Dang Anh Tuan

Nguyen Minh Chuong

Youri V. Egorov

Egorov, Youri V.

Minh Chuong, Nguyen

Anh Tuan, Dang

Numérisation de Documents Anciens Mathématiques

ebonne,mspacesdeérentielsentpeut êtreuvelles des’existenceC. R. Acad. Sci. Paris, Ser. I 337 (2003) 451–456Partial Differential EquationsA semilinear non-classical pseudodifferential boundary valuproblem in the Sobolev spacesYouri V. Egorova, Nguyen Minh Chuongb, Dang Anh Tuanca Laboratoire des mathématiques pour l’industrie et la physique, UMR 5640, UFRMIG, Université Paul Sabatier, 118, route de Nar31062 Toulouse, Franceb Institute of Mathematics, NCST, PO Box 631, BoHo, 10000 Hanoi, Viet Namc Faculty of Mathematics–Mechanics–Informatics, Hanoi National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Viet NaReceived 31 July 2003; accepted 30 August 2003Presented by Louis NirenbergAbstractWe study the solvability of a semilinear non-classical pseudodifferential boundary value problem in the SobolevHl,p,q , 1< p <∞, depending on a complex parameterq. To cite this article: Y.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I337 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.RésuméUn problème aux limites semilinéaire non-classique pour des opérateurs pseudodifférentiels dans les espacesSobolev.On étudie la résolution d’un problème aux limites semilinéaire non-classique pour des opérateurs pseudodiffdans les espaces de SobolevHl,p,q , 1< p < ∞, munis de normes dépendant d’un paramètre complexeq. Pour citer cetarticle : 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.Version française abrégéeNous étudions des problèmes aux limites elliptiques non-linéaires dépendant d’un paramètre complexq pourdes opérateurs pseudodifférentiels dans les espaces de SobolevHl,p,q , 1< p < ∞, munis des normes dépendade q . Nous considérons le cas quand dans les conditions au bord contient un champ de vecteurs quitangent au bord sur une sous-variété. En établissant des résultats pour le cas linéaire dans des classes noopérateurs pseudodifférentiels et en utilisant le théorème des points fixés de Schauder on peut démontrer let 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 rightsreserved.doi:10.1016/j.crma.2003.08.006452 Y.V. Egorov et al. / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 451–456manymay bemin [8]rs in therom thee1. Oblique derivative problemThe oblique derivative problem for an elliptic differential equation of second order was investigated byauthors, in particular, Egorov and Kondratiev studied in [7] the problem in the case when the vector fieldtangent to the bounded smooth boundaryΩ on a smooth submanifoldΓ0. Trung discussed in [12] the problefor singular integro-differential operators of Agranovich type [3]. Egorov and Nguyen Minh Chuong studiedsome semilinear classical and non-classical boundary value problems for the mentioned above operatoSobolev spacesHl,2.Note that the assumptions and the proof of our main Theorem 5.2. here are new and quite different fcorresponding ones in [8].2. DefinitionsLetp, l ∈ R, 1<p <∞. LetF , F ′ be the Fourier transforms with respect tox = (x ′, xn) ∈ Rn, x ′, respectively.Denote byHl,p(Rn) the completion of the spaceC∞0 (Rn) with respect to the norm‖u‖l,p = ‖u‖l,p,Rn =( ∫Rnξ(1+ |ξ |)pl∣∣Fu(ξ)∣∣p dξ)1/p.The spacesHl,p(Rn+),Hl,p(Ω),Hl,p(∂Ω),Hl,p(Γ0) are defined in the standard way (see [1,10,11]). SetQ ={q ∈ C | α0  argq  β0}. By Hl,p,q we denote the above spaces depending on a complex parameterq ∈ Q:‖|u‖|l,p = (‖u‖pl,p + |q|lp‖u‖p0,p)1/p. We use the functionsKαβ(x, z)=Hαβ(z)+ kαβ(x, z), where(i) Hαβ(z), kαβ(x, z) are positively homogeneous of degree (−n) with respect toz;(ii) kαβ(x, z) ∈ C∞(Rn) with respect tox,Dγx kαβ(x, z)→ 0, as|x| → ∞, for all γ ;(iii)∫|z|=1Hαβ(z)dσ(z)= 0,∫|z|=1 kαβ(x, z)dσ(z)= 0;(iv)∫|z|=1 |Hαβ(z)|2 dσ(z) <+∞,∫Rnx∫|z|=1 |kαβ(x, z)|2 dσ(z)dx <+∞, whereσ(z) is an area element of thsphere{z ∈ Rn | |z| = 1}.Let s ∈ Z+. Let us consider the following class of pseudodifferential operators inC∞0 (Rn) with symbolsσA(x, ξ, q):(a) Au(x, q)= (2π)−n/2 ∫Rnξei(x,ξ)σA(x, ξ, q)Fu(ξ)dξ,where(b) σA(x, ξ, q)= ∑|α|+βs qβgαβ(x, ξ)ξα ;(c) gαβ(x, ξ) = g0αβ(x, ξ)+ g1αβ(ξ),with(d) g0αβ(x, ξ) = (2π)−n/2∫Rnze−i(z,ξ)kαβ(x, z)dz;(e) g1αβ(ξ) = (2π)−n/2∫Rne−i(z,ξ)Hαβ(z)dz.zY.V. Egorov et al. / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 451–456 453ened2,9]).tTheorem 2.1.The operatorA can be extended to a bounded linear operator fromHl,p,q(Rn) toHl−s,p,q(Rn) andthe estimate‖|Au‖|l−s,p  C‖|u‖|l,p, u ∈ Hl,p,q(Rn)(1)holds true, whereC is a constant not depending onu, q.Theorem 2.2.LetB0, B, B ′ be the Banach spaces with the norms‖ · ‖0, ‖ · ‖, ‖ · ‖′, respectively. Assume that thembeddingB ↪→ B0 is compact andA : B → B ′ is a bounded linear operator. Thendim(KerA) < +∞ and ImAis closed inB ′ if and only if‖u‖  C(‖Au‖′ + ‖u‖0), u ∈ B, whereC is a constant not depending onu.Proof. See [4]. ✷3. Linear boundary value problemNow consider the linear boundary value problemAu(x, q)= f (x, q), in Ω, (2)Bju(x, q)= gj (x, q), on∂Ω, j = 1, . . . , s, (3)whereA, Bj are pseudodifferential operators of orders 2s, mj , respectively. It is assumed that the problem defiby (A,Bj ) is elliptic that isA is an elliptic operator and the Shapiro–Lopatinski condition is satisfied (see [Moreover they areadmissible, that is, for instance, the symbol of the principal partA0 of the operatorA has theform: σA0(x′,0, η′, ηn) = ∑2sk=0σA0,k (x ′, η′)ηkn, σA0,k is positively homogeneous of degree 2s − k in η′, σA0,2s isindependent ofη′. Here the directionxn is normal.Let 1 < p < ∞, '  max{2s,mj + 1}, and put U = (A,Bj |∂Ω), H',p,q(Ω, ∂Ω) = H'−2s,p,q(Ω) ×∏sj=1H'−mj−1/p,p,q(∂Ω) with the norm‖|(f, g)‖|',p = ‖|(f, g1, . . . , gs)‖|',p = ‖|f ‖|'−2s,p +s∑j=1‖|gj‖|'−mj−1/p,p.If the problem (2), (3) is elliptic, the operatorU 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 fromH',p,q(Ω) to H',p,q(Ω, ∂Ω) and has a parametrix;(ii) U is a Noether operator fromH',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, whereC is a constant nodepending onu, q .4. Linear non-classical pseudodifferential boundary value problemConsider a linear non-classical pseudodifferential boundary value problemAu(x, q)= f (x, q), in Ω, (4)Bj(Dνu(x, q)) = gj (x, q), on∂Ω, j = 1,2, . . . , s, (5)whereA, Bj are admissible pseudodifferential operators of orders 2s, mj − 1, respectively.454 Y.V. Egorov et al. / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 451–456ehodsces, andIf Dν is a vector field that can be tangent to∂Ω on a smooth manifoldΓ0 ⊂ ∂Ω of dimensionn− 2 we supposethat it is not tangent toΓ0. We use here the classification ofΓ0 given in [7]. If Γ0 belongs to the first class, then wadd the following conditionsDknu(x, q)= u0k(x, q), onΓ0, k = 0,1, . . . , s − 1. (6)Let 1< p < ∞, '  max{2s,mj + 1}. Denote byΠl,p,q(Ω),Gl,p,q,Λl,p,q the spaces introduced in [7], witnorms depending on complex parameterq ∈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), anduis a solution of problem(4), (5) (or (4)–(6))andAu ∈ Hl−2s+2,p,q(Ω), BjDνu|∂Ω ∈ Hl−mj+2−1/p,p,q(∂Ω) (oradditionallyDknu|Γ0 ∈ Hl−k+1−1/p,p,q(Γ0)), thenu ∈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 eachε > 0, there exists a small enough neighborhoQp of P ∈ Γ0 such that foru ∈ H',p,q(Ω) and u(x, q) = 0 outside ofQp the following estimate hold‖|χ(z) ∫ z10 u(η, z2, . . . , zn, q)dη‖|',p < ε‖|u‖|',p, where χ is the characteristic function of the support ofu,supposing thatΓ0 is situated in the planex1 = 0.Proof. See [5,7]. ✷In the sequel we denoteSu(z, q)= χ(z) ∫ z10 u(η, z2, . . . , zn, q)dη.Lemma 4.3.LetΓ0 belong to the first class. LetUp be a neighborhood ofP ∈ Γ0. LetR be the parametrix of theelliptic problem: Au = f in Up , Bju = gj on ∂Up, j = 1, . . . , s. SetL2s = ADν − DνA,L0w = R(L2sw,0).Then, for the neighborhoodUp with small enough diameter we have the following representation: L0Sw =L1w + L′1w, SL0w = L2w + L′2w whereL1, L2 are operators fromHl,p,q(Up) to Hl,p,q(Up) having norms<12, andL′1, L′2 are bounded operators fromHl,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 spathe Sobolev embedding theorems inHl,p,q one can prove the following theorems on parametrix.Theorem 4.4.If Γ0 belongs to the first class, then the operatorV = (A,BjDν |∂Ω,Dj−1n |Γ0) :Πl,p,q → Λl,p,q,defined by(4)–(6), possesses a right parametrixR :Λl,p,q →Πl,p,q , that isVR = I + T , T :Λl,p,q → Λl,p,q is acompact operator.Theorem 4.5.If Γ0 belongs to the third class, then the operatorV = (A,BjDν |∂Ω) :Πl,p,q(Ω)→Πl−2s,p,q(Ω)×Πsj=1Gl−mj−1/p,p,q(∂Ω), defined by(4)–(5)possesses a right parametrixR :Πl−2s,p,q(Ω)×s∏j=1Gl−mj−1/p,p,q(∂Ω)→Πl,p,q(Ω),Y.V. Egorov et al. / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 451–456 455obolevdaryediatelyons.that isVR = I + T ,T :Πl−2s,p,q(Ω)×s∏j=1Gl−mj−1/p,p,q(∂Ω)→Πl−2s,p,q(Ω)×s∏j=1Gl−mj−1/p,p,q(∂Ω)is a compact operator.5. Semilinear non-classical pseudodifferential boundary value problemNow we consider a semilinear non-classical pseudodifferential boundary value problem. Using the Sspaces depending on a complex parameterq ∈ Q, for the above linear non-classical pseudodifferential bounvalue problem, from the estimates in smoothness theorems and theorems on parametrix we get immuniqueness 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.Letf ∈ Π'−2s,p,q(Ω), gj ∈ G'−mj−1/p,p,q(∂Ω), andu0k ∈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 toΠl,p,q(Ω).Let us now consider the semilinear non-classical pseudodifferential boundary value problemAu(x, q)= f (x, q,u, . . . ,D2s−1), in Ω, (7)Bj(Dνu(x, q)) = gj (x, q,u, . . . ,Dmj−1u), on∂Ω, j = 1, . . . , s, (8)whereA, Bj are pseudodifferential operators in (4), (5). IfΓ0 belongs to the first class then we add the conditiDknu(x, q)= u0k(x, q), k = 0, . . . , s − 1, onΓ0. (9)Using Theorem 5.1 and the Schauder’s fixed point theorem, one can prove the following main theoremTheorem 5.2.Let Γ0 be the manifold of first class. Assume that the operator(A,Bj ) is elliptic, the functionsf ,gj are measurable and satisfy(in a local coordinate system atx)(i) |E(u, ξ, q)| M(h1(ξ, q)+ ∑2s−1k=0 |q|k|ξ |2s−1−k|Fu(ξ, q)|),(ii) |E(u, ξ, q)−E(v, ξ, q)| M ∑2s−1k=0 |q|k|ξ |2s−1−k|Fu(ξ, q)−Fv(ξ, q)|,(iii) |Gj(u, ξ ′, q)| M(h2j (ξ ′, q)+ ∑mj−1k=0 |q|k|ξ ′|mj−1−k|F ′u(ξ ′, q)|),(iv) |Gj(u, ξ ′, q)−Gj(v, ξ ′, q)| M ∑mj−1k=0 |q|k|ξ ′|mj−1−k|F ′u(ξ ′, q)−F ′v(ξ ′, q)|, withE(u, ξ, q)= F (f (x, q,u(x, q), . . .,D2s−1u(x, q)))(ξ, q),Gj (u, ξ′, q)= F ′(gj (x ′, q,u(x ′,0, q), . . . ,Dmj−1u(x ′,0, q)))(ξ ′, q),whereM is a constant,h1(ξ, q) 0, h2j (ξ ′, q) 0,∫Rn(1+ |ξ | + |q|)(l−2s)p∣∣h1(ξ, q)∣∣p dξ < Lp,∫Rn−1(1+ |ξ ′| + |q|)(l−mj )p∣∣h2j (ξ ′, q)∣∣p dξ ′ <Lp.Then, ifu0k ∈ Hl−k−1/p,p,q(Γ0), k = 0,1, . . . , s − 1, the problem(7)–(9)has a unique solutionu ∈ Πl,p,q, forsufficiently large|q|.456 Y.V. Egorov et al. / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 451–456nntnbmittedted forProof. For eachω ∈Πl,p,q , the problemAu(x, q)= f (x, q,ω, . . . ,D2s−1ω), x ∈Ω , BjDνu(x, q)= gj (x, q,ω,. . . ,Dmj−1ω), x ∈ ∂Ω , j = 1, . . . , s, Djnu(x, q) = u0j (x, q), x ∈ Γ0, j = 0,1, . . . , s − 1, possesses a solutioJω ∈ Πl,p,q, for sufficiently large|q|. We obtain‖Jω‖Πl,p,q  C1(‖f ‖Πl−2s,p,q +s∑j=1‖gj‖Gl−mj−1/p,p,q +s−1∑j=0‖|u0j‖|l−j−1/p,p,Γ0) C2(L+ ‖ω‖Πl−1,p,q +s∑j=1(L+ ‖ω‖Πl−1,p,q) +s−1∑j=0‖|u0j‖|l−j−1/p,p,Γ0) C3 +C4‖ω‖Πl−1,p,q  C3 +C4|q|−1‖ω‖Πl,p,q . (10)Note that‖Jω1 − Jω2‖Πl,p,q  C5(∥∥f (x, q,ω1, . . . ,D2s−1ω1) − f (x, q,ω2, . . . ,D2s−1ω2)∥∥Πl−2s,p,q+s∑j=1∥∥gj (x ′, q,ω1, . . . ,Dmj−1ω1) − gj (x ′, q,ω2, . . . ,Dmj−1ω2)∥∥Gl−mj−1/p,p,q) C6‖ω1 −ω2‖Πl−1,p,q . (11)From (10), there exists a positive numberR such thatJ :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 embeddingS ↪→ Πl−1,p,q is compact,{ωn} is relatively compact inΠl−1,p,q . Consequently by (11),{Jωn} is relatively compact inΠl,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 poiv ∈ S, which is the solution of (7)–(9). By (11) the fixed point ofJ is unique, i.e.,v ∈ S is thus a unique solutioof (7)–(9). ✷In the same way one can study the case whenΓ0 belongs to the third class.References[1] R.A. Adams, Sobolev Spaces, Academic Press, 1975.[2] S. Agmon, A. Douglis, L. Nirenberg, Comm. Pure Appl. Math. (1959) 624–727.[3] M.S. Agranovich, Uspekhi Mat. Nauk 20 (5) (1965) 3–120.[4] Nguyen Minh Chuong, Dang Anh Tuan, Preprint 2002/28, Institute of Mathematics, Hanoi, 2002.[5] Nguyen Minh Chuong, Tran Tri Kiet, Preprint 2003/04, Institute of Mathematics, Hanoi, 2003, Uspekhi Mat. Nauk (in Russian), sufor publication.[6] K. Deimling, Nonlinear Analysis, Springer-Verlag, 1985.[7] Yu.V. Egorov, V.A. Kondratiev, Mat. Sb. 78(120) (1) (1969) 148–176.[8] Yu.V. Egorov, Nguyen Minh Chuong, Uspekhi Mat. Nauk 53 324 (6) (1998) 249–250.[9] Yu.V. Egorov, Mat. Sb. 73 (1967) 356–374.[10] Tran Tri Kiet, Preprint 2003/03, Institute of Mathematics, Hanoi, 2003, Differentsial’nye Uravneniya (in Russian), submitpublication.[11] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Plenum Press, New York, 1982.[12] Le Quang Trung, Uspekhi Mat. Nauk 44 (5) (1989) 169–170.

A semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces

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A semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces

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