New estimates for the div-curl-grad operators and elliptic problems with L1-data in the half-space

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1 New estimates for the div-curl-grad operators and elliptic problems with L1-data in the half-space Chérif Amrouche, Huy Hoang Nguyen To cite this version: Chérif Amrouche, Huy Hoang Nguyen. New estimates for the div-curl-grad operators and elliptic problems with L1-data in the half-space. Applied Mathematics Letters, Elsevier, 2011, 24 (5), pp < /j.aml >. <hal > HAL Id: hal Submitted on 21 Dec 2010 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 New estimates for the div-curl-grad operators and elliptic problems with L 1 -data in the half-space Chérif Amrouche a, Huy Hoang Nguyen b a Laboratoire de Mathématiques Appliquées, Université de Pau et des Pays de l Adour IPRA, Avenue de l Université Pau France b Departamento de Matematica, IMECC, Universidade Estadual de Campinas Caixa Postal 6065, Campinas, SP , BRAZIL Abstract In this Note, we study some properties of the div-curl-grad operators and elliptic problems in the half-space. We consider data in weighted Sobolev spaces and in L 1. Keywords: spaces div-curl-grad operators, elliptic, half-space, weighted Sobolev 1. Introduction The purpose of this Note is to give some estimates for div-curl-grad operators and elliptic problems with L 1 -data in the half-space. We know that if f L n (), there exists u W 0 (Rn +) such that div u = f holds, but does u W 0 (Rn +) L () hold? This is indeed true. In section 2, we give new estimates for L 1 -vector fields, which improve estimates for the solutions of elliptic systems in with data in L 1. In this Note, we use bold type characters to denote vector distributions or spaces of vector distributions with n components and C > 0 usually denotes a generic constant (the value of which may change from line to line). For any q N, P q (respectively, Pq ) stands for the space of polynomials (respectively harmonic polynomials) of degree q. If q is strictly negative integer, we set by convention P q = {0}. Let Ω be an open subset in the n-dimensional real Euclidean space. A typical point in R n is denoted by x = (x 1,..., x n ) and its norm is given by r = x = (x x 2 n) 1 2. We define the weight function ρ(x ) = 1 + r. For each p R and 1 < p <, the conjugate exponent p is given by the relation 1 p + 1 p W 1,p 0 (Ω) = {u D (Ω), = 1. We now define the weighted Sobolev space u w 1 L p (Ω), u L p (Ω)}, where w 1 = 1 + r if p n addresses: cherif.amrouche@univ-pau.fr (Chérif Amrouche a ), nguyen@ime.unicamp.br (Huy Hoang Nguyen b ) Preprint submitted to Applied Mathematics Letters December 8, 2010

3 and w 1 = (1 + r) ln(2 + r) if p = n. This space is a reflexive Banach space when endowed with the norm: u W 1,p 0 (Ω) = ( u w 1 p L p (Ω) + u p L p (Ω) )1/p. We also introduce the space W 2,p 0 (Ω) = {u D (Ω), u w 2 L p (Ω), L p (Ω), D 2 u L p (Ω)}, where w 2 = (1 + r) 2 if p / { n, n} and w 2 = (1 + r) 2 ln(2 + r) in the remaining case. This space is a reflexive Banach space endowed with its natural norm given by u W 2,p 0 (Ω) = ( u w 2 p L p (Ω) + u w 1 p L p (Ω) + D2 u p L p (Ω) )1/p. n We note that the logarithmic weight only appears if p = n or p = and all the local properties of W 1,p 0 (Ω) (respectively, W 2,p 0 (Ω)) coincide with those of the corresponding classical Sobolev space W 1,p (Ω) (respectively, W 2,p (Ω)). For m = 1 or m = 2, we set space of W m, p 0 (Ω) = D(Ω) W m, p 0 (Ω) and we denote the dual W m, p 0 (Ω) by W m,p 0 (Ω), which is the space of distributions. When Ω = R n, we have W m,p 0 (R n ) = W m, p 0 (R n ) (see [2] for more details). 2. The div-grad-curl operators and elliptic systems First of all, we set B a + = {x ; x < a} with a > 0. In this section, we consider the case n 2. We introduce the following theorem. Theorem 2.1. Let f L n (). Then there exists u W 0 (Rn +) L () such that div u = f. Moreover, we have the following estimate u u L ( ) + u W 0 ( ) C f Ln ( ). (1) Proof. Let (f k ) k N D() converges towards f L n () and B r + k such that supp f k B r + k. We set g(x ) = r k f(r k x ). Thanks to Theorem 3 [4], there exists v W 0 (B 1 + ) L (B 1 + ) satisfying div v = g in B+ 1 and the following estimate v L (B + 1 ) + v L n (B + 1 ) C g L n (B + 1 ). We now set u k (x ) = v( x r k ) with x B + r k. It is easy to show that u k W 0 (B + r k ) L (B + r k ) satisfies div u k = f k and we have the following estimate u k L (B + r k ) + u k L n (B + r k ) C f k L n (B + r k ) C f k Ln ( ) where C depends only on n. By extending u k in by zero outside B r + k and denoting ũ k its extended function, we have ũ k L ( ) = u k L (B r + k ) and w 1 2

4 ũ k W 0 ( ) = u k W 0 (B + r k ). Then (ũ k) k is bounded in W 0 (Rn +) L () and we can deduce that there exists a subsequence, again denoted by (ũ k ) k such that ũ k u in W 0 (Rn +) and ũ k u in L (). Hence, we have div u = f and the estimate (1). The conclusion of Theorem 2.1 is equivalent to the estimate u L n/() (), u L n/() ( ) C n u L 1 ( )+W /() 0 ( ). (2) Indeed, let us consider the following unbounded operator A = : L n/() () L 1 () + W /() 0 (). Then the domain D(A) is dense in L n/() () and A is closed. Because the adjoint operator A = div : W 0 () L () L n () is surjective, we deduce the estimate (2). It follows from (2) and that range A is closed in L 1 ()+W /() 0 (). Then (Ker A ) = Im A and as consequence we obtain the following version of De Rham s Theorem: for any f belonging to the space L 1 () + W /() 0 () and satisfying the following compatibility condition v W 0 () L (), with div v = 0, < f, v > = 0, (3) then there exists a unique π L n/() () such that f = π. The following theorem, given without proof, is proved by J. Bourgain - H. Brézis [5] (see also [6]) in the case where is replaced by a bounded smooth domain Ω. Theorem 2.2. i) Let ϕ W 0 (). Then there exists ψ = (ψ 1,..., ψ n ) W 0 () L () and η W 2,n 0 () such that ψ n = 0 on Γ = R {0} satisfying ϕ = ψ + η and the following estimate holds ψ L ( ) + ψ W 0 ( ) + η W 2,n 0 ( ) C ϕ W 0 ( ). (4) ii) If ϕ W 0 (Rn +), then the same result holds with ψ W 1, n 0 () L () and η W 2,n 0 (Rn +) and the corresponding estimate. Define the space X() = { v L 1 (); div v W 2,n/() 0 () }. Let f X(). Thanks to Theorem 2.2, the linear operator F : ϕ f ϕ dx satisfies ϕ D(), < F, ϕ > C f X ϕ W. 0 3

5 As D() is dense in W 0 (Rn +) and by applying Hahn-Banach Theorem, we can uniquely extend F by an element F W /() 0 () satisfying F /() W 0 ( ) C f X(R n + ). Besides, the following linear operator f F from X() into W /() 0 () is continuous and injective. Therefore, X() can be identified to a subspace of W /() 0 (R n ) with continuous and dense embedding. Thanks to [3], we can then deduce that for any f X(), the following problem u = f in and u = 0 on Γ, (5) has a unique solution u W /() 0 () satisfying the following estimate u W /() 0 ( ) C f X(R n + ). The first statement of the following corollary is the equivalent of Theorem 2.1 for the curl operator and the second statement gives a Helmholtz decomposition. Corollary 2.3. i) Let f L 3 (R 3 +) such that div f = 0 in R 3 + and f 3 = 0 on Γ. Then there exists ψ W 0 (R3 +) L (R 3 +) such that f = curl ψ and we have the following estimate ψ W 0 (R3 + ) + ψ L (R 3 + ) C f L 3 (R 3 + ). (6) ii) Let f L 3 (R 3 +). Then there exist ϕ W 0 (R3 +) L (R 3 +) and π W 0 (R 3 +) unique up to an additive constant and satisfying Moreover, we have the following estimate f = curl ϕ + π. (7) ϕ W 0 (R3 + ) + ϕ L (R 3 + ) + π L 3 (R 3 + ) C f L 3 (R 3 + ). (8) In the following proposition, we improve the estimate given by Corollary 1.4 of Van Schaftingen [8]. Proposition 2.4. Let f L 1 (R 3 +) such that div f = 0. following estimate Then we have the ϕ W 0 (R3 +), < f, ϕ > C f L 1 (R 3 + ) curl ϕ L 3 (R 3 + ). (9) Proof. First remark that from the hypothesis, we deduce f W /2 0 (R 3 +). Let ϕ W 0 (R3 +). Then we have curl ϕ L 3 (R 3 +). Thanks to Corollary 2.3, there exists ψ W 0 (R3 +) L (R 3 +) such that curl ψ = curl ϕ with the following estimate ψ W 0 (R3 + ) + ψ L (R 3 + ) C curl ϕ L3 (R 3 + ). (10) 4

6 Besides, there exists η W 2,3 0 (R 3 +) such that ϕ = ψ + η in R 3 +. Then we have < f, ϕ > /2 = f ψ + < f, η > W 0 (R 3 + ) W 0 (R3 + ) R 3 /2 W 0 (R 3 + ) W 0 (R3 + ) + = f ψ. R 3 + Therefore, the estimate (9) is deduced from the estimate (10). Here is a variant of Theorem 2.2. Theorem 2.5. Let ϕ W 0 (R3 +). Then there exist ψ W 0 (R3 +) L (R 3 +) and η W 2,3 0 (R3 +) such that ϕ = ψ + curl η with div 2 η = 0 in R 3 +. However, we have the following estimate Proof. ψ W 0 (R3 + ) + ψ L (R 3 + ) + η W 2,3 0 (R3 + ) C ϕ W 0 (R3 + ). From the hypothesis, we have div ϕ L 3 (R 3 +). Thanks to Theorem 2.1, there exists ψ in W 0 (R3 +) L (R 3 +) such that div ψ = div ϕ and we have the following estimate ψ L (R 3 + ) + ψ W 0 (R3 + ) C div ϕ L3 (R 3 + ). We set f = ϕ ψ, then f W 0 (R3 +) and div f = 0. We extend f to R 3 as follows { f (x f (x, x 3 ) if x 3 > 0,, x 3 ) = (f 1 (x, x 3 ), f 2 (x, x 3 ), f 3 (x, x 3 )) if x 3 < 0. Then we have f W 0 (R3 ) and div f = 0 in R 3. Thanks to [2], there exists y W 2,3 0 (R3 ) such that y = curl f in R 3. As div y W 0 (R 3 ) and is harmonic, then div y = a where a is a constant. Thus, we have curl (curl y f ) = 0. Therfore, we deduce (curl y) = curl curl f = f, it means that f curl y W 0 (R3 ) and again, f curl y = b where b is a constant vector in R 3. Thanks to Lemma 3.1 [7], there exists a polynomial s P 1 such that curl s = b and div s = a. The function z := y + s belongs to W 2,3 0 (R3 ) and f = curl z in R 3 with div z = 0 in R 3. Let w be the vector field defined on R 3 by w(x, x 3 ) = ( z 1 (x, x 3 ), z 2 (x, x 3 ), z 3 (x, x 3 )) with x 3 < 0. It is easy to show curl w = curl z in R 3. Then we have w = z + θ in R 3 (11) with θ D (R 3 ). As θ belongs to W 0 (R3 ), we can show θ W 2,3 0 (R 3 ). Let θ 0 W 2,3 0 (R 3 +) (cf. [3]) such that θ 0 = 0 in R 3 + and θ 0 = θ on Γ. We 5

7 set ζ = z 1 2 θ 0. Then we have curl ζ = f and div ζ = 0 in R 3 +. Remark now that from (11), we deduce 2z = θ = θ 0 on Γ, i.e., ζ = 0 on Γ. Let χ W 3,3 0 (R 3 +) be a solution of the system: 2 χ = 0 in R 3 +, χ = 0 and χ = ζ 3 on Γ. We set h = ζ χ. Then h W 2,3 0 x (R3 +) such that 3 f = curl h and div h = 0 in R 3 +, h = 0 on Γ. We know that there exists µ W 3,3 0 (R 3 +) satisfying 3 µ = 0 in R 3 µ +, = 0 and 2 µ x 3 x 2 3 = h 3 x 3 on Γ. We set η = h µ. Then η W 2,3 0 (R3 +), curl η = f in R 3 + and η = 0 on Γ. It is easy to show that η = 0 on Γ and then η W 2,3 0 x (R3 +). The 3 proof is finished. We introduce the following proposition. Proposition 2.6. Let f L 1 (R 3 +) such that div f = 0 in R 3 +. Then there exists a unique ϕ L 3/2 (R 3 +) such that curl ϕ = f, div ϕ = 0 in R 3 + and ϕ 3 = 0 on Γ satisfying the following estimate ϕ L 3/2 (R 3 + ) C f L1 (R 3 + ). Proof. It is easy to show the uniqueness of ϕ. We now try to prove the existence of ϕ. We know that f W /2 0 (R 3 +). Then, there exists a unique z W /2 0 (R 3 +) satisfying z = f, with div z = 0 in R 3 +. The function ϕ = curl z is the required function. A variant of this result can be obtained. If f L 1 (R 3 +) such that div f = 0 in R 3 + and f 3 = 0 on Γ, then we can prove the existence of a unique ϕ L 3/2 (R 3 +) such that f = curl ϕ with div ϕ = 0 in R 3 + and ϕ = 0 on Γ. More generally, we can prove the following result. Theorem 2.7. i) Let f L 1 (R 3 +) + W /2 0 (R 3 +) such that div f = 0. Then there exists a unique ϕ L 3/2 (R 3 +) such that curl ϕ = f with div ϕ = 0 in R 3 + and ϕ 3 = 0 on Γ satisfying the following estimate ϕ L 3/2 (R 3 + ) C f L1 (R 3 + )+W /2 0 (R 3 + ). ii) Let f X(R 3 +). Then there exists a unique ϕ L 3/2 (R 3 +) such that div ϕ = 0 with ϕ 3 = 0 on Γ and a unique π L 3/2 (R 3 +) satisfying f = curl ϕ + π with the corresponding estimate. Combining the above results with [1] and [3], we can solve the follow elliptical systems. Theorem 2.8. i) Let g L 1 (Γ) and g n W 1+ 1 n, satisfying the compatibility condition Γ g = 0 and < g n, 1 > = 0. If div g W 2+ 1 n, n, then the system u = 0 in and u = g on Γ (12) n 6

8 has a unique very weak solution u L n/() (). ii) Let f L 1 (), g L 1 (Γ) and g n W 1+ 1 n, n satisfying the compatibility condition f + Γ g = 0 and f R n n + < g n, 1 > = 0. If + then the system f ξ + [ f, g R ] = sup n + Γ g ξ < ξ W 2,n 0 ( ), ξ 0 ξ W 2,n 0 ( ) u = f in and has a unique solution u W /() 0 (). u = g on Γ (13) iii) Let f L 1 () such that div f [W 2,n 0 () W 1 (Rn +)] and Then f W /() 0 () and the system f n = 0. u = f in ; u = 0 and has a unique solution u W /() 0 (). iv) Let f L 1 () such that f = 0. If u n = 0 on Γ (14) holds, then the system [ f ] = sup ξ D(R n ξ + ), =0 on Γ xn f ξ < ξ W 2,n 0 ( ) u = f in, u n = 0 and u = 0 on Γ has a unique solution u W /() 0 (). Proof. We will prove only the first point i). Step 1: The proof will be started by showing g W 1+ 1 n, n. Let µ D(Γ) and ϕ W 0 () such that ϕ = µ on Γ. Thanks to Theorem 2.2, there exist ψ W 0 () L () and η W 2,n 0 () such that ψ n = 0 on Γ satisfying ϕ = ψ + η and the estimate ψ L ( ) + ψ W 0 ( ) + η W 2,n 0 ( ) C ϕ W 0 ( ). (15) Then < g, µ > D (Γ) D(Γ) = Γ g ψ < div g, η >W 2+ 1 n, n W 2 1 n,n + < g n, ϕ n >W 1+ 1 n, n W 1 1 n,n 7

9 and < g, µ > D (Γ) D(Γ) g L 1 (Γ) ψ L ( ) + div g W 2+ 1 n, n η W 2,n 0 ( ) + g n W 1+ n 1, n ϕ n W 1 1 n,n. Thanks to the density of D(Γ) in W 1 1 n,n and (15), we can deduce g W 1+ 1 n, n. Step 2: The system (13) is equivalent to the following one: Find u belonging L n/() () such that for all v W 2,n 0 () W 1 (Rn +), u v = < g, v n > W 1+ n 1, n W 1 n 1,n But for all F L n (), there exists v W 2,n 0 () W 1 (Rn +), unique up to an element of x n R n, such that v = F in, v = 0 on Γ and the following estimate holds v W 2,n 0 ( )/xnrn C F Ln ( ). Then, we have for all a R n, < g, v > = < g, (v + ax n) > C g W 1+ n 1, n W 1+ n 1, n W 1 n 1,n Consequently, taking the infinum, we have < g, v > W 1+ n 1, n W 1 n 1,n Then, the linear operator T : F < g, v > W 1+ n 1, n W 1 n 1,n 0 (R n ). + v + ax n W 2,n C g W 1+ n 1, n F L n ( ). W 1+ n 1, n W 1 n 1,n is continuous on L n () and thanks to the Riesz reprensentation theorem, there exists a unique u L n/() () such that T (F ) = u F, i.e., u is the solution of (13). Acknowledgments. We would like to thank Jean Van Schaftingen for many helpful comments and discussions. HHN s research has been supported in part by FAPESP grant #2009/ and #2007/

10 References [1] C. Amrouche, The Neumann problem in the half-space, Comptes Rendus de l Académie des Sciences 335, Série I, , (2002). [2] C. Amrouche, V. Girault, J. Giroire, Weighted Sobolev spaces for the Laplace equation in R n, Journal de Mathmatiques Pures et Appliques, 73 (6), , (1994). [3] C. Amrouche, S. Nečasová, Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition, Mathematica Bohemica, 126 (2), , (2001). [4] J. Bourgain, H. Brézis, On the equation div Y = f and application to control of phases, Journal of the American Mathematical Society, 16 (2), , (2002). [5] J. Bourgain, H. Brézis, New estimates for elliptic equations and Hodge type systems, Journal of the European Mathematical Society, 9 (2), , (2007). [6] H. Brézis, J. V. Schaftingen, Boundary extimates for elliptic systems with L 1 -data, Calculus of Variations and Partial Differential Equations, 30 (3), , (2007). [7] V. Girault, The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of R 3, Journal of the faculty of science, The University of Tokyo, 39 (Sec. IA) (2), , (1992). [8] J. Van Schaftingen, Estimates for L 1 -vector fields, Comptes Rendus de l Académie des Sciences 339, Série I, , (2004). 9