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Pointwise Inequalities for Elliptic Boundary Value Problems

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Pointwise Inequalities for Elliptic Boundary Value Problems Pointwise Inequalities for Elliptic Boundary Value Problems Guo Luoa,∗, Vladimir Maz’yab aDepartment of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong bDepartment of Mathematics, Link¨opingUniversity, Link¨oping, Sweden Abstract We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on several examples, including a scalar second-order elliptic equation, the 3D Lame´ system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic bound- ary value problems provided that the corresponding weighted integral inequalities are satisfied. Keywords: Pointwise inequality, Weighted positivity 1. Introduction An important open problem in the mathematical theory of linear elasticity is whether solutions of the elasticity system, when supplemented with homogeneous Dirichlet boundary conditions and sufficiently smooth right-hand side data, are uniformly bounded in arbitrary domains. A similar question stands in the theory of hydrostatics, where the uniform boundedness of solutions of the Stokes system in general domains remains unknown. For bounded domains Ω with smooth boundaries @Ω, inequalities of the form k a b j j 1 ≤ j j j j u L (Ω) CΩ D u Lq(Ω) Lu Wl;p(Ω) (1) can often be obtained, by combining appropriate a priori estimates with Sobolev inequalities. The problem of such inequalities is that the constant CΩ generally depends on the smoothness of the domain Ω, and can blow up if the boundary of Ω contains geometric singularities. Efforts have been devoted to the study of inequalities of the type (1) with constants independent of the domain Ω. In Xie [14], a sharp pointwise bound 2 1 juj 1 ≤ jDuj 2 j∆uj 2 (2) L (Ω) 2π L (Ω) L (Ω) was obtained for functions u with zero-trace and with L2-integrable gradient Du and Laplacian ∆u on arbi- trary three-dimensional domains Ω. The constant (2π)−1 was shown to be the best possible. In Xie [15], a similar inequality with a slightly different best constant ((3π)−1 instead of (2π)−1) was conjectured for the Stokes system on arbitrary domains Ω ⊆ R3, and was proved in the special case Ω = R3. Further develop- ment and results along these lines can be found in Heywood [2] and the references there in. Regarding the ∗Corresponding author. Tel.: +852 3442 4191. Email addresses: [email protected] (Guo Luo), [email protected] (Vladimir Maz’ya) Preprint submitted to Journal of Mathematical Analysis and Applications May 9, 2015 3D Lame´ system, estimates of the form (2) seem to be less well studied, and we are not aware of any results or conjectures similar to (2). In this paper, we introduce a new approach to obtaining pointwise inequalities of the form (1) with constants independent of the domain Ω. The method works for elliptic operators L satisfying a weighted integral inequality Z Lu · Φu dx ≥ 0; (3) Ω where the weight Φ is either the fundamental solution or Green’s function of L. Weighted inequalities of the form (3) were first established for second-order scalar operators (see Section 3.1 below for a prototyp- ical derivation), and were later generalized to certain higher-order scalar operators [12] and second-order systems [5, 6]. They have important applications in the regularity theory of boundary points, and have been studied extensively in Maz’ya [7, 8, 11, 12], Maz’ya and Tashchiyan [9], Eilertsen [1]. By utilizing a slightly modified version of (3) (see (7) below), we shall show that the pointwise estimate (1) follows almost immediately from the weighted positivity of L, and the constant thus obtained is independent of the domain Ω. The method is illustrated on several concrete problems, including a scalar second-order elliptic equation, the 3D Lame´ system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic boundary value problems provided that the corresponding weighted integral inequalities are satisfied. k;q In what follows, W0 (Ω) denotes the usual Sobolev space of zero-trace functions, equipped with the norm 1=q X β q juj k;q := jD uj q : W0 (Ω) L (Ω) jβ|≤k Theorem 1. Let L be a second-order elliptic operator, @ Lu = −Di(ai j(x)D ju); Di = ; @xi n defined in a bounded domain Ω ⊂ R (n ≥ 3) with real, measurable coefficients ai j(x). Suppose L satisfies the strong ellipticity condition 2 2 λjξj ≤ ai j(x)ξiξ j ≤ Λjξj ; Λ ≥ λ > 0; n for almost all x 2 Ω and all ξ = (ξ1; : : : ; ξn) 2 R . Let s < n=(n − 2); p = s=(s − 1); q = (n − 2)s and let 2 1;q 2 p u W0 (Ω) be such that Lu L (Ω). Then a 1−a 1 juj 1 ≤ CjLuj p jDuj q ; a = ; L (Ω) L (Ω) L (Ω) n − 1 where C is an absolute constant depending only on λ, Λ; n, and s. Theorem 2. Let L be the 3D Lam´esystem, Lu = −∆u − α grad div u = −Dkkui − αDkiuk; i = 1; 2; 3; (4) where α = 1=(1−2ν) > −1 and ν is Poisson’s ratio. Let α 2 (α−; α+) ≈ (−0:194; 1:524); q < 3; p = q=(q−1) 2 1;q 2 p 3 and let u W0 (Ω) be such that Lu L (Ω), where Ω is an arbitrary bounded domain in R . Then 1=2 1=2 j j 1 ≤ j j j j u L (Ω) C Lu Lp(Ω) Du Lq(Ω); where C is an absolute constant depending only on α and q. 2 Theorem 3. Let L be an elliptic operator of order 2m, jαj m X α+β α @ L = (−1) aαβD ; D = ; @xα1 @xα2 ··· @xαn jαj=jβj=m 1 2 n n defined in R (n > 2m) with real, constant coefficients aαβ. Suppose L satisfies the strong ellipticity condition 2m α β 2m λjξj ≤ aαβξ ξ ≤ Λjξj ; Λ ≥ λ > 0; n 0 for all ξ = (ξ1; : : : ; ξn) 2 R . Let F be the fundamental solution of L. Let q < n=(n−2m); q = q=(q−1); k = − 2 k;q 2 q0 n n 2m and let u W0 (Ω) be such that Lu L (Ω), where Ω is an arbitrary bounded domain in R . If F is homogeneous of order 2m − n and L is weighted positive with the weight F, then 2 k 0 jujL1(Ω) ≤ CjD ujLq(Ω)jLujLq (Ω); where C is an absolute constant depending only on λ, Λ; m; n and q. The proofs of Theorems 1–3 are given in Section 3. 2. The Notion of Weighted Positivity n N Let Ω be a domain in R and let L = (Li)i=1 be a scalar elliptic operator (N = 1) or an elliptic system (N > 1) defined on Ω. Without making further structural assumptions on L, we recall first the abstract notion of weighted positivity. 2 N Definition 1 (Weighted positivity). Assume that 0 Ω and that Ψ(x) = (Ψi j(x))i; j=1 is a given (matrix) N function that is sufficiently regular except possibly at x = 0. The operator L = (Li)i=1 is said to be weighted positive (N = 1) or weighted positive definite (N > 1) with the weight Ψ if Z Z Lu · Ψu dx = (Lu)iΨi ju j dx ≥ 0; (5) Ω Ω N 2 1 n f g for all real-valued, smooth vector functions u = (ui)i=1 C0 (Ω 0 ). R · The concept of weighted positivity is often more useful when the integral Ω Lu Ψu dx in (5) has a positive lower bound. This motivates the following definition. 2 N ≥ Definition 2 (Strong weighted positivity). Assume that 0 Ω; L = (Li)i=1 is of order 2m; m 1, and that N Ψ(x) = (Ψi j(x))i; j=1 is a given (matrix) function that is sufficiently regular except possibly at x = 0. The operator L is said to be strongly weighted positive (N = 1) or strongly weighted positive definite (N > 1) with the weight Ψ if, for some c > 0, Z Xm Z Lu · Ψu dx ≥ c jDkuj2jxj2k−2mjΨj dx; (6) Ω k=1 Ω N 2 1 nf g k for all real-valued, smooth vector functions u = (ui)i=1 C0 (Ω 0 ). Here D denotes the gradient operator k α 2 PN 2 of order k, i.e. D = fD g with jαj = k, and jΨj stands for the Frobenius norm of Ψ, i.e. jΨj = i; j=1jΨi jj . 3 Among all possible candidates of the weight function Ψ, the special choice Ψ = Φ where Φ is the fun- damental solution or Green’s function of L is of the most interest. In particular, if L has constant coefficients and is strongly weighted positive in the sense of (6), with the weight Ψ = Φ, then it can be shown that, for all x 2 Ω and for the same constant c as given in (6), Z 1 Xm Z jDkuj2 Lu · Φ(x − y)u dy ≥ ju(x)j2 + c jΦ(x − y)j dy; (7) 2 jx − yj2m−2k Ω k=1 Ω N 2 1 for all real-valued, smooth vector functions u = (ui)i=1 C0 (Ω) [see, for example, 12]. Estimate (7) is significant because it provides a pointwise bound for the test function u, and it serves as the basis of the estimates to be derived below in Section 3. 3. Proofs of the Main Theorems 3.1.
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