C R Acad Sci Paris, Ser I 337 (2003) 451–456 Partial Differential Equations A semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces Youri V Egorov a , Nguyen Minh Chuong b , Dang Anh Tuan c a Laboratoire des mathématiques pour l’industrie et la physique, UMR 5640, UFRMIG, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse, France b Institute of Mathematics, NCST, PO Box 631, BoHo, 10000 Hanoi, Viet Nam c Faculty of Mathematics–Mechanics–Informatics, Hanoi National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam Received 31 July 2003; accepted 30 August 2003 Presented by Louis Nirenberg Abstract We study the solvability of a semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces Hl,p,q , < p < ∞, depending on a complex parameter q To cite this article: Y.V Egorov et al., C R Acad Sci Paris, Ser I 337 (2003) 2003 Académie des sciences Published by Éditions scientifiques et médicales Elsevier SAS All rights reserved Résumé Un problème aux limites semilinéaire non-classique pour des opérateurs pseudodifférentiels dans les espaces de Sobolev On étudie la résolution d’un problème aux limites semilinéaire non-classique pour des opérateurs pseudodifférentiels dans les espaces de Sobolev Hl,p,q , < p < ∞, munis de normes dépendant d’un paramètre complexe q Pour citer cet article : Y.V Egorov et al., C R Acad Sci Paris, Ser I 337 (2003) 2003 Académie des sciences Published by Éditions scientifiques et mộdicales Elsevier SAS All rights reserved Version franỗaise abrégée Nous étudions des problèmes aux limites elliptiques non-linéaires dépendant d’un paramètre complexe q pour des opérateurs pseudodifférentiels dans les espaces de Sobolev Hl,p,q , < p < ∞, munis des normes dépendant de q Nous considérons le cas quand dans les conditions au bord contient un champ de vecteurs qui peut être tangent au bord sur une sous-variété En établissant des résultats pour le cas linéaire dans des classes nouvelles des opérateurs pseudodifférentiels et en utilisant le théorème des points fixés de Schauder on peut démontrer l’existence et l’unicité des solutions du problème Cette Note est la continuation des articles publiés [7,8,12] E-mail address: egorov@mip.ups-tlse.fr (Y.V Egorov) 1631-073X/$ – see front matter 2003 Académie des sciences Published by Éditions scientifiques et médicales Elsevier SAS All rights reserved doi:10.1016/j.crma.2003.08.006 452 Y.V Egorov et al / C R Acad Sci Paris, Ser I 337 (2003) 451–456 Oblique derivative problem The oblique derivative problem for an elliptic differential equation of second order was investigated by many authors, in particular, Egorov and Kondratiev studied in [7] the problem in the case when the vector field may be tangent to the bounded smooth boundary Ω on a smooth submanifold Γ0 Trung discussed in [12] the problem for singular integro-differential operators of Agranovich type [3] Egorov and Nguyen Minh Chuong studied in [8] some semilinear classical and non-classical boundary value problems for the mentioned above operators in the Sobolev spaces Hl,2 Note that the assumptions and the proof of our main Theorem 5.2 here are new and quite different from the corresponding ones in [8] Definitions Let p, l ∈ R, < p < ∞ Let F , F be the Fourier transforms with respect to x = (x , xn ) ∈ Rn , x , respectively Denote by Hl,p (Rn ) the completion of the space C0∞ (Rn ) with respect to the norm u l,p = u l,p,Rn + |ξ | = pl F u(ξ ) p 1/p dξ Rnξ The spaces Hl,p (Rn+ ), Hl,p (Ω), Hl,p (∂Ω), Hl,p (Γ0 ) are defined in the standard way (see [1,10,11]) Set Q = {q ∈ C | α0 arg q β0 } By Hl,p,q we denote the above spaces depending on a complex parameter q ∈ Q: p p |u |l,p = ( u l,p + |q|lp u 0,p )1/p We use the functions Kαβ (x, z) = Hαβ (z) + kαβ (x, z), where (i) Hαβ (z), kαβ (x, z) are positively homogeneous of degree (−n) with respect to z; γ (ii) kαβ (x, z) ∈ C ∞ (Rn ) with respect to x, Dx kαβ (x, z) → 0, as |x| → ∞, for all γ ; (iii) |z|=1 Hαβ (z) dσ (z) = 0, |z|=1 kαβ (x, z) dσ (z) = 0; (iv) |z|=1 |Hαβ (z)|2 dσ (z) < +∞, Rn |z|=1 |kαβ (x, z)|2 dσ (z) dx < +∞, where σ (z) is an area element of the x sphere {z ∈ Rn | |z| = 1} Let s ∈ Z+ Let us consider the following class of pseudodifferential operators in C0∞ (Rn ) with symbols σA (x, ξ, q): (a) Au(x, q) = (2π)−n/2 Rnξ ei(x,ξ )σA (x, ξ, q)F u(ξ ) dξ, where (b) σA (x, ξ, q) = |α|+β s q β gαβ (x, ξ )ξ α ; (ξ ), (x, ξ ) + gαβ (c) gαβ (x, ξ ) = gαβ with (d) gαβ (x, ξ ) = (2π)−n/2 (e) −i(z,ξ ) k (x, z) dz; αβ Rnz e −n/2 −i(z,ξ ) gαβ (ξ ) = (2π) Hαβ (z) dz Rn e z Y.V Egorov et al / C R Acad Sci Paris, Ser I 337 (2003) 451–456 453 Theorem 2.1 The operator A can be extended to a bounded linear operator from Hl,p,q (Rn ) to Hl−s,p,q (Rn ) and the estimate |Au |l−s,p C |u |l,p , u ∈ Hl,p,q Rn (1) holds true, where C is a constant not depending on u, q Theorem 2.2 Let B0 , B, B be the Banach spaces with the norms · , · , · , respectively Assume that the embedding B → B0 is compact and A : B → B is a bounded linear operator Then dim(Ker A) < +∞ and Im A C( Au + u ), u ∈ B, where C is a constant not depending on u is closed in B if and only if u Proof See [4] ✷ Linear boundary value problem Now consider the linear boundary value problem Au(x, q) = f (x, q), Bj u(x, q) = gj (x, q), in Ω, (2) j = 1, , s, on ∂Ω, (3) where A, Bj are pseudodifferential operators of orders 2s, mj , respectively It is assumed that the problem defined by (A, Bj ) is elliptic that is A is an elliptic operator and the Shapiro–Lopatinski condition is satisfied (see [2,9]) Moreover they are admissible, that is, for instance, the symbol of the principal part A0 of the operator A has the k form: σA0 (x , 0, η , ηn ) = 2s k=0 σA0,k (x , η )ηn , σA0,k is positively homogeneous of degree 2s − k in η , σA0,2s is independent of η Here the direction xn is normal Let < p < ∞, max{2s, mj + 1}, and put U = (A, Bj |∂Ω ), H ,p,q (Ω, ∂Ω) = H −2s,p,q (Ω) × s H (∂Ω) with the norm −m −1/p,p,q j j =1 s |(f, g) | ,p = |(f, g1 , , gs ) | ,p = |f | −2s,p + |gj | −mj −1/p,p j =1 If the problem (2), (3) is elliptic, the operator U is called elliptic The following theorem (see [4,5]) is used to prove theorems in the next sections Theorem 3.1 If U is elliptic then (i) U is a bounded linear operator from H ,p,q (Ω) to H ,p,q (Ω, ∂Ω) and has a parametrix; (ii) U is a Noether operator from H ,p,q (Ω) to H ,p,q (Ω, ∂Ω); (iii) The estimate |u | ,p,q C( |Uu | ,p,q + |u |0,p,q ), u ∈ H ,p,q (Ω), holds true, where C is a constant not depending on u, q Linear non-classical pseudodifferential boundary value problem Consider a linear non-classical pseudodifferential boundary value problem Au(x, q) = f (x, q), in Ω, Bj Dν u(x, q) = gj (x, q), (4) on ∂Ω, j = 1, 2, , s, where A, Bj are admissible pseudodifferential operators of orders 2s, mj − 1, respectively (5) 454 Y.V Egorov et al / C R Acad Sci Paris, Ser I 337 (2003) 451–456 If Dν is a vector field that can be tangent to ∂Ω on a smooth manifold Γ0 ⊂ ∂Ω of dimension n − we suppose that it is not tangent to Γ0 We use here the classification of Γ0 given in [7] If Γ0 belongs to the first class, then we add the following conditions Dnk u(x, q) = u0k (x, q), on Γ0 , k = 0, 1, , s − (6) Let < p < ∞, max{2s, mj + 1} Denote by Πl,p,q (Ω), Gl,p,q , Λl,p,q the spaces introduced in [7], with norms depending on complex parameter q ∈ Q Theorem 4.1 If Γ0 belongs to the second or third class (or Γ0 belongs to the first class), u ∈ Hl,p,q (Ω), and u is a solution of problem (4), (5) (or (4)–(6)) and Au ∈ Hl−2s+2,p,q (Ω), Bj Dν u|∂Ω ∈ Hl−mj +2−1/p,p,q (∂Ω) (or additionally Dnk u|Γ0 ∈ Hl−k+1−1/p,p,q (Γ0 )), then u ∈ Hl+1,p,q (Ω) To prove Theorem 4.1 we need the following lemmas Lemma 4.2 Let Γ0 belong to the first class For each ε > 0, there exists a small enough neighborhood Qp of P ∈ Γ0 such that for u ∈ H ,p,q (Ω) and u(x, q) = outside of Qp the following estimate holds z |χ(z) u(η, z2 , , zn , q) dη | ,p < ε |u | ,p , where χ is the characteristic function of the support of u, supposing that Γ0 is situated in the plane x1 = Proof See [5,7] ✷ In the sequel we denote Su(z, q) = χ(z) z1 u(η, z2 , , zn , q) dη Lemma 4.3 Let Γ0 belong to the first class Let Up be a neighborhood of P ∈ Γ0 Let R be the parametrix of the elliptic problem: Au = f in Up , Bj u = gj on ∂Up , j = 1, , s Set L2s = ADν − Dν A, L0 w = R(L2s w, 0) Then, for the neighborhood Up with small enough diameter we have the following representation: L0 Sw = L1 w + L1 w, SL0 w = L2 w + L2 w where L1 , L2 are operators from Hl,p,q (Up ) to Hl,p,q (Up ) having norms < 12 , and L1 , L2 are bounded operators from Hl,p,q (Up ) to Hl+1,p,q (Up ) Using Lemmas 4.2, 4.3 and Theorem 3.1 we get Theorem 4.1 Using now theory of classical boundary value problems, theory of Fredholm operators in Banach spaces, and the Sobolev embedding theorems in Hl,p,q one can prove the following theorems on parametrix Theorem 4.4 If Γ0 belongs to the first class, then the operator j −1 V = A, Bj Dν |∂Ω , Dn |Γ0 : Πl,p,q → Λl,p,q , defined by (4)–(6), possesses a right parametrix R : Λl,p,q → Πl,p,q , that is V R = I + T , T : Λl,p,q → Λl,p,q is a compact operator Theorem 4.5 If Γ0 belongs to the third class, then the operator V = (A, Bj Dν |∂Ω ) : Πl,p,q (Ω) → Πl−2s,p,q (Ω)× Πjs=1 Gl−mj −1/p,p,q (∂Ω), defined by (4)–(5) possesses a right parametrix s R : Πl−2s,p,q (Ω) × Gl−mj −1/p,p,q (∂Ω) → Πl,p,q (Ω), j =1 Y.V Egorov et al / C R Acad Sci Paris, Ser I 337 (2003) 451–456 455 that is V R = I + T , s T : Πl−2s,p,q (Ω) × s Gl−mj −1/p,p,q (∂Ω) → Πl−2s,p,q (Ω) × j =1 Gl−mj −1/p,p,q (∂Ω) j =1 is a compact operator Semilinear non-classical pseudodifferential boundary value problem Now we consider a semilinear non-classical pseudodifferential boundary value problem Using the Sobolev spaces depending on a complex parameter q ∈ Q, for the above linear non-classical pseudodifferential boundary value problem, from the estimates in smoothness theorems and theorems on parametrix we get immediately uniqueness and existence theorems (with large enough |q|) as below Theorem 5.1 Let Γ0 be a manifold of the first or of the third class Assume that the operator (A, Bj ) is elliptic Let f ∈ Π −2s,p,q (Ω), gj ∈ G −mj −1/p,p,q (∂Ω), and u0k ∈ H −k−1/p,p,q (Γ0 ), if Γ0 belongs to the first class Then for large enough |q|, there exists a unique solution of the problem (4), (5) or (4)–(6) belonging to Πl,p,q (Ω) Let us now consider the semilinear non-classical pseudodifferential boundary value problem Au(x, q) = f x, q, u, , D 2s−1 , in Ω, Bj Dν u(x, q) = gj x, q, u, , D mj −1 u , (7) on ∂Ω, j = 1, , s, (8) where A, Bj are pseudodifferential operators in (4), (5) If Γ0 belongs to the first class then we add the conditions Dnk u(x, q) = u0k (x, q), k = 0, , s − 1, on Γ0 (9) Using Theorem 5.1 and the Schauder’s fixed point theorem, one can prove the following main theorem Theorem 5.2 Let Γ0 be the manifold of first class Assume that the operator (A, Bj ) is elliptic, the functions f , gj are measurable and satisfy (in a local coordinate system at x) 2s−1 k 2s−1−k |F u(ξ, q)|), k=0 |q| |ξ | 2s−1 k 2s−1−k |F u(ξ, q) − F v(ξ, q)|, k=0 |q| |ξ | mj −1 , q)| M(h2j (ξ , q) + k=0 |q|k |ξ |mj −1−k |F u(ξ , q)|), mj −1 , q) − Gj (v, ξ , q)| M k=0 |q|k |ξ |mj −1−k |F u(ξ , q) − F v(ξ (i) |E(u, ξ, q)| M(h1 (ξ, q) + (ii) |E(u, ξ, q) − E(v, ξ, q)| M (iii) |Gj (u, ξ (iv) |Gj (u, ξ , q)|, with E(u, ξ, q) = F f x, q, u(x, q), , D 2s−1 u(x, q) (ξ, q), Gj (u, ξ , q) = F gj x , q, u(x , 0, q), , D mj −1 u(x , 0, q) (ξ , q), where M is a constant, h1 (ξ, q) + |ξ | + |q| Rn (l−2s)p 0, h2j (ξ , q) h1 (ξ, q) p 0, + |ξ | + |q| dξ < Lp , (l−mj )p p h2j (ξ , q) dξ < Lp Rn−1 Then, if u0k ∈ Hl−k−1/p,p,q (Γ0 ), k = 0, 1, , s − 1, the problem (7)–(9) has a unique solution u ∈ Πl,p,q , for sufficiently large |q| 456 Y.V Egorov et al / C R Acad Sci Paris, Ser I 337 (2003) 451–456 Proof For each ω ∈ Πl,p,q , the problem Au(x, q) = f (x, q, ω, , D 2s−1 ω), x ∈ Ω, Bj Dν u(x, q) = gj (x, q, ω, j , D mj −1 ω), x ∈ ∂Ω, j = 1, , s, Dn u(x, q) = u0j (x, q), x ∈ Γ0 , j = 0, 1, , s − 1, possesses a solution J ω ∈ Πl,p,q , for sufficiently large |q| We obtain s Jω Πl,p,q C1 f Πl−2s,p,q s−1 + gj j =1 Gl−mj −1/p,p,q + |u0j |l−j −1/p,p,Γ0 j =0 s C2 L + ω Πl−1,p,q s−1 L+ ω + Πl−1,p,q + j =1 C3 + C4 ω Πl−1,p,q C3 + C4 |q|−1 ω |u0j |l−j −1/p,p,Γ0 j =0 Πl,p,q (10) Note that J ω1 − J ω2 Πl,p,q f x, q, ω1 , , D 2s−1 ω1 − f x, q, ω2 , , D 2s−1 ω2 C5 s + Πl−2s,p,q gj x , q, ω1 , , D mj −1 ω1 − gj x , q, ω2 , , D mj −1 ω2 j =1 C6 ω1 − ω2 Πl−1,p,q Gl−mj −1/p,p,q (11) From (10), there exists a positive number R such that J : S → S, S = {ω ∈ Πl,p,q | ω Πl,p,q R}, and by (11), J : S → S is continuous Assume that {ωn } ⊂ S Because the embedding S → Πl−1,p,q is compact, {ωn } is relatively compact in Πl−1,p,q Consequently by (11), {J ωn } is relatively compact in Πl,p,q It follows that J : S → S is compact, while S is convex, closed, bounded, so by Schauder’s fixed point theorem (see [6], p 60), J possesses a fixed point v ∈ S, which is the solution of (7)–(9) By (11) the fixed point of J is unique, i.e., v ∈ S is thus a unique solution of (7)–(9) ✷ In the same way one can study the case when Γ0 belongs to the third class References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] R.A Adams, Sobolev Spaces, Academic Press, 1975 S Agmon, A Douglis, L Nirenberg, Comm Pure Appl Math (1959) 624–727 M.S Agranovich, Uspekhi Mat Nauk 20 (5) (1965) 3–120 Nguyen Minh Chuong, Dang Anh Tuan, Preprint 2002/28, Institute of Mathematics, Hanoi, 2002 Nguyen Minh Chuong, Tran Tri Kiet, Preprint 2003/04, Institute of Mathematics, Hanoi, 2003, Uspekhi Mat Nauk (in Russian), submitted for publication K Deimling, Nonlinear Analysis, Springer-Verlag, 1985 Yu.V Egorov, V.A Kondratiev, Mat Sb 78(120) (1) (1969) 148–176 Yu.V Egorov, Nguyen Minh Chuong, Uspekhi Mat Nauk 53 324 (6) (1998) 249–250 Yu.V Egorov, Mat Sb 73 (1967) 356–374 Tran Tri Kiet, Preprint 2003/03, Institute of Mathematics, Hanoi, 2003, Differentsial’nye Uravneniya (in Russian), submitted for publication F Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Plenum Press, New York, 1982 Le Quang Trung, Uspekhi Mat Nauk 44 (5) (1989) 169–170 ... is a compact operator Semilinear non-classical pseudodifferential boundary value problem Now we consider a semilinear non-classical pseudodifferential boundary value problem Using the Sobolev spaces. .. Using now theory of classical boundary value problems, theory of Fredholm operators in Banach spaces, and the Sobolev embedding theorems in Hl,p,q one can prove the following theorems on parametrix... where C is a constant not depending on u, q Linear non-classical pseudodifferential boundary value problem Consider a linear non-classical pseudodifferential boundary value problem Au(x, q) =