Báo cáo hóa học: "CLASSES OF ELLIPTIC MATRICES" docx

CLASSES OF ELLIPTIC MATRICES ANTONIO TARSIA Received 12 December 2005; Revised 20 February 2006; Accepted 21 February 2006 The equivalence between some conditions concerning elliptic matrices is shown, namely, the Cordes condition, a generalized form of Campanato’s condition, and a generalized form of a condition of Buic ˘ a. Copyright © 2006 Antonio Tarsia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let Ω be an open 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 problem: u ∈ H 2,2 ∩H 1,2 0 (Ω), n  i, j=1 a ij (x) D ij u(x) = f (x), a.e. x ∈Ω. (1.1) If f ∈ L 2 (Ω), it is known (see the counterexamples in [6]) that problem (1.1)isnotwell posed with the only hypothesis of uniform ellipticity on the matrix A(x): there exists a positive constant ¯ ν such that n  i, j=1 a ij (x) η i η j ≥ ¯ ν η 2 n ,a.e.inΩ, ∀η =  η 1 , ,η n  ∈ R n . (1.2) It is therefore essential, in order to be a ble to solve Problem (1.1), to assume some hy- potheses on A(x)strongerthan(1.2). In this paper we consider some of these ones and compare them. More precisely, we will consider the following conditions and show that they are equivalent. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 74171, Pages 1–8 DOI 10.1155/JIA/2006/74171 2 Classes of elliptic matrices Condition 1.1 (the Cordes condition, see [5, 8]). A(x) R n 2 = 0, a.e. in Ω, and there exists ε ∈ (0, 1) such that   n i, j =1 a ii (x)  2  n i, j =1 a 2 ij (x) ≥ n −1+ε,a.e.inΩ. (1.3) Condition 1.2 (Condition A xp ). There exist four real constants σ, γ, δ, p with σ>0, γ>0, δ ≥ 0, γ + δ<1, p ≥ 1, and a function a(x) ∈L ∞ (Ω), with a(x) ≥σ a.e. in Ω,suchthat      n  i=1 ξ ii −a(x) n  i, j=1 a ij (x) ξ ij      p ≤ γξ p n 2 + δ      n  i=1 ξ ii      p (1.4) for all ξ ={ξ ij } i, j=1, ,n ∈ R n 2 ,a.e.inΩ. When p = 1, the above condition will be simply 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 operators of Campanato’s condition A, (see [4]), which was defined for nonlinear operators. A particular version of Condition A xp , that is, with p = 2and(x) constant, is stated in [7] for nonlinear operators. Condition 1.3 (Condition B x ). There exist four real positive real constants σ, c 1 , c 2 , c 3 and a function β ∈ L ∞ (Ω)suchthat (i) 0 <c 1 −c 2 −c 3 < 1, (ii) β(x) ≥ σ a.e. in Ω, and moreover β(x) n  i, j=1 a ij (x) ξ ij n  i=1 ξ ii ≥ c 1  n  i=1 ξ ii  2 −c 2      n  i=1 ξ ii       ξ n 2 −c 3 ξ 2 n 2 (1.5) for all ξ ={ξ ij } i, j=1,···,n ∈ R n 2 ,a.e.inΩ. If β(x)isconstantonΩ, we will denote this condition as Condition B; it has been defined by Buic ˘ ain[2]. The importance of Conditions A xp or B x is in the fact that they allow to show in a relatively simple manner, by means of near operators theor y (see [4, 9]) or weakly near operators theory (see [1–3]), that problem (1.1) is well posed. The usefulness of showing the equivalence among these conditions is due to the fact that to verify whether a matrix satisfies Condition A xp or B x is very complicated, even if n = 2, while to verify whether it satisfies the Cordes condition is much simpler. Antonio Tarsia 3 2. A procedure of decomposition for matrices In this section we consider a short procedure of decomposition of the matrices A and I which has been developed in [10]. We set Ω 0 =  x ∈Ω : there exists b(x) ∈R such that b(x)A(x) =I  ; Ω 1 = Ω\Ω 0 . (2.1) Remark 2.1. Set M = sup 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 meas Ω 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)betheprojectionsofI on the lines through the zero vector of R n 2 and tangent to Σ x .Moreoverletv 2 be the projection of I on the line through the zero vector of R n 2 and per pendicular to v 1 ,andletw 1 betheprojectionofI onthelinethrough the zero vector of R n 2 and perpendicular 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 plane W(x). Then I = v 1 + v 2 = w 1 + w 2 , and there are L ∞ functions a i = a i (x)andb i = b i (x), i =1,2, such that A(x) = a 1 (x) v 1 (x)+a 2 (x) v 2 (x) = b 1 (x) w 1 (x)+b 2 (x) w 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) =  n i, j =1 b ij c ij .) Set Q v (x, ν,τ) =  ξ ∈R n 2 : ξ =sv 1 + tv 2 ,0< ν ≤ s, t ≤τ  , Q w (x, ν,τ) =  ξ ∈R n 2 : 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(x)suchthat∃z ∈ Σ x , ∃t>0 for which v = tz  , C ρ (x) =  v : v ∈ C  Σ x  : ∃t>0suchthatI −tv<ρ  ,0<ρ<1. (2.2) The following propositions are proved in [10]. Proposition 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) Proposition 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  ⊂ C ρ (x) . (2.4) 3. Condition B x Proposition 3.1. Condition A x and Condition B x are equivalent. 4 Classes of elliptic matrices Proof. We assume that A satisfies Condition A x .Itfollows(from(1.4)withp = 1) by squaring both members  I |ξ  2 −2a(x)  A | ξ  I |ξ  ≤ γ 2 ξ 2 +2γδ|  I |ξ  | ξ+ δ 2  I |ξ  2 (3.1) then 2a(x)  A | ξ  I |ξ  ≥ (1 −δ 2 )  I |ξ  2 −2γδ|  I |ξ  | 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(x)A(x) and assume that Condition B holds for A,thenwe will show that A also satisfies Condition A x . To this purpose we w rite Condition B in the following form: there exist four real positive constants M, c 1 , c 2 , c 3 with 0 <c 1 −c 2 −c 3 < 1, sup x∈Ω A(x) ≤M such that  A(x) |ξ  I |ξ  ≥ c 1  I |ξ  2 −c 2    I |ξ     ξ−c 3 ξ 2 , (3.3) for all ξ ∈ R n 2 ,a.e.inΩ. Then we obtain the thesis by using the decomposition 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(x)suchthatb(x)A(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 particulcular 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) = t 2 1 (n −1) + t 2 2 , then  A | ξ  =  a 1 (x) v 1 + a 2 (x) v 2 | t 1 v 1 + t 2 v 2  = a 1 t 1 (n −1) + a 2 t 2 ,  I |ξ  =  v 1 + v 2 | t 1 v 1 + t 2 v 2  = t 1 (n −1) + t 2 . (3.4) Now, (3.4) and the above remarks yield the following form of Condition B:foreachξ ∈ W(x),  A | ξ  I |ξ  =  a 1 t 1 (n −1) + a 2 t 2 ][t 1 (n −1) + t 2  ≥ c 1  t 1 (n −1) + t 2  2 −c 2  t 1 (n −1) + t 2   t 2 1 (n −1) + t 2 2 −c 3  t 2 1 (n −1) + t 2 2  . (3.5) Put F  t 1 ,t 2  =  a 1 t 1 (n −1) + a 2 t 2  t 1 (n −1) + t 2  − c 1  t 1 (n −1) + t 2  2 + c 2  t 1 (n −1) + t 2   t 2 1 (n −1) + t 2 2 + c 3  t 2 1 (n −1) + t 2 2  . (3.6) Remark that F  t 1 ,t 2  ≥ 0, ∀  t 1 ,t 2  ∈ R 2 (by (3.5)). (3.7) Antonio Tarsia 5 In particular F  1 √ n −1 ,0  = a 1 (n −1) −c 1 (n −1) + c 2 √ n −1+c 3 ≥ 0 (3.8) 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 implies 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 implies, by Proposition 2.2, A(x) ∈ R(x, ν 0 ,τ 0 ), then by Proposition 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 proposition 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 example states that Condition B is stronger than Condition A x and there- fore is also stronger than the Cordes condition. Example 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(x) = ⎧ ⎨ ⎩ A 1 ,ifx ∈Ω 1 , A 2 ,ifx ∈Ω 2 , A 1 =  10 01  , A 2 =  200 −150 −150 200  . (3.11) A is uniformly elliptic on Ω and, since n = 2, this implies the Cordes condition and there- fore 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  A 1 | ξ  I |ξ  ≥ c 1  I |ξ  2 −c 2    I |ξ     ξ−c 3 ξ 2 (3.12) for each ξ ∈ R 4 , that is,  1 −c 1  I |ξ  2 + c 2    I |ξ     ξ+ c 3 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 nonneg- ative if (1 −c 1 )c 3 ≥ c 2 2 /4. In particular it must hold c 1 < 1. Otherwise if A(x) satisfied Condition B on Ω 2 it would be  A 2 | ξ  I |ξ  ≥ c 1  I |ξ  2 −c 2    I |ξ     ξ−c 3 ξ 2 , (3.14) 6 Classes of elliptic matrices where c 1 , c 2 , c 3 are the above determined constants for the matrix A 1 . Now we consider the matrix ξ =  − 10 −20  , (3.15) byreplacingitin(3.14), we obtain −100 ≥ c 1 −c 2 √ 5 −5c 3 , that is, c 2 ( √ 5 −1) + 4c 3 ≥ c 1 −c 2 −c 3 + 100; that implies (because by hypothesis it holds c 1 >c 2 + c 3 )4c 1 > 4(c 2 + c 3 ) ≥ 100, then c 1 ≥ 25. This contradicts what we have obtained for A 1 , that is, c 1 < 1. 4. Condition A xp We prove equiv alence be tween the Cordes condition and Condition A xp in the same way used in [10] for the proof of equivalence between Condition A and the Cordes condition. The first step is following. Lemma 4.1. Condition A xp with δ =0 is equivalent to Cordes Condition. Proof (see also [10]). We can write Condition A xp ,ifδ = 0, as follows:    I −a(x)A(x) |ξ    ≤ γ 1/p ξ (4.1) for all ξ ∈ R n 2 ,andp ≥1. This is just Condition A x with δ = 0 and, accordingly to what proved in [10], this is equivalent to the Cordes condition.  The second step for the achievement of our goal is following. Lemma 4.2. If A(x) satisfies Condition A xp for some function a(x) and some constants σ, γ, δ, then it satisfies the same condition with δ = 0 and possibly different σ, γ, a(x). Proof. We proceed on the line of the proof of [10, Lemma 3.3]. We follow the notations of Section 2. Condition A xp ,withδ =0, yields Condition A xp with δ = 0, by replacing the coefficient a(x) of the first condition withanewcoefficient ¯ a(x), defined by ¯ a(x) = ⎧ ⎨ ⎩ b(x), if x ∈Ω 0 , c(x), if x ∈ Ω 1 . (4.2) If x ∈ Ω 0 ,thenCondition A xp with δ = 0 is trivially satisfied. Moreover, by Remark 2.1, 1/M ≤ b(x) ≤ 1/ ¯ ν.Nowletx ∈Ω 1 . We prove the existence of a function c(x) by means of the decomposition of matrices A(x), I stated in Section 2 and replacing the expressions obtained in Condition A xp :    I −a(x)A(x) |ξ    p =    v 1 + v 2 −a(x)  a 1 v 1 + a 2 v 2  | ξ    p =  take ξ = v i , i = 1,2  =    v 1 + v 2 −a(x)  a 1 v 1 + a 2 v 2  | v i    p =      v i   2 −a(x) a i   v i   2    p =   1−a(x)a i   p   v i   2p ≤ γ   v i   p +δ  v 1 +v 2 | v i  p = γ   v i   p +δ   v i   2p . (4.3) Antonio Tarsia 7 From this 1 a(x) ⎛ ⎝ 1 − p  γ + δv i  p   v i   ⎞ ⎠ ≤ a i ≤ 1 a(x) ⎛ ⎝ 1+ p  γ + δ   v i   p   v i   ⎞ ⎠ . (4.4) We observe that 1 −(γ + δ) 1/p ≤ 1 − p  γ + δ   v i   p   v i   ,1+ p  γ + δ   v i   p   v i   ≤ 1+(γ + δ) 1/p . (4.5) Using v 1 = √ n −1, v 2 = 1, we can write γ + δ   v i   p   v i   p ≤ γ + δ, i = 1,2. (4.6) We conclude, from (4.4), by setting M 1 = sup Ω a(x), ν = 1 M 1  1 −(γ + δ) 1 p  , τ = 1 σ  1+(γ + δ) 1/p  (4.7) for all x ∈ Ω 1 , A(x) ∈ Q v (x, ν,τ). Then by taking ξ =w i (i =1,2) in Condition A xp ,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 Proposition 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 Proposition 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(x)A(x)   ≤ ρ. (4.8) (This inequality also implies ( √ 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 xp are equivalent. This theorem and Corollary 3 .2 imply the following. Corollary 4.4. Condition B x and Condition A xp are equivalent. Theorem 4.3 and Corollary 3 .2,bytheresultsprovedin[10], imply the following. Corollary 4.5. Let n = 2. Then every uniformly elliptic symmetric matrix satisfies Condi- tion A xp and Condition B x . References [1] A. Buic ˘ a, Some properties preserved by weak nearness, Seminar on Fixed Point Theory Cluj Napoca 2 (2001), 65–70. [2] , Existence of strong solutions of fully nonlinear elliptic equations, Proceedings of Confer- ence on Analysis and Optimization of Differential Systems, Constanta, September 2002. [3] A. Buic ˘ aandA.Domokos,Nearness, accretivity, and the solvability of nonlinear equations,Nu- merical Functional Analysis and Optimization 23 (2002), no. 5-6, 477–493. 8 Classes of elliptic matrices [4] S. Campanato, A Cordes type condition for nonlinear nonvariational systems, Rendiconti Ac- cademia Nazionale delle Scienze detta dei XL. Serie V. Memorie di Matematica. Parte I 13 (1989), no. 1, 307–321. [5] H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, Proceed- ings of Symposium in Pure Math., vol. 4, American Mathematical Society, Rhode Island, 1961, pp. 157–166. [6] O.A.LadyzhenskayaandN.N.Ural’tseva,Linear and Quasilinear Elliptic Equations,Academic Press, New York, 1968. [7] A. Maugeri, D. K. Palagachev, and L. G. Softova, Elliptic and Parabolic Equations with Discontin- uous Coefficients, Wiley-VCH, Berlin, 2000. [8] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Annali di Matematica Pura ed Applicata. Serie Quarta 69 (1965), 285–304. [9] A. Tarsia, Some topological properties preserved by nearness between operators and applications to P.D.E, Czechoslovak Mathematical Journal 46 (1996), no. 4, 607–624. [10] , On Cordes and Campanato conditions, Archives of Inequalities and Applications 2 (2004), no. 1, 25–39. Antonio Tarsia: Dipartimento di Matematica “L. Tonelli,” Universit ` adiPisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy E-mail address: tarsia@dm.unipi.it . condition with δ = 0 and possibly different σ, γ, a(x). Proof. We proceed on the line of the proof of [10, Lemma 3.3]. We follow the notations of Section 2. Condition A xp ,withδ =0, yields Condition. Cluj Napoca 2 (2001), 65–70. [2] , Existence of strong solutions of fully nonlinear elliptic equations, Proceedings of Confer- ence on Analysis and Optimization of Differential Systems, Constanta, September. v 1 ,w 2 ∈ W(x)betheprojectionsofI on the lines through the zero vector of R n 2 and tangent to Σ x .Moreoverletv 2 be the projection of I on the line through the zero vector of R n 2 and per pendicular