DSpace at VNU: A semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces

Partial Differential EquationsA semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces aLaboratoire des mathématiques pour l’industrie et la physique, U

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Partial Differential Equations

A semilinear non-classical pseudodifferential boundary value

problem in the Sobolev spaces

aLaboratoire des mathématiques pour l’industrie et la physique, UMR 5640, UFRMIG, Université Paul Sabatier, 118, route de Narbonne,

31062 Toulouse, France

bInstitute of Mathematics, NCST, PO Box 631, BoHo, 10000 Hanoi, Viet Nam

cFaculty 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

H l,p,q , 1 < 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).

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 H l,p,q , 1 < 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).

Version française 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 H l,p,q , 1 < 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).

doi:10.1016/j.crma.2003.08.006

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1 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 H l,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]

2 Definitions

Let p, l ∈ R, 1 < p < ∞ Let F , Fbe the Fourier transforms with respect to x = (x, x n )∈ Rn , x, respectively.

Denote by H l,p (Rn ) the completion of the space C∞

0 (Rn ) with respect to the norm

u l,p = u l,p,Rn=

Rn ξ

1+ |ξ|plF u(ξ )p

dξ

1/p

.

The spaces H l,p (Rn

+), H l,p (Ω), H l,p (∂Ω), H l,p (Γ0) are defined in the standard way (see [1,10,11]) Set Q=

{q ∈ C | α0 argq β0} By H l,p,q we denote the above spaces depending on a complex parameter q ∈ Q:

|u| l,p = (u p

l,p + |q| lp u p

0,p ) 1/p We use the functions K αβ (x, z) = H αβ (z) + k αβ (x, z), where (i) H αβ (z), k αβ (x, z) are positively homogeneous of degree ( −n) with respect to z;

(ii) k αβ (x, z) ∈ C∞(Rn ) with respect to x, D x γ k αβ (x, z) → 0, as |x| → ∞, for all γ ;

(iii)

|z|=1 H αβ (z) dσ (z)= 0,|z|=1 k αβ (x, z) dσ (z)= 0;

(iv)

|z|=1 |H αβ (z)|2dσ (z) <+∞,Rn

x

|z|=1 |k αβ (x, z)|2dσ (z) dx < +∞, where σ(z) is an area element of the

sphere{z ∈ R n | |z| = 1}.

Let s∈ Z+ Let us consider the following class of pseudodifferential operators in C∞

σ A (x, ξ, q):

(a) Au(x, q) = (2π) −n/2

Rn

ξ ei(x,ξ ) σ A (x, ξ, q)F u(ξ ) dξ,

where

(b) σ A (x, ξ, q)=|α|+βs q β g αβ (x, ξ )ξ α;

(c) g αβ (x, ξ ) = g0

αβ (x, ξ ) + g1

αβ (ξ ),

with

(d) g αβ0 (x, ξ ) = (2π) −n/2

Rn

ze−i(z,ξ) k

αβ (x, z) dz;

(e) g αβ1 (ξ ) = (2π) −n/2

Rne−i(z,ξ) H αβ (z) dz.

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Theorem 2.1 The operator A can be extended to a bounded linear operator from Hl,p,q (Rn ) to H l −s,p,q (Rn ) and the estimate

|Au| l −s,p C|u| l,p , u ∈ H l,p,q

Rn

(1)

holds true, where C is a constant not depending on u, q.

Theorem 2.2 Let B0, B, Bbe the Banach spaces with the norms · 0, · , · , respectively Assume that the

embedding B " → B0is compact and A : B → Bis a bounded linear operator Then dim(Ker A) < +∞ and Im A

is closed in Bif and only if u C(Au+ u0), u ∈ B, where C is a constant not depending on u.

Proof See [4]. ✷

3 Linear boundary value problem

Now consider the linear boundary value problem

where A, B j are pseudodifferential operators of orders 2s, m j, respectively It is assumed that the problem defined

by (A, B j ) 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 A0of the operator A has the form: σ A0(x, 0, η, η

n )=2s

k=0σ A 0,k (x, η)η k

n , σ A 0,k is positively homogeneous of degree 2s − k in η, σ

A 0,2s is

independent of η Here the direction x nis normal.

Let 1 < p < ∞, ' max{2s, m j + 1}, and put U = (A, B j|∂Ω ), H ',p,q (Ω, ∂Ω) = H ' −2s,p,q (Ω)× s

j=1H ' −m j −1/p,p,q (∂Ω) with the norm

|(f, g)| ',p = |(f, g1, , g s )|',p = |f | ' −2s,p+

s

j=1

|g j|' −m j −1/p,p

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

(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

4 Linear non-classical pseudodifferential boundary value problem

Consider a linear non-classical pseudodifferential boundary value problem

B j

D ν u(x, q)

where A, B j are admissible pseudodifferential operators of orders 2s, m j− 1, respectively

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If D ν is a vector field that can be tangent to ∂Ω on a smooth manifold Γ0⊂ ∂Ω of dimension n − 2 we suppose

that it is not tangent to Γ0 We use here the classification of Γ0given in [7] If Γ0belongs to the first class, then we add the following conditions

Let 1 < p < ∞, ' max{2s, m j + 1} Denote by Π l,p,q (Ω), G l,p,q , Λ l,p,q the spaces introduced in [7], with

norms depending on complex parameter q ∈ Q.

Theorem 4.1 If Γ0belongs to the second or third class (or Γ0belongs to the first class), u ∈ H l,p,q (Ω), and u

is a solution of problem (4), (5) (or (4)–(6)) and Au ∈ H l −2s+2,p,q (Ω), B j D ν u|∂Ω ∈ H l −m j +2−1/p,p,q (∂Ω) (or additionally D k n u|Γ0∈ H l −k+1−1/p,p,q (Γ0)), then u ∈ H l +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

Q p of P ∈ Γ0 such that for u ∈ H ',p,q (Ω) and u(x, q) = 0 outside of Q p the following estimate holds

|χ(z)z1

0 u(η, z2, , z n , q) dη|',p < ε |u| ',p , where χ is the characteristic function of the support of u, supposing that Γ0is situated in the plane x1= 0.

Proof See [5,7]. ✷

In the sequel we denote Su(z, q) = χ(z)z1

0 u(η, z2, , z n , q) dη.

Lemma 4.3 Let Γ0belong to the first class Let U p be a neighborhood of P ∈ Γ0 Let R be the parametrix of the elliptic problem: Au = f in U p , B j u = g j on ∂U p , j = 1, , s Set L 2s = AD ν − D ν A, L0w = R(L 2s w, 0) Then, for the neighborhood U p with small enough diameter we have the following representation: L0Sw=

L1w + L

1w, SL0w = L2w + L

2w where L1, L2 are operators from H l,p,q (U p ) to H l,p,q (U p ) having norms

<12, and L

1, L

2are bounded operators from H l,p,q (U p ) to H l +1,p,q (U p ).

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 H l,p,q one can prove the following theorems on parametrix

Theorem 4.4 If Γ0belongs to the first class, then the operator

V =A, B j D ν|∂Ω , D j−1

n |Γ0

: Π l,p,q → Λ l,p,q , defined by (4)–(6), possesses a right parametrix R : Λ l,p,q → Π l,p,q , that is V R = I + T , T : Λ l,p,q → Λ l,p,q is a compact operator.

Theorem 4.5 If Γ0belongs to the third class, then the operator V = (A, B j D ν|∂Ω ) : Π l,p,q (Ω) → Π l −2s,p,q (Ω)×

Π j s=1G l −m j −1/p,p,q (∂Ω), defined by (4)–(5) possesses a right parametrix

R : Π l −2s,p,q (Ω)×

s

j=1

G l −m j −1/p,p,q (∂Ω) → Π l,p,q (Ω),

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that is V R = I + T ,

T : Π l −2s,p,q (Ω)×

s

j=1

G l −m j −1/p,p,q (∂Ω) → Π l −2s,p,q (Ω)×

s

j=1

G l −m j −1/p,p,q (∂Ω)

is a compact operator.

5 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 Γ0be a manifold of the first or of the third class Assume that the operator (A, B j ) is elliptic Let f ∈ Π ' −2s,p,q (Ω), g j ∈ G ' −m j −1/p,p,q (∂Ω), and u 0k ∈ H ' −k−1/p,p,q (Γ0), if Γ0belongs to the first class Then for large enough |q|, there exists a unique solution of the problem (4), (5) or (4)–(6) belonging to

Π l,p,q (Ω).

Let us now consider the semilinear non-classical pseudodifferential boundary value problem

Au(x, q) = fx, q, u, , D 2s−1

B j

D ν u(x, q)

= g j

x, q, u, , D m j−1u

where A, B j are pseudodifferential operators in (4), (5) If Γ0belongs to the first class then we add the conditions

Using Theorem 5.1 and the Schauder’s fixed point theorem, one can prove the following main theorem

Theorem 5.2 Let Γ0be the manifold of first class Assume that the operator (A, B j ) is elliptic, the functions f ,

g j are measurable and satisfy (in a local coordinate system at x)

(i) |E(u, ξ, q)| M(h1(ξ, q)+2s−1

k=0 |q| k |ξ| 2s −1−k |F u(ξ, q)|),

(ii) |E(u, ξ, q) − E(v, ξ, q)| M2s−1

k=0 |q| k |ξ| 2s −1−k |F u(ξ, q) − F v(ξ, q)|,

(iii) |G j (u, ξ, q) | M(h 2j (ξ, q)+m j−1

k=0 |q| k |ξ|m j −1−k |Fu(ξ, q) |),

(iv) |G j (u, ξ, q) − G j (v, ξ, q) | Mm j−1

k=0 |q| k |ξ|m j −1−k |Fu(ξ, q) − Fv(ξ, q) |, with

E(u, ξ, q) = Ff

x, q, u(x, q), , D 2s−1u(x, q)

(ξ, q),

G j (u, ξ, q) = Fg j

x, q, u(x, 0, q), , D m j−1u(x, 0, q)

(ξ, q),

where M is a constant, h1(ξ, q) 0, h 2j (ξ, q) 0,

Rn

1+ |ξ| + |q|(l −2s)ph1(ξ, q)p

dξ < L p ,

Rn−1

1+ |ξ| + |q|(l −m j )ph 2j (ξ, q)p

dξ< L p

Then, if u 0k ∈ H l −k−1/p,p,q (Γ0), k = 0, 1, , s − 1, the problem (7)–(9) has a unique solution u ∈ Π l,p,q , for sufficiently large |q|.

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Proof For each ω ∈ Π l,p,q , the problem Au(x, q) = f (x, q, ω, , D 2s−1ω), x ∈ Ω, B j D ν u(x, q) = g j (x, q, ω, , D m j−1ω), x ∈ ∂Ω, j = 1, , s, D j

n u(x, q) = u 0j (x, q), x ∈ Γ0, j = 0, 1, , s − 1, possesses a solution

J ω ∈ Π l,p,q , for sufficiently large |q| We obtain

J ω Π l,p,q C1

f Π l −2s,p,q +

s

j=1

g jG l −mj −1/p,p,q +

s−1

j=0

|u 0j|l −j−1/p,p,Γ0

C2

L + ω Π l −1,p,q+

s

j=1

L + ω Π l −1,p,q

+

s−1

j=0

|u 0j|l −j−1/p,p,Γ0

C3+ C4ω Π l −1,p,q C3+ C4|q|−1ω Π l,p,q (10) Note that

J ω1− J ω2Π l,p,q C5

f x,q,ω

1, , D 2s−1ω

1

− fx, q, ω2, , D 2s−1ω

2

Π l −2s,p,q

+

s

j=1

g j

x, q, ω

1, , D m j−1ω

1

− g j

x, q, ω

2, , D m j−1ω

2

G l −mj −1/p,p,q

C6ω1− ω2Π l −1,p,q (11)

From (10), there exists a positive number R such that J : S → S, S = {ω ∈ Π l,p,q | ω Π l,p,q R}, and by (11),

Assume that {ω n } ⊂ S Because the embedding S "→ Π l −1,p,q is compact, {ω n} is relatively compact in

Π l −1,p,q Consequently by (11), {J ω n } is relatively compact in Π l,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 Γ0belongs 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), submitted for 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), submitted for publication.

[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.

[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,... of Mathematics, Hanoi, 2003, Differentsial’nye Uravneniya (in Russian), submitted for publication.

[11] F Treves, Introduction to Pseudodifferential and Fourier Integral... 2002.

[5] Nguyen Minh Chuong, Tran Tri Kiet, Preprint 2003/04, Institute of Mathematics, Hanoi, 2003, Uspekhi Mat Nauk (in Russian), submitted for publication.

[6]

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