Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 13, 1-12; http://www.math.u-szeged.hu/ejqtde/

Fredholmness of an Abstract Differential Equation of Elliptic Type

Aissa Aibeche Département de Mathématiques, Université Ferhat Abbes Route de Scipion 19000 Setif, Algeria E-mail; aibeche@wissal.dz

Abstract

In this work, we obtain algebraic conditions which assure the Fredholm solvability of an abstract differential equation of elliptic type. In this respect, our work can be considered as an extension of Yakubov’s results to the case of boundary conditions containing a linear operator. Although essential technical, this extension is not straight forward as we show it below. The obtained abstract result is applied to a non regular boundary value problem for a second order partial differential equation of an elliptic type in a cylindrical domain. It is interesting to note that the problems considered in cylindrical domains are not coercive.

Keywords:Ellipticity, Abstract Differential Equation, Fredholmness, Non regular elliptic problems.

AMS Subject Classification:35J05, 35P20, 34L10, 47E05.

Contents

1 Introduction 1

2 Preliminaries 2

3 Homogeneous Abstract differential equation 2

4 Non homogeneous abstract differential equation 5

5 Fredholmness for an abstract differential equation 7

6 Fredholmness for non regular elliptic problems 9

1 Introduction

Many works are devoted to the study of hyperbolic or parabolic abstract equations [12, 13, 7]. In [12, 15] regular boundary value problems for elliptic abstract equations are considered. A few works are concerned with non regular problems, essentially done by Yakubov but he did not consider boundary conditions containing linear operators. We shall be interested to this type of problems, in particular, differential operator equation with a linear operator in the boundary conditions, for which we shall prove the Fredholmness in spite of non regularity. The non regularity here means that the boundary conditions do not satisfy the Shapiro-Lopatinski conditions, because they are not local, the problem is in fact a two-points problem.

Consider in the space L_{p}(0, 1; H) where H is a Hilbert space, the boundary value problem for the second order abstract differential equation.

L(D)u = −u′′(x) + Au(x) + A(x)u(x) = f(x)  x ∈ (0, 1),  (1)
\begin{aligned}L_{1}u & = & δu(0) & = & f_{1} \\ L_{2}u & = & u′(1) + Bu(0) & = & f_{2} \end{aligned}  (2)

A, A(x), B are linear operators and δ is a complex number. The idea is to write (1), (2) as a sum of two problems, where the first one is a principal problem with a parameter which gives an isomorphism and we establish a non coercive estimates for its solution and the second one is a compact perturbation of the first. Then we use the Kato perturbation theorem to conclude. Finally, we apply the abstract results obtained to a non regular boundary value problem for a concrete partial differential equation in a cylinder. This paper is organized as follows. In the second section, we give some preliminaries. In the third section, we study an homogeneous abstract differential equation, we prove the isomorphism and the non coercive estimates for the solution, an estimates which is not explicit with respect to the spectral parameter. In the fourth section, we consider a non homogeneous abstract differential equation, splitting the solution into two parts, the first part is the solution of a homogeneous problem and the second part is a solution of an abstract differential equation on the whole axis. Using results of the previous section and the Fourier multipliers we obtain analogous results as in the previous section. In the fifth section, we consider a general problem, then we use the Kato perturbation theorem. In the sixth section, we apply the obtained abstract results to a boundary value problem for a concrete partial differential equation.

2 Preliminaries

Let H_{1}, H be Hilbert spaces such that H_{1} ⊂ H with continuous injection. We define the space

W_{p}^{2}(0, 1; H_{1}, H) = {u; u ∈ L_{p}(0, 1, H_{1}), u” ∈ L_{p}(0, 1, H)}

provided with the finite norm

∥u∥_{W_{p}^{2}(0,1;H_{1},H)} = ∥u∥_{L_{p}(0,1;H_{1})} + ∥u′′∥_{L_{p}(0,1;H)}

Let E_{0}, E_{1} be two Banach spaces, continuously embedded in the Banach space E, the pair {E_{0}, E_{1}} is said an interpolation couple. Consider the Banach space

E_{0} + E_{1} = {u : u ∈ E, ∃u_{j} ∈ E_{j}, j = 0, 1, with u = u_{0} + u_{1}} ,

∥u∥_{E_{0}+E_{1}} = inf_{u=u_{0}+u_{1}, u_{j}∈E_{j}} (∥u_{0}∥_{E_{0}} + ∥u_{1}∥_{E_{1}}) ,

and the functional

K(t, u) = inf_{u=u_{0}+u_{1}, u_{j}∈E_{j}} (∥u_{0}∥_{E_{0}} + t ∥u_{1}∥_{E_{1}}) .

The interpolation space for the couple {E_{0}, E_{1}} is defined, by the K-method, as follows

(E_{0}, E_{1})_{θ,p} = \{u : u ∈ E_{0} + E_{1}, ∥u∥_{θ,p} = (∫_{0}^{∞} t^{−1−θp}K^{p}(t, u)dt)^{\frac{1}{p}} < ∞\} ,

0 < θ < 1, 1 ≤ p ≤ ∞.

(E_{0}, E_{1})_{θ,∞} = {u : u ∈ E_{0} + E_{1}, ∥u∥_{θ,∞} = sup_{t∈(0,∞)} t^{−θ}K(t, u)dt < ∞} ,

0 < θ < 1.

Let A be a closed operator in H. H(A) is the domain of A provided with the hilbertian norm

∥u∥_{H(A)}^{2} = ∥Au∥^{2} + ∥u∥^{2} , u ∈ D(A).

3 Homogeneous Abstract differential equation

Looking to the principal part of the problem (1), (2) with a parameter

L(D)u = −u′′(x) + (A + λI)u(x) = 0  x ∈ (0, 1),  (3)
\begin{aligned}L_{1}u & = & δu(0) & = & f_{1} \\ L_{2}u & = & u′(1) + Bu(0) & = & f_{2} \end{aligned}  (4)

Theorem 3.1 Assume that the following conditions are satisfied

1. A is positive linear closed operator in H.

2. δ ≠ 0.

3. B is continuous from H(A) in H(A^{1/2}) and from H(A^{1/2}) in H.

Then the problem (3), (4) for f_{1} ∈ (H, H(A))_{1− \frac{1}{2p} ,p}, f_{2} ∈ (H, H(A))_{\frac{1}{2} − \frac{1}{2p} ,p} and λ

such that |arg λ| ≤ ϕ < π, |λ| ⟶ ∞, has a unique solution in the space W_{p}^{2}(0, 1; H(A), H), with p ∈ (1, ∞), and for the solution of the problem (3), (4) the following non coercive estimate holds

∥u′′∥_{L_{p}(0,1;H)} + ∥Au∥_{L_{p}(0,1;H)}
≤ C(λ) (∥f_{1}∥_{(H,H(A))_{1− \frac{1}{2p} ,p}} + ∥f_{2}∥_{f_{2}∈(H,H(A))_{\frac{1}{2} − \frac{1}{2p} ,p}}) .  (5)

Proof. The solution u, belonging to W_{p}^{2}(0, 1; H(A), H), of the equation (3) is in the form

u(x) = e^{−xA \frac{1}{2} ; λ} g_{1} + e^{−(1−x)A \frac{1}{2} ; λ} g_{2}  (6)

with A_{λ} = A+λI, g_{1} and g_{2} belong to (H, H(A))_{1−1/2p,p}. Indeed let u ∈ W_{p}^{2}(0, 1; H(A), H) be a solution of (3). Then we have

(D − Aλ^{\frac{1}{2}} ) (D + Aλ^{\frac{1}{2}} ) u(x) = 0.

Note by

v(x) = (D + Aλ^{\frac{1}{2}} ) u(x).

From [17, p.168] v ∈ W_{p}^{1}(0, 1; H(A^{\frac{1}{2}} ), H) and

(D − Aλ^{\frac{1}{2}} ) v(x) = 0.  (7)

So

v(x) = e^{−(1−x)A \frac{1}{2} ; λ} v(1).  (8)

where, according to [14, p.44],

v(1) ∈ (H(A^{\frac{1}{2}} ), H)_{\frac{1}{p} ,p} = (H, H(A^{\frac{1}{2}} ))_{1− \frac{1}{p} ,p} .

From (7) , (8) we have

u(x) = e^{−xA \frac{1}{2} ; λ} u(0) + ∫_{0}^{x} e^{−(x−y)A \frac{1}{2} ; λ} e^{−(1−y)A \frac{1}{2} ; λ} v(1)dy

= e^{−xA \frac{1}{2} ; λ} u(0) + \frac{1}{2}A_{λ}^{− \frac{1}{2}} \{e^{−(1−x)A \frac{1}{2} ; λ} − e^{−xA \frac{1}{2} ; λ} e^{−A \frac{1}{2} ; λ} \} v(1),

where u(0) ∈ (H(A), H)_{\frac{1}{2p} ,p} [14, p.44]. Now,

A^{\frac{1}{2}} : (H, H(A))_{\frac{1}{2} − \frac{1}{2p} ,p} → (H, H(A))_{\frac{p−1}{2p} ,p} = (H, H(A))_{1− \frac{1}{2p} ,p}

is an isomorphism. Consequently the last inequality is in the form (6).

Let us show the reverse, i.e. the function u in the form (6) with g_{1} and g_{2} in (H, H(A))_{1−1/2p,p}, belongs to W_{p}^{2}(0, 1; H(A), H). From interpolation spaces properties see [11], [14, p.96] and the expression (6) of the function u we have

∥u∥_{W_{p}^{2}(0,1;H(A),H)} ≤ (∥AA_{λ}^{−1}∥ + 1) \{(∫_{0}^{1} ∥A_{λ}e^{−xA \frac{1}{2} ; λ} g_{1}∥^{p} dx)^{\frac{1}{p}}

+ (∫_{0}^{1} ∥A_{λ}e^{−(1−x)A \frac{1}{2} ; λ} g_{2}∥^{p} dx)^{\frac{1}{p}} \}

≤ C (∥g_{1}∥_{(H,H(A_{λ}))_{1−1/2p,p}} + ∥g_{2}∥_{(H,H(A_{λ}))_{1−1/2p,p}})
≤ C(λ) (∥g_{1}∥_{(H,H(A))_{1−1/2p,p}} + ∥g_{2}∥_{(H,H(A))_{1−1/2p,p}}) .  (9)

The function u satisfies the boundary conditions (2) if

\{ \begin{matrix}δg_{1} + δe^{−A \frac{1}{2} ; λ} g_{2} & = & f_{1} \\ −Aλ^{\frac{1}{2}} e^{−A \frac{1}{2} ; λ} g_{1} + Bg_{1} + Aλ^{\frac{1}{2}} g_{2} + Be^{−A \frac{1}{2} ; λ} g_{2} & = & f_{2} \end{matrix}

which we can write in matrix form as:

[( \begin{matrix}δI & 0 \\ B & Aλ^{\frac{1}{2}} \end{matrix} ) + ( \begin{matrix}0 & δe^{−A \frac{1}{2} ; λ} \\ −Aλ^{\frac{1}{2}} e^{−A \frac{1}{2} ; λ} & Be^{−A \frac{1}{2} ; λ} \end{matrix} )] ( \begin{matrix}g_{1} \\ g_{2} \end{matrix} ) = ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) .  (10)

The first matrix of operators is invertible, its inverse is

( \begin{matrix}\frac{1}{δ}I & 0 \\ −\frac{1}{δ}A_{λ}^{− \frac{1}{2}} B & A_{λ}^{− \frac{1}{2}} \end{matrix} ) .  (11)

Multiplying the two members of (10) by the matrix inverse (11), we get the following system:

\{ \begin{matrix}g_{1} + e^{−A \frac{1}{2} ; λ} g_{2} & = & \frac{1}{δ}f_{1} \\ e^{−A \frac{1}{2} ; λ} g_{1} + g_{2} & = & −\frac{1}{δ}A_{λ}^{− \frac{1}{2}} Bf_{1} + A_{λ}^{− \frac{1}{2}} f_{2} \end{matrix}

we can solve it by Cramer’s method, because the coefficients of the linear system are

bounded linear operators. The determinant is given by I + e^{−2A \frac{1}{2} ; λ} which is invertible

as a little perturbation of unity, in fact ∥e^{−2A \frac{1}{2} ; λ} ∥ ≤ q < 1.

Hence the solution is written as

\{ \begin{matrix}g_{1} & = & \frac{1}{δ}f_{1} + R_{11}(λ)f_{1} + R_{12}(λ)f_{2} \\ g_{2} & = & −\frac{1}{δ} (I + T (λ)) A_{λ}^{− \frac{1}{2}} Bf_{1} + (I + T (λ)) A_{λ}^{− \frac{1}{2}} f_{2} + R_{21}(λ)f_{1} \end{matrix}  (12)

where R_{ij}(λ) are given by

\{ \begin{matrix}R_{11}(λ) & = & −\frac{1}{δ} (I + T (λ)) e^{−2A \frac{1}{2} ; λ} + \frac{1}{δ} (I + T (λ)) A_{λ}^{− \frac{1}{2}} e^{−A \frac{1}{2} ; λ} B \\ R_{12}(λ) & = & − (I + T (λ)) A_{λ}^{− \frac{1}{2}} e^{−A \frac{1}{2} ; λ} \\ R_{21}(λ) & = & \frac{1}{δ} (I + T (λ)) e^{−A \frac{1}{2} ; λ} \end{matrix}

and satisfy ∥R_{ij}(λ)∥ → 0 when |λ| → ∞. (I + T (λ)) is the inverse of I +e^{−2A \frac{1}{2} ; λ} obtained from the corresponding Neumann series. Finally the solution u is given by

u(x) = e^{−xA \frac{1}{2} ; λ} (\frac{1}{δ}f_{1} + R_{11}(λ)f_{1} + R_{12}(λ)f_{2})

+e^{−(1−x)A \frac{1}{2} ; λ} (−\frac{1}{δ} (I + T (λ)) A_{λ}^{− \frac{1}{2}} Bf_{1} + (I + T (λ)) A_{λ}^{− \frac{1}{2}} f_{2} + R_{21}(λ)f_{1}) .

From the assumptions of theorem (3.1) and the properties of interpolation spaces, the following applications are continuous,

(I + T (λ)) A_{λ}^{− \frac{1}{2}} B : (H, H(A))_{1−1/2p,p} ⟼ (H, H(A))_{1−1/2p,p} ,
(I + T (λ)) A_{λ}^{− \frac{1}{2}} : (H, H(A))_{1/2−1/2p,p} ⟼ (H, H(A))_{1−1/2p,p} .

Then we have the estimates

∥(I + T (λ)) A_{λ}^{− \frac{1}{2}} Bf_{1}∥_{(H,H(A))_{1−1/2p,p}} ≤ C ∥f_{1}∥_{(H,H(A))_{1−1/2p,p}}

and

∥(I + T (λ)) A_{λ}^{− \frac{1}{2}} f_{2}∥_{(H,H(A))_{1−1/2p,p}} ≤ C ∥f_{2}∥_{(H,H(A))_{1/2−1/2p,p}} .

This give a bound of g_{k} in function of f_{k},namely

∥g_{k}∥_{(H,H(A))_{1−1/2p,p}} ≤ C (∥f_{1}∥_{(H,H(A))_{1−1/2p,p}} + ∥f_{2}∥_{(H,H(A))_{1/2−1/2p,p}}) .

Finally, taking into account of the estimate (9) we obtain the following non coercive estimate

∥u∥_{W_{p}^{2}(0,1;H(A),H)} ≤ C (λ) (∥f_{1}∥_{(H,H(A))_{1−1/2p,p}} + ∥f_{2}∥_{(H,H(A))_{1/2−1/2p,p}}) .

4 Non homogeneous abstract differential equation

Consider, now, the principal problem for the non homogeneous equation with a parameter

L_{0}(λ, D)u = −u′′(x) + A_{λ}u(x) = f(x)  x ∈ (0, 1),  (13)
\begin{aligned}L_{10}u & = & δu(0) & = & f_{1} \\ L_{20}u & = & u′(1) + Bu(0) & = & f_{2} \end{aligned}  (14)

We have the result.

Theorem 4.1 Suppose the following conditions satisfied

1. A is a positive linear closed operator in H.

2. B is continuous from H(A) in H(A^{1/2}) and from H(A^{1/2}) in H.

3. δ ≠ 0.

Then the problem (13),(14), for f, f_{1} and f_{2} in L_{p}(0, 1; H), (H, H(A))_{1− \frac{1}{2p} ,p} and (H, H(A))_{\frac{1}{2} − \frac{1}{2p} ,p}

respectively, and for λ such that |arg λ| ≤ ϕ < π, |λ| ⟶ ∞, has a unique solution belonging to the space W_{p}^{2}(0, 1; H(A), H), for p ∈ (1, ∞), and the following non coercive estimate holds

∥u′′∥_{L_{p}(0,1;H)} + ∥Au∥_{L_{p}(0,1;H)} ≤
C(λ) \{∥f∥_{L_{p}(0,1;H)} + ∥f_{1}∥_{(H,H(A))_{1− \frac{1}{2p} ,p}} + ∥f_{2}∥_{f_{2}∈(H,H(A))_{\frac{1}{2} − \frac{1}{2p} ,p}}\} (15)

Proof. In the theorem (3.1) we proved the uniqueness. Let us show, now, that the solution of the problem (13), (14) belonging to W_{p}^{2}(0, 1; H(A), H) can be written in the form u(x) = u_{1}(x) + u_{2}(x), u_{1}(x) is the restriction to [0, 1] of ũ_{1}(x) where ũ_{1}(x) is the solution of the equation

L_{0}(λ, D)ũ_{1}(x) = f̃(x),  x ∈ ℝ,  (16)

with f̃(x) = f(x) if x ∈ [0, 1] and f̃(x) = 0 otherwise. u_{2}(x) is the solution of the problem

L_{0}(λ, D)u_{2} = 0, L_{10}u_{2} = f_{1} − L_{10}u_{1}, L_{20}u_{2} = f_{2} − L_{20}u_{1}.  (17)

The solution of the equation (16) is given by the formula

û_{1}(x) = \frac{1}{\sqrt{2π}} ∫_{ℝ} e^{iμx}L_{0}(λ, iμ)^{−1}ˆf̃(μ)dμ  (18)

where ˆ f̃is the Fourier transform of the function f̃(x), L_{0}(λ, s) is the characteristic pencil of the equation (16) i.e. L_{0}(λ, s) = −s^{2}I + A + λI. From (18) it follows that:

\left∥{ \atop u_{1}}\right∥_{W_{p}^{2}(0,1;H(A),H)} ≤ \left∥{ \atop u_{1}}\right∥_{L_{p}(R;H(A))} + ∥u′′_{1}∥_{L_{p}(ℝ;H)}

≤ ∥F^{−1}L_{0}(λ, iμ)^{−1}F f̃(μ)∥_{L_{p}(R;H(A))} + ∥F^{−1}(iμ)^{2}L_{0}(λ, iμ)^{−1}F f̃(μ)∥_{L_{p}(ℝ;H)}

≤ ∥F^{−1}AL_{0}(λ, iμ)^{−1}F f̃(μ)∥_{L_{p}(R;H)} + ∥F^{−1}(iμ)^{2}L_{0}(λ, iμ)^{−1}F f̃(μ)∥_{L_{p}(ℝ;H)} (19)

where F is the Fourier transform. Let’s prove that the functions

T_{k+1}(λ, μ) = (iμ)^{2}A^{1−k}L_{0}(λ, iμ)^{−1}, k = 0, 1;  (20)

are Fourier multipliers of type (p, p) in H. From condition (1) of theorem (4.1) for |arg λ| ≤ ϕ < π, |λ| → ∞ and μ ∈ ℝ we have

∥L_{0}(λ, iμ)^{−1}∥ = ∥(A + λI + μ^{2}I)^{−1}∥ ≤ C(1 + |λ + μ^{2}|)^{−1} ≤ C |μ|^{−2}  (21)
∥AL_{0}(λ, iμ)^{−1}∥ = ∥A(A + λI + μ^{2}I)^{−1}∥ ≤ C  (22)

From (20) , (21) , (22) we have

∥T_{1}(λ, μ)∥_{B(H)} ≤ C ∥AL_{0}(λ, iμ)^{−1}∥_{B(H)} ≤ C  (23)

∥T_{2}(λ, μ)∥_{B(H)} ≤ Cμ^{2} ∥L_{0}(λ, iμ)^{−1}∥_{B(H)} ≤ C  (24)

Since

\frac{∂}{∂μ}T_{k+1}(λ, μ) = 2ki^{2k}μ^{2k−1}A^{1−k}L_{0}(λ, iμ)^{−1}

−i^{2k+1}μ^{2k}A^{1−k}L_{0}(λ, iμ)^{−1} \frac{∂}{∂μ}L_{0}(λ, iμ)L_{0}(λ, iμ)^{−1},

then

∥\frac{∂}{∂μ}T_{k+1}(λ, μ)∥_{B(H)} ≤ C |μ|^{−1} .  (25)

According to Mikhlin-Schwartz theorem [6, p.1181], from (23) , (24) , (25) it follows that the functions μ → T_{k+1}(λ, μ) are Fourier multipliers of type (p, p) in H. Then, from (19) we have

\left∥{ \atop u_{1}}\right∥_{W_{p}^{2}(ℝ;H(A),H)} ≤ C ∥f̃∥_{L_{p}(ℝ,H)}

hence u_{1} ∈ W_{p}^{2}(0, 1; H(A), H) and from [14, p.44] we have

u′_{1}(0) ∈ (H(A), H)_{\frac{1}{2} + \frac{1}{2p} ,p} = (H, H(A))_{\frac{1}{2} − \frac{1}{2p} ,p},
u_{1}(0) ∈ (H, H(A))_{1− \frac{1}{2p} ,p}.

Therefore L_{10}u_{1} ∈ (H, H(A))_{1− \frac{1}{2p} ,p} and L_{20}u_{1} ∈ (H, H(A))_{\frac{1}{2} − \frac{1}{2p} ,p}. From theorem (3.1), the problem (17) when |arg λ| ≤ ϕ < π, |λ| → ∞ has a solution u_{2}(x) which is in W_{p}^{2}(0, 1; H(A), H). So, from the theorem (3.1) and the estimate (5) we obtain the desired inequality (15).

5 Fredholmness for an abstract differential equation

We can, now, find conditions for Fredholm solvability the problem (1), (2). It is more practical to formulate this in terms of unbounded Fredholm operators in Banach spaces.

Let’s define 𝕃 by

\begin{aligned}𝕃 : & W_{p}^{2}(0, 1; H(A), H) & → & L_{p}(0, 1; H) ⊕ (H, H(A))_{1− \frac{1}{2p}} ⊕ (H, H(A))_{\frac{1}{2} − \frac{1}{2p}} \\ & u & → & (L(D)u, L_{1}u, L_{2}u) \end{aligned}

where

D (𝕃) = {u; u ∈ W_{p}^{2}(0, 1; H(A), H); L(D)u ∈ L_{p}(0, 1; H),
L_{1}u ∈ (H, H(A))_{1− \frac{1}{2p} ,p}, L_{2}u ∈ (H, H(A))_{\frac{1}{2} − \frac{1}{2p} ,p}\}

L(D)u, L_{1}u and L_{2}u are defined previously..

Theorem 5.1 Suppose the following conditions satisfied

1. A is a linear closed positive operator in H. The injection of H(A) in H is compact.

2. B is continuous from H(A) in H(A^{1/2}) and from H(A^{1/2}) in H.

3. δ ≠ 0.

4. A(x) : H(A) → H(A^{\frac{1}{2}} ) is compact for almost all x ∈ [0, 1] .

∀ε > 0, for almost all x ∈ [0, 1] ,

∥A(x)u∥_{H(A^{\frac{1}{2}} )} ≤ ε ∥u∥_{H(A)} + C(ε) ∥u∥_{H} , u ∈ D(A).

The functions A(x)u, when u ∈ H, are measurable from [0, 1] in H(A^{\frac{1}{2}} ).

Then

a - ∀u ∈ W_{p}^{2}(0, 1; H(A), H), and for |arg λ| ≤ ϕ < π, |λ| ⟶ ∞, p ∈ (1, ∞), the non coercive estimate holds

∥u′′∥_{L_{p}(0,1;H)} + ∥Au∥_{L_{p}(0,1;H)} ≤ C(λ) {∥L(D)u∥_{L_{p}(0,1;H)}

+ ∥L_{1}u∥_{(H,H(A))_{1− \frac{1}{2p} ,p}} + ∥L_{2}u∥_{f_{2}∈(H,H(A))_{\frac{1}{2} − \frac{1}{2p} ,p}} + ∥u∥_{L_{p}(0,1;H)}\} . (26)

b - 𝕃 : u ⟶ (L(D)u, L_{1}u, L_{2}u) from W_{p}^{2}(0, 1; H(A), H) to L_{p}(0, 1; H)⊕(H, H(A))_{1− \frac{1}{2p}} ⊕ (H, H(A))_{\frac{1}{2} − \frac{1}{2p}} is a Fredholm.

Proof. a- Let u be a solution of (1), (2) belonging to W_{p}^{2}(0, 1; H(A), H). Then u is a solution of the problem

L_{0}(λ, D)u = f(x) + λu(x) − A(x)u(x)  x ∈ (0, 1),  (27)
\begin{aligned}L_{10}u & = & δu(0) & = & f_{1} \\ L_{20}u & = & u′(1) + Bu(0) & = & f_{2}. \end{aligned}  (28)

From theorem (4.1) we get the estimate

∥u′′∥_{L_{p}(0,1;H)} + ∥Au∥_{L_{p}(0,1;H)} ≤ C(λ) {∥f(.) + λu(.) − A(.)u(.)∥_{L_{p}(0,1;H)}

+ ∥f_{1}∥_{(H,H(A))_{1− \frac{1}{2p} ,p}} + ∥f_{2}∥_{f_{2}∈(H,H(A))_{\frac{1}{2} − \frac{1}{2p} ,p}}\} .  (29)

Condition (4) of theorem and [16, lemme 1.2, p.162] yield the compacity of the operator

u → λu − A(.)u
W_{p}^{2}(0, 1; H(A), H) → L_{p}(0, 1; H).

And from [16, lemme 2.7, p.53] ∀ε > 0 we have

∥f(.) + λu(.) − A(.)u(.)∥_{L_{p}(0,1;H)} ≤
C [∥f∥_{L_{p}(0,1;H)} + ε (∥u∥_{L_{p}(0,1;H)} + ∥Au∥_{L_{p}(0,1;H)}) + C(ε) ∥u∥_{L_{p}(0,1;H)}] .

Substituting in (29) we obtain the desired estimate. b- The operator 𝕃 can be written in the form

𝕃 = 𝕃_{0λ} + 𝕃_{1λ}  (30)

where

D (𝕃_{0λ}) = {u; u ∈ W_{p}^{2}(0, 1; H(A), H); L_{0}(λ, D)u ∈ L_{p}(0, 1; H),
L_{10}u ∈ (H, H(A))_{1− \frac{1}{2p} ,p}, L_{20}u ∈ (H, H(A))_{\frac{1}{2} − \frac{1}{2p} ,p}\}
𝕃_{0λ}u = (L_{0}(λ, D)u, L_{10}u, L_{20}u)
L_{0}(λ, D)u = −u′′(x) + (A + λI)u(x), x ∈ [0, 1]
L_{10}u = δu(0)
L_{20}u = u′(1) + Bu(0)

and

𝕃_{1λ}u = (−λu(x) + A(x)u(x), 0, 0) ,

condition (4) implies that D (𝕃) = D (𝕃_{0λ}) . According to the theorem (4.1), when λ → ∞,the operator

𝕃_{0λ} : W_{p}^{2}(0, 1; H(A), H) → L_{p}(0, 1; H) ⊕ (H, H(A))_{1− \frac{1}{2p}} ⊕ (H, H(A))_{\frac{1}{2} − \frac{1}{2p}}

has an inverse. Always, from condition (4) of the theorem and from [16, lemme 1.2, p.162] we show that the operator 𝕃_{1λ}u from W_{p}^{2}(0, 1; H(A), H) in L_{p}(0, 1; H)⊕(H, H(A))_{1− \frac{1}{2p}} ⊕ (H, H(A))_{\frac{1}{2} − \frac{1}{2p}} is compact. Applying the perturbation theorem on Fredholm operators (see [12] or [10, p.238]), to the operator (30) we get the desired result.

6 Fredholmness for non regular elliptic problems

The Fredholm property for regular elliptic boundary value problems has been proved, in particular in, [1, 2, 3, 8]. In the case of non regular problems it has been done for a class of elliptic problems which still coercive, in [4, 5]. Here, we establish the same property, but for a class of elliptic problems which are not coercive. In [17, 18] other non regular boundary conditions for the same type of equations are considered and analogous results are proved.

Consider in the domain [0, 1] × [0, 1] the non regular boundary value problem for a second order elliptic equation

L(x, y, D_{x}, D_{y})u = D_{x}^{2}u(x, y) + D_{y}(a(y)D_{y}u(x, y)) + A(x)u(x, .)|_{y}
= f(x, y), (x, y) ∈ [0, 1] × [0, 1] ,  (31)

P_{1}u = D_{y}^{m_{1}}u(x, 0) + ∑_{j≤m_{1}−1} a_{j}D_{y}^{j}u(x, 0) = 0, x ∈ [0, 1] ,  (32)
P_{2}u = D_{y}^{m_{2}}u(x, 1) + ∑_{j≤m_{2}−1} b_{j}D_{y}^{j}u(x, 1) = 0, x ∈ [0, 1] ,

L_{1}u = δu(0, y) = f_{1}(y), y ∈ [0, 1] ,  (33)
L_{2}u = D_{x}u(1, y) + Bu(0, y) = f_{2}(y), y ∈ [0, 1] ,

where 0 ≤ m_{1}, m_{2} ≤ 1, a_{j}, b_{j}, δ ∈ ℂ , D_{x} = \frac{∂}{∂x}, D_{y} = \frac{∂}{∂y} . Note, as in [14, p.186,317], (L_{2} (G) , W_{2}^{2}(G))_{θ,p} = B_{2,p}^{2θ} (G) where G ⊂ ℝ^{r} is a bounded domain.

Theorem 6.1 Suppose

1. a ∈ C^{1} (0, 1) , ∃ρ > 0 such that a(y) ≥ ρ.

2. A(x) : W_{2}^{2}(0, 1; P_{s}u = 0) in W_{2}^{1}(0, 1; P_{s}u = 0, m_{s} = 0) are compact for almost all x ∈ [0, 1] ; ∀ε > 0 and for almost all x ∈ [0, 1] ∥A(x)u∥_{W_{2}^{1}(0,1)} ≤ ε ∥u∥_{W_{2}^{2}(0,1)} + C(ε) ∥u∥_{L_{2}(0,1)} ∀u ∈ W_{2}^{2}(0, 1; P_{s}u = 0); the function x → A(x)u from [0, 1] in W_{2}^{1}(0, 1) are measurable.

3. B is continuous from H(A) in H(A^{1/2}) and from H(A^{1/2}) in H.

Then the operator

L : u → (L(x, y, D_{x}, D_{y})u, L_{1}u, L_{2}u)

from

W_{p}^{2}(0, 1; W_{2}^{2}(0, 1; P_{s}u = 0), L_{2}(0, 1))

in

L_{p}(0, 1; W_{2}^{1}(0, 1; P_{s}u = 0, m_{s} = 0))

⨁_{k=1}^{2} B_{2,p}^{k+1− \frac{1}{p}} (0, 1; P_{s}u = 0, m_{s}< k+\frac{1}{2}−\frac{1}{p}; P_{s}(a(y)u′(y))′ = 0, m_{s}< k−\frac{3}{2}−\frac{1}{p})

is a Fredholm.

Proof.

Set H = L_{2}(0, 1), and define the operators A and 𝔸(x) by the equalities

D_{A} = W_{2}^{2}(0, 1; P_{s}u = 0) Au = −D_{y}(a(y)D_{y}u(y)),
D_{𝔸(x)} = W_{2}^{1}(0, 1; P_{s}u = 0, m_{s} = 0) 𝔸(x)u = A(x)u|_{y}.

The problem (31) , (32) , (33) can be written as

u”(x) − Au(x) + 𝔸(x)u(x) = f(x),  (34)
δu′(0) = f_{1}
u′(1) + Bu(0) = f_{2}  (35)

with u(x) = u(x, .), f(x) = f(x, .) and f_{k} = f_{k}(.) are functions with values in the Hilbert space H = L_{2}(0, 1).

Applying the theorem (5.1) to the problem (34) (35) . From condition (1) the operator A is positively defined in H and from [14, p.344] the injection W_{2}^{2}(0, 1; P_{s}u = 0) ⊂ L_{2}(0, 1) is compact, then the condition (1) of the theorem (5.1) is satisfied. From [9] (see also [14, p.321]) we have

(H, H(A))_{θ,p} = (L_{2}(0, 1), W_{2}^{2}(0, 1; P_{s}u = 0))_{θ,p}
= B_{2,p}^{2θ} (0, 1; P_{k}u = 0, m_{k} < 2θ − 1/2) .

Therefore

(H, H(A))_{1/2} = B_{2,2}^{1} (0, 1; P_{k}u = 0, m_{k} = 0)
= W_{2}^{1} (0, 1; P_{k}u = 0, m_{k} = 0) ,

and

(H, H(A))_{\frac{2p−jp−1}{2p} ,p} = B_{2,p}^{2−j−1/p} (0, 1; P_{k}u = 0, m_{k} < 3/2 − j − 1/p) .

So, for the problem (34) (35) , all the conditions of theorem (5.1) are satisfied. From which it may be concluded the assertion of the theorem (6.1) .

Consider, now, in the cylindrical domain [0, 1]×G where G ⊂ ℝ^{r} is a bounded domain, the boundary value problem for the second order elliptic equation

L(x, y, D_{x}, D_{y})u = D_{x}^{2}u(x, y) + ∑_{i,j=1}^{r} D_{i}(a_{ij}(y)D_{j}u(x, y)) + A(x)u(x, .)|_{y}
= f(x, y), (x, y) ∈ [0, 1] × G,  (36)

P u = ∑_{|η|≤m} b_{η}D_{y}^{η}u(x, y′) = 0, (x, y′) ∈ [0, 1] × Γ,  (37)
L_{1}u = δu(0, y) = f_{1}(y), y ∈ [0, 1] ,  (38)
L_{2}u = D_{x}u(1, y) + Bu(0, y) = f_{2}(y), y ∈ [0, 1] ,

where m ≤ 1; δ ∈ ℂ ^{∗}, D_{x} = \frac{∂}{∂x}, Γ = ∂G is the boundary of G. D_{x} = \frac{∂}{∂x}D_{y}^{η} = D_{1}^{η_{1}} ⋯ D_{r}^{η_{r}}, D_{j}^{η_{j}} = \frac{∂}{∂y_{j}} , η_{=} (η_{1}, ⋯, η_{r}) , |η| = η_{1} + ⋯ + η_{r}. Theorem 6.2 Suppose

1. a_{ij} ∈ C^{1} (\overline{G}) , b_{η} ∈ C^{2−m} (\overline{G}) , Γ ∈ C^{2}.

2. a_{ij}(y) = \overline{a_{ji}(y)}; ∃γ > 0 such that ∑_{k,j=1}^{r} a_{kj}(y)σ_{k}σ_{j} ≥ γ ∑_{k=1}^{r} σ_{k}^{2}, y ∈ \overline{G}, σ ∈ ℝ^{r}

3. A(x) : W_{2}^{2}(G; P u = 0) in W_{2}^{1}(G; P u = 0, m = 0) are compact for almost all x ∈ G; ∀ε > 0 and for almost all x ∈ [0, 1] we have ∥A(x)u∥_{W_{2}^{1}(G)} ≤ ε ∥u∥_{W_{2}^{2}(G)} + C(ε) ∥u∥_{L_{2}(G)} ∀u ∈ W_{2}^{2}(G; P u = 0); the functions x → A(x)u from [0, 1] in W_{2}^{1}(G) are measurable.

4. B is continuous from H(A) in H(A^{1/2}) and from H(A^{1/2}) in H. Then the operator

L : u → (L(x, y, D_{x}, D_{y})u, L_{1}u, L_{2}u)

from

W_{p}^{2}(0, 1; W_{2}^{2}(G; P u = 0), L_{2}(G))

in

L_{p}(0, 1; W_{2}^{1}(G; P u = 0, m = 0))

⨁_{k=1}^{2} B_{2,p}^{k+1− \frac{1}{p}} (G; P u = 0, m < k+\frac{1}{2}−\frac{1}{p}; P ∑_{i,j=1}^{r} D_{i}(a_{ij}(y)D_{j}u(y)) = 0, m< k−\frac{3}{2}−\frac{1}{p})

is a Fredholm. Proof.

Set H = L_{2}(G), the operators A are 𝔸(x) defined by the equalities

D_{A} = W_{2}^{2}(G; P u = 0) Au = − ∑_{i,j=1}^{r} D_{i}(a_{ij}(y)D_{j}u(y)),

D_{𝔸(x)} = W_{2}^{1}(G; P u = 0, m = 0) 𝔸(x)u = A(x)u|_{y}.

The problem (36) , (37) , (38) can be written in the form (34) , (35) i.e.

u”(x) − Au(x) + 𝔸(x)u(x) = f(x),

δu′(0) = f_{1}
u′(1) + Bu(0) = f_{2}

with u(x) = u(x, .), f(x) = f(x, .) and f_{k} = f_{k}(.) are functions with values in the Hilbert space H = L_{2}(0, 1). The rest of the proof is similar of the one of the theorem (6.1) .

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(Received December 6, 2002)