and assume that Condition B holds for A,thenwe will show that A also satisfies Condition A x. To this urose we write Condition B in the following form: there exist four real ositive constants M, c 1, c 2, c 3 with 0 <c 1 c 2 c 3 < 1, su x Ω A(x M such that ( A(x ξ c1 2 c2 ξ c3 ξ 2, (3.3 for all ξ R n2,a.e.inω. Then we obtain the thesis by using the decomosition of A and I stated in Section 2. For this we distinguish two cases: x Ω 0 and x Ω 1. If x Ω 0, that is, there exists b(xsuchthatb(xa(x = I,thenCondition A x is trivially true (take in (1.4 a(x = b(x. Instead, if x Ω 1,withmeasΩ 1 > 0, we observe that (3.3 holds in articulcular for ξ W(x. So we can write ξ as a linear combination of the basis {v 1 (x,v 2 (x}. Now,let t 1,t 2 R be such that ξ = t 1 v 1 (x+t 2 v 2 (x, accordingly ξ 2 = (ξ ξ = t1(n t2, 2 then ( ( A ξ = a1 (xv 1 + a 2 (xv 2 t 1 v 1 + t 2 v 2 = a1 t 1 (n 1 + a 2 t 2, ( ( (3.4 I ξ = v1 + v 2 t 1 v 1 + t 2 v 2 = t1 (n 1 + t 2. Now, (3.4 and the above remarks yield the following form of Condition B: foreachξ W(x, ( A ξ = a1 t 1 (n 1 + a 2 t 2 ]t 1 (n 1 + t 2 ] Put Remark that c 1 t1 (n 1 + t 2 ] 2 c2 t1 (n 1 + t 2 ] t 2 1(n 1 + t 2 2 c 3 t 2 1 (n 1 + t 2 2]. (3.5 F ( t 1,t 2 = a1 t 1 (n 1 + a 2 t 2 ] t1 (n 1 + t 2 ] c1 t1 (n 1 + t 2 ] 2 ] + c 2 t1 (n 1 + t 2 t1(n t2 2 + c 3 t 2 1 (n 1 + t2 2 ] (3.6. F ( t 1,t 2 0, ( t1,t 2 R 2 (by (3.5. (3.7

5 Antonio Tarsia 5 In articular ( 1 F,0 = a 1 (n 1 c 1 (n 1 + c 2 n 1+c3 0 (3.8 n 1 from which a 1 (x c 1 c 2 n 1 c 3 n 1 c 1 c 2 c 3 > 0. (3.9 While the inequality F(0,1 = a 2 (x c 1 + c 2 + c 3 0 imlies a 2 (x c 1 c 2 c 3 > 0. In the same way, by taking the system of orthogonal vectors {w 1,w 2 } as basis of W(x, it follows that b i (x c 1 c 2 c 3 > 0, i = 1,2, x Ω 1. (3.10 So we have shown (see Section 2 thata(x Q v (x,ν,τ Q w (x,ν,τ. This imlies, by Proosition 2.2, A(x R(x,ν 0,τ 0, then by Proosition 2.3, A(x C ρ (x, which is equivalent to say that Condition A x is valid with δ = 0. Taking into account this roosition and the equivalence between the Cordes condition and Condition A x, shown in 10], we have the following. Corollary 3.2. Condition B x and the Cordes condition are equivalent. The following examle states that Condition B is stronger than Condition A x and therefore is also stronger than the Cordes condition. Examle 3.3. Let Ω = Ω 1 Ω 2,whereΩ 1 ={(x 1,x 2 R 2 :0<x 1 < 1, 0 <x 2 1} and Ω 2 ={(x 1,x 2 R 2 :0<x 1 < 1, 1 <x 2 < 2},moreover A 1, ifx Ω 1, A(x = A 2, ifx Ω 2, A 1 = ( 1 0, A = ( ( A is uniformly ellitic on Ω and, since n = 2, this imlies the Cordes condition and therefore also Condition A x (see 10]. Nevertheless A does not satisfy Condition B. Indeed, we consider x Ω 1,thenA(x = A 1.WeobservethatifA 1 satisfied Condition B, it would be ( A1 ξ c 1 2 c2 ξ c3 ξ 2 (3.12 for each ξ R 4, that is, ( 1 c1 2 + c2 ξ + c3 ξ 2 0. (3.13 The bilinear form Φ(X,Y = (1 c 1 X 2 + c 2 XY + c 3 Y 2,where(X,Y R 2, is nonnegative if (1 c 1 c 3 c 2 2/4. In articular it must hold c 1 < 1. Otherwise if A(x satisfied Condition B on Ω 2 it would be ( A2 ξ c 1 2 c2 ξ c3 ξ 2, (3.14

6 6 Classes of ellitic matrices where c 1, c 2, c 3 are the above determined constants for the matrix A 1. Now we consider the matrix ( 1 0 ξ =, ( byrelacingitin(3.14, we obtain 100 c 1 c 2 5 5c3, that is, c 2 ( c 3 c 1 c 2 c ; that imlies (because by hyothesis it holds c 1 >c 2 + c 3 4c 1 > 4(c 2 + c 3 100, then c This contradicts what we have obtained for A 1, that is, c 1 < Condition A x We rove equivalence between the Cordes condition and Condition A x in the same way used in 10] for the roof of equivalence between Condition A and the Cordes condition. The first ste is following. Lemma 4.1. Condition A x with δ = 0 is equivalent to Cordes Condition. Proof (see also 10]. We can write Condition A x,ifδ = 0, as follows: ( I a(xa(x ξ γ 1/ ξ (4.1 for all ξ R n2,and 1. This is just Condition A x with δ = 0 and, accordingly to what roved in 10], this is equivalentto thecordes condition. The second ste for the achievement of our goal is following. Lemma 4.2. If A(x satisfies Condition A x for some function a(x and some constants σ, γ, δ, then it satisfies the same condition with δ = 0 and ossibly different σ, γ, a(x. Proof. We roceed on the line of the roof of 10, Lemma 3.3]. We follow the notations of Section 2. Condition A x,withδ 0, yields Condition A x with δ = 0, by relacing the coefficient a(x of the first condition withanewcoefficient ā(x, definedby b(x, if x Ω 0, ā(x = (4.2 c(x, if x Ω 1. If x Ω 0,thenCondition A x with δ = 0 is trivially satisfied. Moreover, by Remark 2.1, 1/M b(x 1/ ν.nowletx Ω 1. We rove the existence of a function c(x by means of the decomosition of matrices A(x, I stated in Section 2 and relacing the exressions obtained in Condition A x : ( I a(xa(x ξ = ( v 1 + v 2 a(x ( a 1 v 1 + a 2 v 2 ξ = ( take ξ = v i, i = 1,2 = ( v 1 + v 2 a(x ( a 1 v 1 + a 2 v 2 vi = v i 2 a(xai v i 2 = 1 a(xa i v i 2 γ v i +δ ( v1 +v 2 v i = γ v i +δ v i 2. (4.3

7 Antonio Tarsia 7 From this 1 γ + δ vi 1 a(x v i ai 1 1+ a(x γ + δ v i v i. (4.4 We observe that 1 (γ + δ 1/ 1 Using v 1 = n 1, v 2 = 1, we can write γ + δ v i γ + δ v i v i, 1+ v i 1+(γ + δ 1/. (4.5 γ + δ v i v i γ + δ, i = 1,2. (4.6 We conclude, from (4.4, by setting M 1 = sua(x, ν = 1 ] 1 (γ + δ 1, τ = 1 ] 1+(γ + δ 1/ (4.7 Ω M 1 σ for all x Ω 1, A(x Q v (x,ν,τ. Then by taking ξ = w i (i = 1,2 in Condition A x,with similar calculations, we obtain for all x Ω 1, A(x Q w (x,ν,τ. Then for all x Ω 1, A(x Q v (x,ν,τ Q w (x,ν,τ. From Proosition 2.2 it follows that there exist ν 0,τ 0,with 0 < ν 0 <τ 0,suchthatA(x R(x,ν 0,τ 0. By Proosition 2.3 there exists ρ (0,1 such that A(x C ρ (x, that is, there exist c(x > 0andρ (0,1 such that I c(xa(x ρ. (4.8 (This inequality also imlies ( n 1/M < c(x < ( n +1/ ν, x Ω 1. From Lemmas 4.1 and 4.2 we have the following. Theorem 4.3. The Cordes condition and Condition A x are equivalent. This theorem and Corollary 3.2 imly the following. Corollary 4.4. Condition B x and Condition A x are equivalent. Theorem 4.3 and Corollary 3.2,by the results roved in10], imly the following. Corollary 4.5. Let n = 2. Then every uniformly ellitic symmetric matrix satisfies Condition A x and Condition B x. References 1] A. Buică, Some roerties reserved by weak nearness, Seminar on Fixed Point Theory Cluj Naoca 2 (2001, ], Existence of strong solutions of fully nonlinear ellitic equations, Proceedings of Conference on Analysis and Otimization of Differential Systems, Constanta, Setember ] A. Buică anda. Domokos, Nearness, accretivity, and the solvability of nonlinear equations, Numerical Functional Analysis and Otimization 23 (2002, no. 5-6,

8 8 Classes of ellitic matrices 4] S. Camanato, A Cordes tye condition for nonlinear nonvariational systems, Rendiconti Accademia Nazionale delle Scienze detta dei XL. Serie V. Memorie di Matematica. Parte I 13 (1989, no. 1, ] H. O. Cordes, Zero order a riori estimates for solutions of ellitic differential equations, Proceedings of Symosium in Pure Math., vol. 4, American Mathematical Society, Rhode Island, 1961, ] O.A.LadyzhenskayaandN.N.Ural tseva,linear and Quasilinear Ellitic Equations, Academic Press, New York, ] A. Maugeri, D. K. Palagachev, and L. G. Softova, Ellitic and Parabolic Equations with Discontinuous Coefficients, Wiley-VCH, Berlin, ] G. Talenti, Sora una classe di equazioni ellittiche a coefficienti misurabili, Annali di Matematica Pura ed Alicata. Serie Quarta 69 (1965, ] A. Tarsia, Some toological roerties reserved by nearness between oerators and alications to P.D.E, Czechoslovak Mathematical Journal 46 (1996, no. 4, ], On Cordes and Camanato conditions, Archives of Inequalities and Alications 2 (2004, no. 1, Antonio Tarsia: Diartimento di Matematica L. Tonelli, Università di Pisa, Largo Bruno Pontecorvo 5, Pisa, Italy address: tarsia@dm.unii.it