On Cordes and Campanato Conditions Antonio Tarsia ∗ Keywords: nonvariational elliptic equations; Cordes condition; Campanato condition. AMS 1991 Subject classifications: 35j60; 35j25 Let Ω be an open bounded set in Rn with a sufficiently regular boundary, and let A(x) = {aij }i,j=1,···,n be a real matrix, with coefficients aij ∈ L∞ (Ω). We consider the following problem: 1,2 u ∈ H 2,2 ∩ H0 (Ω) (1) Pn a.e. in Ω. i,j=1 aij (x)Di Dj u(x) = f (x), If f ∈ L2 (Ω), it is known1 that problem (1) is not well posed with only hypothesis of uniform ellipticity on the matrix A(x): there exists a positive constant ν such that n X aij (x)ηi ηj ≥ νkηk2Rn , a.e. in Ω, ∀η = (η1 , . . . , ηn ) ∈ Rn . (2) i,j=1 When n > 2 , the proof of existence and uniqueness of the solution for problem (1.1) needs hypotheses on A(x) stronger than uniform ellipticity. In this paper we compare some of these hypotheses: namely Campanato and Cordes ones. We display below these conditions. Condition 1 (Condition of Cordes, see [10], [15]). Let A(x) = {aij (x)}i,j=1,···,n be a matrix such that kA(x)k n2 6= 0 , a.e. in R Ω. We say that A(x) satisfies the Condition of Cordes if there exists ε ∈ (0, 1) such that Pn 2 ( i=1 aii (x)) Pn ≥ n − 1 + ε, a.e. in Ω. (3) 2 i,j=1 aij (x) Condition 2 (Condition A, see [2]). ∗ Dipartimento di Matematica, Università di Pisa, via F. Buonarroti, 2. 56127 PISA, ITALY. e-mail: [email protected] 1 See the counterexamples in [11] and in [14]. 1 There exist three real constants α, γ, δ with α > 0, γ > 0, δ ≥ 0 and γ +δ < 1 such that: 1/2 n n n n X X X X 2 ξii − α aij (x)ξij ≤ γ ξij +δ ξii , (4) i=1 i,j=1 i,j=1 i=1 2 ∀ξ = {ξij }i,j=1,···,n ∈ Rn , a.e. in Ω. Condition 3 (Condition Ax , see [2] and [5]). There exist three real constants σ, γ, δ and a function a(x) ∈ L∞ (Ω), with σ > 0 , γ > 0, δ ≥ 0, γ + δ < 1, a(x) ≥ σ > 0 , such that 1/2 n X n n X X X n 2 ξii − a(x) aij (x)ξij ≤ γ ξij + δ ξii , i=1 i,j=1 i,j=1 i=1 (5) 2 ∀ξ = {ξij }i,j=1,···,n ∈ Rn , a.e. in Ω. In my paper I will show that Condition Ax is equivalent to the Cordes Condition (see Section 3), while Condition A is stronger than Condition Ax , even if n = 2 (see Section 2). If n = 2, we will show that Condition Ax is equivalent to the uniform ellipticity (see Section 2). So if n = 2 a matrix is unformly elliptic if and only if satisfies Cordes Condition. The usefulness of these results is in the fact that to verify when a matrix satisfies Condition Ax can be very hard, while it is quite easy to verify whether a matrix satisfies Cordes Condition. On the other side Condition Ax is useful for showing in a simple manner, namely by means of Campanato Theory of Near Operators (see [2] and [7] ), existence and uniqueness (and also regularity) of solution of Problem (1); the reader can also see [12], for more results and details about the theory of nonvariational elliptic equations . References [1] O. Arena , A. Maugeri, Perturbazione di operatori parabolici di ordine 2n con termini di ordine inferiore, Boll. Unione Mat. Ital., IV Ser. 9, (1974), 169-184. [2] S. Campanato, A Cordes type Condition for nonlinear non-variational Systems, Rend. Accad. Naz. Sci. Detta XL, V Ser. 13, No 1,(1989), 307-321. [3] S. Campanato, A history of Cordes Condition for second order elliptic operators, Lions, J-L (ed) et al., Boundary value problems for partial differential equations and applications. Dedicated to E. Magenes on the occasion of his 70th birthday. Paris: Masson. Res. Notes Appl. Math 29, 1983, pp. 319-325. 2 [4] S. Campanato, Attuale formulazione della teoria degli operatori ellittici e attuale definizione di operatore ellittico, Matematiche 51, No.2, (1996), 291-298. [5] S. Campanato, Nonvariational basic parabolic systems of second order, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX Ser., Rend. Lincei, Mat. Appl. 2, No.2, (1991), 129-136. [6] S. Campanato, Sul problema di Cauchy Dirichlet per equazioni paraboliche del secondo ordine, non variazionali, a coefficienti discontinui. Rend. Semin. Mat. Univ. Padova 41, (1968), 153-163. [7] S. Campanato, Sistemi differenziali del secondo ordine di tipo ellittico, Quaderno n.1 del Dottorato di Ricerca in Mat. Univ. di Catania (1991). [8] S. Campanato, Non variational differential systems. A condition for local existence and uniqueness , Ric. Mat. 40, suppl. , (1991), 129-140. [9] S. Campanato, Nonvariational basic elliptic systems of second order, Rend. Semin. Mat. Fis. Milano 60, (1991),113-131. [10] H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, Proc. Sympos. Pure Math. 4, (1961), 157-166. [11] O. A. Ladyzhenskaya, N. N. Ural’ceva , Linear and quasi linear elliptic equations , Academic Press, New York, 1968. [12] A. Maugeri, D. K. Palagachev, L. G. Softova, Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research,109, WileyVCH, Weinheim, 2000. [13] C. Miranda, Su di una particolare equazione ellittica del secondo ordine a coefficienti discontinui, An. Sti. Univ. Al. I. Cura Jasi, N. Ser.,Sec. Ia 11B, (1965), 209-215. [14] C. Pucci, Limitazioni per soluzioni di equazioni ellittiche , Ann. Mat. Pura Appl. , IV Ser. 74, (1966), 15-30. [15] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti discontinui, Ann. Mat. Pura Appl., IV Ser. 69 , (1965), 285-304. 3