Linear Second Order Ordinary Differential Equations with L^{p}-coefficients

Manfred Goebel Martin-Luther-Universität Halle-Wittenberg FB Mathematik und Informatik D – 06099 Halle (Saale), Germany

1991 Math. Subject Classification: 34A30, 34B24, 34B27 Key words: Cauchy problem, two-point boundary value problem, Dirichlet boundary conditions, Fredholm alternative, Green function Running head title: Second Order Differential Equation

Mailing address: Prof. Dr. M. Goebel Martin-Luther-Universität Halle-Wittenberg FB Mathematik und Informatik D – 06099 Halle (Saale), Germany

E-mail: goebel@maja.mathematik.uni-halle.de

Linear Second Order Ordinary Differential Equations with L^{p}-coefficients

By Manfred Goebel of Halle (Saale)

(Received December 24, 1995)

Abstract. The paper deals with a Dirichlet boundary value problem for a linear second order ordinary differential operator, whose coefficients belong to certain L^{p}-spaces. Its solution is to be understood in the sense of Sobolev, so that the Fredholm alternative holds. The main purpose of this paper is, in case of unique solvability, to introduce a Green function by means of which the solution can be given explicitly by integrals. We give the precise definition of the Green function via Riesz’ Representation Theorem and establish some of its basic properties. As a preliminary tool the Cauchy initial value problem is considered, too.

Contents

1. Introduction 1

2. Notations and Preliminaries 3

3. Differential Equation 6 3.1. Nonhomogeneos Differential Equation . . . . . . . . . . . . . . . . . . . 6 3.2. Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3. Homogeneous Differential Equation . . . . . . . . . . . . . . . . . . . . . 9

4. Dirichlet Boundary Value Problem 12 4.1. Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2. Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1. Introduction

The subject of the paper is the linear Dirichlet boundary value problem (P)

𝔏y = f − g′ in (0, 1), y(0) = 0, y(1) = 0,

and the related homogenous boundary value problem (hP)

𝔏y = 0 in (0, 1), y(0) = 0, y(1) = 0,

where 𝔏 denotes a linear second order ordinary differential operator of the form

(𝔏y)(x) = −\frac{d}{dx}[a_{1}(x)y′(x) + a_{2}(x)y(x)] + [b_{1}(x)y′(x) + b_{2}(x)y(x)] (1)

(d/dx = ′), with given coefficients a_{1} ∈ L^{∞}(0, 1), a_{2}, b_{1} ∈ L^{2}(0, 1) and b_{2} ∈ L^{1}(0, 1). We assume that there exists a constant γ > 0 with a_{1}(x) ≥ γ for a.e. x ∈ (0, 1), so that we consider the elliptical case only. At the right-hand side the functions f ∈ L^{1}(0, 1) and g ∈ L^{2}(0, 1) are given. Throughout the whole paper a solution to the boundary value problem (P), and of course also in the special case (hP), will be understood in the sense of Sobolev. So we will work in the Sobolev space H_{0}^{1}(0, 1). It is well known and readily verified that then for (P) the Fredholm Alternative does hold (cf. Theorem 5 below). This will be supplemented by our statement that the solution set to (hP) is at most onedimensional. However, the main purpose of this paper is, under the assumption that (hP) has only the trivial solution, to define a Green function G = G(x, ξ), x, ξ ∈ (0, 1), by means of which the unique solution y ∈ H_{0}^{1}(0, 1) to (P) can be represented by

y(x) = ∫_{0}^{1} G(x, ξ)f(ξ) dξ + ∫_{0}^{1} G_{ξ}(x, ξ)g(ξ) dξ, a.e. x ∈ (0, 1),

and its first derivative

y′(x) = ∫_{0}^{1} G_{x}(x, ξ)f(ξ) dξ + ∫_{0}^{1} G_{ξx}(x, ξ)g(ξ) dξ + \frac{g(x)}{a_{1}(x)}, a.e. x ∈ (0, 1),

We prove the existence, uniqueness and some basic properties of the Green function.

There is any amount of papers, textbooks and monographs dealing with Green functions for ordinary differential equations; only a few of them are quoted in the references. In E. Kamke [10, § 26] the reader can find the classical Green function. In N. Dunford, J. T. Schwartz [6, Chap. XIII] the Green function is studied for a differential equation with C^{∞}-coefficients and a solution to the boundary value problem is almost a classical one. In the framework of a L^{2}-theory for a differential equation with coefficients belonging to certain Sobolev spaces the Green function is considered in J. Locker [11, Chap. III]. In both monographs general boundary value problems are treated as a part of Functional Analysis; in particular, the Green function is defined by means of Riesz’ Representation Theorem. An interesting book for us is V. E. Lyantse, O. G. Storozh [13]. There, in § 5 of part II for the special case

−y′′(x) + p(x)y(x) = f′(x), x ∈ (0, 1), y(a) = 0, y(b) = 0,

with p ∈ L^{∞}(a, b), p(x) ≥ p_{0} > 0 for a.e. x ∈ (a, b), and f ∈ L^{2}(a, b), the Green function is introduced and some of its basic properties are derived. Thereby, like in this paper, a solution to the considered boundery value problem is in the sense of Sobolev.

The plan of the paper is as follows. In Section 2 we start with some notations and preliminaries, among which a version of du Bois-Reymond’s fundamental lemma of Calculus of Variations is most important for our paper. An elementary approach to Sobolev spaces of functions of one variable is given also in this section. Section 3.1 deals with the differential equation 𝔏y = f −g′ and Section 3.2 with the Cauchy problem for it. In Section 3.3 we consider the homogeneous equation 𝔏y = 0. Section 4.1 contains some facts on the Fredholm Alternative for the bondary value problem (P). The last Section 4.2 is the main part of the paper. Here we present the definition of the Green function to the differential operator L and establish some of its basic properties. Though the theory of Green funtions is a very old topic our results seem to be new.

We want to emphasize that throughout the whole paper we work in the framework of second order differential equation (in the sense of Sobolev) without rewriting it into an equivalent system of two first order differential equations. Among others this has the great advantage that we get at once a Green function and not a Green matrix.

Eventually, we want to mention that our interest in the Green function stems from dealing with optimal control problems for nonlinear second order ordinary differential equations, cf. [8, 9]. There we needed a sharp estimation for y′_{ε}(x), x ∈ (0, 1), where y′_{ε} ∈ H_{0}^{1}(0, 1) is the unique solution to (P), but with coefficients and a right-hand side depending on a parameter ε > 0. We have suceeded by using for y′_{ε}(x) the above given formula with, of course, a Green function depending also on the parameter ε.

2. Notations and Preliminaries

Throughout the whole paper c denotes a generic positive constant. The Norm in the n-dimensional Euclidean space ℝ^{n}, n ≥ 1, will be denoted by | · |. Let α, β ∈ ℝ, α < β, be fixed. C[α, β] is space of all continuous functions over [α, β]. The scalar product and the norm in the Lebesgue space L^{2}(α, β) will be denoted by (·, ·) and ∥·∥, respectively. For 1 ≤ p < ∞, p ≠ 2, the symbol ∥·∥_{p} will be used for the norm in L^{p}(α, β). The Sobolev spaces W^{1,p}(α, β) and W_{0}^{1,p}(α, β), 1 ≤ p < ∞, are defined by

W^{1,p}(α, β) = {y ∈ L^{p}(α, β) : ∃ z ∈ L^{p}(α, β), ∃ c ∈ ℝ (2)

such that y(x) = ∫_{α}^{x} z(t) dt + c a.e. x ∈ (α, β)}

and

(3) W_{0}^{1,p}(α, β) = {y ∈ L^{p}(α, β) : ∃ z ∈ L^{p}(α, β) such that

y(x) = ∫_{α}^{x} z(t) dt a.e. x ∈ (α, β) and ∫_{α}^{β} z(t) dt = 0},

respectively. By the defining relation (2) an element y ∈ L^{p}(α, β) belongs to W^{1,p}(α, β) if and only if it is equivalent to the function ỹ(x) = ∫_{α}^{x} z(t)dt + c, x ∈ [α, β], which is absolutely continuous over the closed interval [α, β]; for properties of such functions the reader is refered to I. P. Natanson [16, Kap. IX]. Both z ∈ L^{p}(α, β) and c ∈ ℝ, and hence also ỹ, are uniquely determined by y. We identify y with the corresponding ỹ, so that we may say y itself is absolutely continuous on [α, β]. In this sense we have the inclusion W^{1,p}(α, β) ⊂ C[α, β] and for each x ∈ [α, β] it makes sense to speak about the value y(x). So we say, y ∈ W^{1,p}(α, β) satisfies the point condition y(x_{0}) = c, where x_{0} ∈ [α, β] and c ∈ ℝ are given, if and only if ỹ(x_{0}) = c. This having in mind W_{0}^{1,p}(α, β) can be redefined by

W_{0}^{1,p}(α, β) = {y ∈ W^{1,p}(α, β) : y(α) = y(β) = 0 } .

Clearly, it holds y′ = z for any y ∈ W^{1,p}(α, β).

Let the case p = 2 be considered. As usual we write H^{1}(α, β) and H_{0}^{1}(α, β) instead of W^{1,2}(α, β) and W_{0}^{1,2}(α, β), respectively. Introducing in H^{1}(α, β) a scalar product by

(y, z)_{H^{1}(α,β)} = (y, z) + (y′, z′), y, z ∈ H^{1}(α, β),

H^{1}(α, β) becomes a separable Hilbert space; the related norm will be denoted by ∥y∥_{H^{1}(α,β)}. H_{0}^{1}(α, β) is a closed subspace of H^{1}(α, β). The existence of a positive constant c with

|y(x)| ≤ c∥y∥_{H^{1}(α,β)} ∀ x ∈ [α, β], ∀ y ∈ H^{1}(α, β). (4)

is easily verified. That means, H^{1}(α, β) is continuously embedded into C[α, β]. One can show this embedding is even compact. In H_{0}^{1}(α, β) we introduce another scalar product by

(y, z)_{0} = (y′, z′), y, z ∈ H_{0}^{1}(α, β),

which is equivalent to the former one; the assigned norm is denoted by ∥ · ∥_{0}. Directly from the definition (3) it follows that

|y(x)| ≤ (β − α)^{1/2}∥y′∥ ∀ x ∈ [α, β], ∀ y ∈ H_{0}^{1}(α, β), (5)

and, consequently, we have the Friedrichs inequality

∥y∥ ≤ (β − α)∥y∥_{0} ∀ y ∈ H_{0}^{1}(α, β). (6)

H^{−1}(α, β) denotes the dual space to H_{0}^{1}(α, β). L(H_{0}^{1}(0, 1)) and L(H_{0}^{1}(0, 1), H^{−1}(α, β)) are the usual spaces of linear bounded operators. The symbol χ_{M} denotes the characteristic function of the subset M ⊂ ℝ.

Our first lemma, whose proof is omitted, gives some elementary properties of the product of functions belonging to Sobolev spaces.

Lemma 1 (i) Let x_{0} ∈ [α, β], i = 1, 2, and

y_{i}(x) = ∫_{x_{0}}^{x} z_{i}(t) dt + c_{i}, a.e. x ∈ (α, β),

with z_{i} ∈ L^{2}(α, β), c_{i} ∈ ℝ. Then for the product y_{1}y_{2} it holds

(y_{1}y_{2})(x) = ∫_{x_{0}}^{x}[z_{1}(t)y_{2}(t) + z_{2}(t)y_{1}(t)] dt + c_{1}c_{2}, a.e. x ∈ (α, β).

(ii) y_{1}y_{2} ∈ H^{1}(α, β) ∀ y_{1}, y_{2} ∈ H^{1}(α, β)

(iii) y_{1}y_{2} ∈ H_{0}^{1}(α, β) ∀ y_{1} ∈ H^{1}(α, β), ∀ y_{2} ∈ H_{0}^{1}(α, β)

The following Theorem 1 is very important in our considerations on boundary value problems for second order ordinary differential equations. Both its formulation and its proof can be traced back to the work of P. du Bois-Reymond [2, 3] on foundations of variational calculus. Numerous variants of it can be found also in the up-to-date literature on this subject. We mention only V. M. Alekseev, V. M. Tihomirov, S. V. Fomin [1, § 4.1.3.] (cf. V. M. Tihomirov [19]), L. Cesari [5, § 2.4], J. L. Troutman [21, Chap. 4] and P. D. Loewen [12, § 1.A].

Theorem 1 (du Bois-Reymond) For any fixed a ∈ L^{2}(α, β), b ∈ L^{1}(α, β) and ξ ∈ [α, β] it holds

∫_{α}^{β}[a(x)z′(x) + b(x)z(x)] dx = 0 ∀ z ∈ H_{0}^{1}(α, β) (7)

if and only if

(8) ∃ c ∈ ℝ such that a(x) = ∫_{ξ}^{x} b(t) dt + c, a.e. x ∈ (α, β).

P r o o f. First we assume (8). Using integration by parts we see that

∫_{α}^{β} a(x)z′(x) dx = ∫_{α}^{β} z′(x) ∫_{ξ}^{x} b(t) dt dx = − ∫_{α}^{β} b(x)z(x) dx ∀ z ∈ H_{0}^{1}(α, β),

i. e., statement (7) is proved. Now we assume (7) to be valid. Since b ∈ L^{1}(α, β) we have

b(x) = \frac{d}{dx} ∫_{ξ}^{x} b(t) dt, a.e. x ∈ (α, β).

Therefore, again using integration by parts we find

∫_{α}^{β} b(x)z(x) dx = ∫_{α}^{β} z(x)\frac{d}{dx} ∫_{ξ}^{x} b(t) dt dx = − ∫_{α}^{β} z′(x) ∫_{ξ}^{x} b(t) dt dx ∀ z ∈ H_{0}^{1}(α, β)

and (7) becomes

∫_{α}^{β} m(x)z′(x) dx = 0 ∀ z ∈ H_{0}^{1}(α, β), (9)

where we have put

m(x) = a(x) − ∫_{ξ}^{x} b(t) dt, x ∈ (α, β), m ∈ L^{2}(α, β).

Introducing for any y ∈ L^{2}(α, β) the constant

c_{y} = \frac{1}{β − α} ∫_{α}^{β} y(x) dx

for the function

z(x) = ∫_{α}^{x} y(t) dt − (x − α)c_{y}, x ∈ (α, β),

we have

z ∈ H_{0}^{1}(α, β), z′(x) = y(x) − c_{y} a.e. x ∈ (α, β),

and from (9) we get

∫_{α}^{β} m(x)(y(x) − c_{y}) dx = 0 ∀ y ∈ H_{0}^{1}(α, β).

Then, in virtue of

c_{y} ∫_{α}^{β} m(x) dx = c_{m} ∫_{α}^{β} y(x) dx

it follows

∫_{α}^{β}(m(x) − c_{m})y(x) dx = 0 ∀ y ∈ L^{2}(α, β),

i. e., for almost all x ∈ (α, β) we have m(x) = c_{m}, which yields assertion (8).

Let us note that in (8) the constant c is given by

c = \frac{1}{β − α} ∫_{α}^{β} (a(x) − ∫_{ξ}^{x} b(t) dt) dξ (10)

and that (8) is equivalent to the statement

a ∈ W^{1,1}(α, β) with a′(x) = b(x) a.e. x ∈ (α, β). (11)

3. Differential Equation

3.1. Nonhomogeneos Differential Equation

We start our treatment of the boundary value problem (P) with considering the inhomogeneous differential equation

𝔏y = f − g′ in (α, β). (12)

Here (α, β), 0 ≤ α < β ≤ 1, is an arbitrarily fixed interval. The linear differential operator L we have defined in (1) at the beginning of the introductory section, where we have also given the assumptions concerning its coefficients a_{1}, ... b_{2} and the funtions f and g on the right-hand side of (12). Throughout the whole paper we assume that a_{1}, ..., g have at least these properties.

To define what we mean by a solution of (12) we introduce two maps Π_{αβ}(·, ·) : H^{1}(0, 1) × H^{1}(0, 1) ⟶ ℝ, < φ, · >_{αβ}: H^{1}(0, 1) ⟶ ℝ by

Π_{αβ}(y, z) (13)

= ∫_{α}^{β} {[a_{1}(x)y′(x) + a_{2}(x)y(x)]z′(x) + [b_{1}(x)y′(x) + b_{2}(x)y(x)]z(x)} dx,

< φ, z >_{αβ} = ∫_{α}^{β} [f(x)z(x) + g(x)z′(x)] dx, (14)

which, because of H^{1}(0, 1) ⊂ C[0, 1] make sense. In case of α = 0 and β = 1 instead of Π_{01}(·, ·) and < φ, · >_{01} we just write Π(·, ·) and < φ, · >, respectively.

Definition 1 A function y ∈ H^{1}(0, 1) is said to be a (generalized) solution to the differential equation 𝔏y = f − g′ in the interval (α, β), if

Π_{αβ}(y, z) = < φ, z >_{αβ} ∀ z ∈ H_{0}^{1}(α, β).

Our first theorem concerning the solution to (12) is an immediate consequence of the du Bois-Reymond Theorem 1. Corollary 1 follows from the remarks after Theorem 1 and justifies, at least particularly, our Definition 1. Theorem 2 together with Corollary 1 shows that we use in principle a definition for the solution to (12), which is very close to those introduced by C. Carathéodory [4, Kap. XI] for first order differential equations and those used, for example, by W. T. Reid [17] in case of a selfadjoint second order differential equation. Corollary 2 is a simple regularity result for solutions to (12). It should be possible to prove further regularity statements along the lines of V. P. Mikhailov [14, Chap. IV, § 2].

Theorem 2 Let ξ ∈ [α, β] be fixed. Then a function y ∈ H^{1}(0, 1) is a solution to (12) if and only if there exists a c ∈ ℝ such that

a_{1}(x)y′(x) + a_{2}(x)y(x) − g(x)
= ∫_{ξ}^{x} [b_{1}(t)y′(t) + b_{2}(t)y(t) − f(t)] dt + c a.e. x ∈ (α, β).

Corollary 1 A function y ∈ H^{1}(0, 1) is a solution to (12) if and only if

(i) a_{1}y′ + a_{2}y − g ∈ W^{1,1}(α, β),

(ii) the differential equation (12) is pointwise satisfied for almost every x ∈ (α, β).

Corollary 2 If a_{2}, g ∈ L^{∞}(α, β) and y ∈ H^{1}(0, 1) solves (12), then it holds y′ ∈ L^{∞}(α, β) with

∥y′∥_{∞} ≤ γ^{−1}[(∥a_{2}∥_{∞} + ∥b_{1}∥ + ∥b_{2}∥_{1})∥y′∥ + ∥f∥_{1} + ∥g∥_{∞} + |c|]

(all norms are taken over the interval (α, β)), where γ is the positive constant introduced in the introduction after formula (1) and c ∈ ℝ is the same as in Theorem 2.

To prove Corollary 2 we have to use (5). Because of

(β − α)c

= ∫_{α}^{β} [a_{1}(x)y′(x) + a_{2}(x)y(x) − g(x) − ∫_{ξ}^{x} (b_{1}(t)y′(t) + b_{2}(t)y(t) − f(t)) dt] dx

(cf. (10)), the constant |c| can be estimated above by

(β − α)|c| ≤ (∥a_{1}∥_{∞} + ∥a_{2}∥_{∞} + ∥b_{1}∥ + ∥b_{2}∥_{1})∥y′∥ + ∥f∥_{1} + ∥g∥_{∞}, (15)

where again all norms are taken over the interval (α, β).

3.2. Cauchy Problem

Now we shall be concerned with the Cauchy initial value problem

𝔏y = f − g′ in (0, 1), (16a)

y(x_{0}) = c_{1}, (a_{1}y′ + a_{2}y − g)(x_{0}) = c_{2}, (16b)

where x_{0} ∈ [0, 1] and c_{1}, c_{2} ∈ ℝ are any given numbers. Particularly, because of Corollary 1 the following definition of a solution to this problem makes sense. Notice that it would be senseless to replace the second initial condition in (16b) by, e. g. , y′(x_{0}) = c_{2}.

Definition 2 A function y ∈ H^{1}(0, 1) is said to be a (generalized) solution to the above Cauchy problem, if it solves equation (16a) in the sense of Definition 1 and if the initial conditions (16b) are satisfied.

Using the given functions a_{1}, ..., g we define for a. e. x, t ∈ (0, 1) the functions

\{ \begin{matrix}K(x, t) & = & h_{2}(x) (a_{2}(x) − b_{1}(x) − ∫_{t}^{x} b_{2}(τ) dτ) , \\ h_{0}(x) & = & h_{2}(x) (g(x) − ∫_{x_{0}}^{x} f(t) dt) , \\ h_{1}(x) & = & h_{2}(x) (−a_{2}(x) + ∫_{x_{0}}^{x} b_{2}(t) dt) , \\ h_{2}(x) & = & 1/a_{1}(x), \end{matrix} (17)

and consider, with the same constants c_{1}, c_{2} as in (16b), the Volterra integral equation of second type

z(x) + ∫_{x_{0}}^{x} K(x, t)z(t) dt = h_{0}(x) + c_{1}h_{1}(x) + c_{2}h_{2}(x), x ∈ (0, 1). (18)

In virtue of K ∈ L^{2}((0, 1) × (0, 1)) and h_{0}, h_{1}, h_{2} ∈ L^{2}(0, 1) this integral equation has to be considered over L^{2}(0, 1). According to F. Smithies [18, Sect. 2.7.] (see also S. Fenyö, H. W. Stolle [7, Abschnitt 13.1.1.]) (18) is always uniquely solvable in L^{2}(0, 1) for x_{0} = 0. However, since it is not difficult to show that this remains valid also in case x_{0} ∈ (0, 1], we have the following statement.

Lemma 2 For arbitrarily given f ∈ L^{1}(0, 1), g ∈ L^{2}(=, 1), c_{1}, c_{2} ∈ ℝ and x_{0} ∈ [0, 1] the Volterra integral equation (18) is uniquely solvable in L^{2}(0, 1).

The next lemma gives the expected interrelations between the solutions of the Cauchy problem (16a,16b) and the solutions of the Volterra integral equation (18).

Lemma 3 (i) If y ∈ H^{1}(0, 1) is a solution to (16a,16b), then z = y′ ∈ L^{2}(0, 1) is a solution to (18).

(ii) If z ∈ L^{2}(0, 1) is a solution to (18), then y ∈ H^{1}(0, 1) defined by

y(x) = ∫_{x_{0}}^{x} z(t) dt + c_{1}, x ∈ [0, 1], (19)

is a solution to (16a,16b).

P r o o f. (i) Let y ∈ H^{1}(0, 1) be a solution to (16a,16b). Then, in virtue of the initial condition y(x_{0}) = c_{1} and Theorem 1 we have the both identities

y(x) = ∫_{x_{0}}^{x} y′(t) dt + c_{1}, a.e. x ∈ (0, 1),

a_{1}(x)y′(x) + a_{2}(x)y(x) − g(x) (20)

= ∫_{x_{0}}^{x} [b_{1}(t)y′(t) + b_{2}(t)y(t) − f(t)] dt + c_{2}, a.e. x ∈ (α, β),

from which, by eliminating y in (20), we get

a_{1}(x)y′(x) = ∫_{x_{0}}^{x} ( − a_{2}(x) + b_{1}(t))y′(t) dt + ∫_{x_{0}}^{x} b_{2}(t) ∫_{x_{0}}^{t} y′(τ) dτ dt
+ g(x) − ∫_{x_{0}}^{x} f(t) dt + c_{1} (−a_{2}(x) + ∫_{x_{0}}^{x} b_{2}(t) dt) + c_{2}, a.e. x ∈ (0, 1).

Via integration by parts of the second term on the right side we come to

a_{1}(x)y′(x) = ∫_{x_{0}}^{x} (−a_{2}(x) + b_{1}(t) + ∫_{t}^{x} b_{2}(τ) dτ) y′(t) dt (21)

+ g(x) − ∫_{x_{0}}^{x} f(t) dt + c_{1} (−a_{2}(x) + ∫_{x_{0}}^{x} b_{2}(t) dt) + c_{2}, a.e. x ∈ (0, 1),

showing z = y′ is a solution to (18). (ii) Let now conversely z ∈ L^{2}(0, 1) be a solution to (18) and y given by (19). Obviously, we have y ∈ H^{1}(0, 1), y(x_{0}) = c_{1}, y′(x) = z(x) for a.e. x ∈ (0, 1) and relation (21) is satisfied by this function. Since from the latter we may conclude that y satisfies also (20), we see, by Theorem 2, that it also solves the differential equation (16a) and, furthermore, fulfils the second initial condition in (16b). Thus, y is a solution to the Cauchy problem (16a,16b).

Putting the last two lemma together we obtain an existence and uniqueness theorem for the Cauchy problem.

Theorem 3 For arbitrarily given f ∈ L^{1}(0, 1), g ∈ L^{2}(0, 1), c_{1}, c_{2} ∈ ℝ and x_{0} ∈ [0, 1] the Cauchy problem (16a,16b) is uniquely solvable in H^{1}(0, 1). In particular, the honogeneous Cauchy problem has only the trivial solution y = 0.

3.3. Homogeneous Differential Equation

Consider now the homogeneous differential equation

𝔏y = 0 in (0, 1) (22)

under the same assumptions as before, so that the Definition 1 (with α = 0 and β = 0) applies. Obviously, the set S_{1} of all solutions y ∈ H^{1}(0, 1) to (22) is linear. Our next aim is to show dim S_{1} ≤ 2. First, for any y_{1}, y_{2} ∈ H^{1}(0, 1) we define the Wronskian determinant by setting

W (x) = W (y_{1}, y_{2})(x) = y_{1}(x)y′_{2}(x) − y_{2}(x)y′_{1}(x), a.e. x ∈ (0, 1),

which in our considerations will play the same role as in the classical theory. Evidently, it holds W ∈ L^{2}(0, 1) and the identity

∫_{0}^{1} [(a_{2}(x) − b_{1}(x))z(x) − a_{1}(x)z′(x)]W (x) dx (23)

= Π(y_{1}, y_{2}z) − Π(y_{2}, y_{1}z) ∀ y_{1}, y_{2}, z ∈ H^{1}(0, 1)

is easily verified. Surprisingly, we have the following Lemma 4.

Lemma 4 (Liouville) For two arbitrary solutions y_{1}, y_{2} ∈ H^{1}(0, 1) to the homogeneous differential equation (22) it holds

a_{1}W ∈ H^{1}(0, 1), (i)

(ii) ∀ ξ ∈ [0, 1] ∃ c ∈ ℝ such that

a_{1}(x)W (x) = c exp \{∫_{ξ}^{x} \frac{b_{1}(t) − a_{2}(t)}{a_{1}(t)} dt\}, a.e. x ∈ (0, 1).

P r o o f. Let ξ ∈ [0, 1] be fixed. Setting

z(x) = ψ(x) exp \{∫_{ξ}^{x} \frac{a_{2}(t) − b_{1}(t)}{a_{1}(t)} dt\}, a.e. x ∈ (0, 1),

with an arbitrary ψ ∈ H_{0}^{1}(0, 1), we have z ∈ H^{1}(0, 1) and

(a_{2}(x) − b_{1}(x))z(x) − a_{1}(x)z′(x) = −a_{1}(x)ψ′(x) exp \{∫_{ξ}^{x} \frac{a_{2}(t) − b_{1}(t)}{a_{1}(t)} dt\},

a.e. x ∈ (0, 1),

so that, using identity (23) and and the assumptions concerning y_{1}, y_{2} ∈ H^{1}(0, 1), we get

∫_{0}^{1} a_{1}(x)W (x) exp \{∫_{ξ}^{x} \frac{a_{2}(t) − b_{1}(t)}{a_{1}(t)} dt\}ψ′(x) dx = 0 ∀ ψ ∈ H_{0}^{1}(0, 1).

Theorem 1 yields the assertion.

Corollary 3 For the Wronskian determinant W = W (x) of two solutions to (22) it holds either W (x) = 0 or W (x) ≠ 0 for a.e. x ∈ (0, 1), where the former is equivalent to ∫_{0}^{1} a_{1}(x)W (x) dx = 0 and the latter to ∫_{0}^{1} a_{1}(x)W (x) dx ≠ 0.

Lemma 5 (i) Any two functions y_{1}, y_{2} ∈ H^{1}(0, 1) with ∫_{0}^{1} a_{1}(x)W (x) dx ≠ 0 are linearly independent.

(ii) Any two solutions y_{1}, y_{2} ∈ H^{1}(0, 1) to (22) are linearly independent if and only if ∫_{0}^{1} a_{1}(x)W (x) dx ≠ 0.

P r o o f. Let there be given y_{1}, y_{2} ∈ H^{1}(0, 1) not vanishing a.e. on (0, 1) and satisfying ∫_{0}^{1} a_{1}(x)W (x) dx ≠ 0. Suppose there exist two constants c_{1}, c_{2} ∈ ℝ, c_{1}^{2} + c_{2}^{2} > 0, such that c_{1}y_{1}(x) + c_{2}y_{2}(x) = 0, a.e. x ∈ (0, 1). (24)

Then we get also

c_{1}y′_{1}(x) + c_{2}y′_{2}(x) = 0, a.e. x ∈ (0, 1),

and, hence, c_{1}W (x) = 0, a.e. x ∈ (0, 1), which implies c_{1} = 0. Since from (24) it follows also c_{2} = 0, statement (i) is proved. It remains to show that for two linearly independent solutions y_{1}, y_{2} ∈ H^{1}(0, 1) to (22) the condition ∫_{0}^{1} a_{1}(x)W (x) dx ≠ 0 is fulfilled. Suppose the contrary ∫_{0}^{1} a_{1}(x)W (x) dx = 0. Then, because of Corollary 3, we have W (x) = 0 for a.e. x ∈ (0, 1) and we may take a x_{0} ∈ (0, 1), for which y_{i}(x_{0}), a_{i}(x_{0}) ∈ ℝ, i = 1, 2, and W (x_{0}) = 0. Using a nontrivial solution c_{1}^{0}, c_{2}^{0} ∈ ℝ to the algebraic equation system

c_{1}y_{1}(x_{0}) + c_{2}y_{2}(x_{0}) = 0, c_{1}y′_{1}(x_{0}) + c_{2}y′_{2}(x_{0}) = 0

we define the function y = c_{1}^{0}y_{1} + c_{2}^{0}y_{2}, y ∈ H^{1}(0, 1), which is a solution to the homogeneous Cauchy problem

𝔏y = 0 in (0, 1), y(x_{0}) = 0, (a_{1}y′ + a_{2}y)(x_{0}) = 0. (25)

Therefore, by Theorem 3, we get y(x) = 0 for a.e. x ∈ (0, 1). This is a contradiction to our assumption.

Theorem 4 The solution set S_{1} ⊂ H^{1}(0, 1) to the homogeneous differential equation (22) has the dimension two.

P r o o f. 1. Let x_{0} ∈ (0, 1) be fixed. Because of Theorem 3 there are unique solutions y_{1}, y_{2} ∈ H^{1}(0, 1) to 𝔏y = 0 in (0, 1) satisfying the initial conditions

y(x_{0}) = 1, (a_{1}y′ + a_{2}y)(x_{0}) = 0,

and

y(x_{0}) = 0, (a_{1}y′ + a_{2}y)(x_{0}) = 1,

respectively. In view of the definition of H^{1}(0, 1) and of Theorem 2, for these solutions we have the formulas

y_{1}(x) = ∫_{x_{0}}^{x} y′_{1}(t) dt + 1, a_{1}(x)y′_{1}(x) + a_{2}(x)y_{1}(x) = ∫_{x_{0}}^{x} (b_{1}(t)y′_{1}(t) + b_{2}(t)y_{1}(t)) dt,

and

y_{2}(x) = ∫_{x_{0}}^{x} y′_{2}(t) dt, a_{1}(x)y′_{2}(x) + a_{2}(x)y_{2}(x) = ∫_{x_{0}}^{x} (b_{1}(t)y′_{2}(t) + b_{2}(t)y_{2}(t)) dt + 1,

valid for a.e. x ∈ (0, 1). Using the formula given in Lemma 1 we may verify that

a_{1}(x)W (x) = (a_{1}(x)y′_{2}(x) + a_{2}(x)y_{2}(x))y_{1}(x) − (a_{1}(x)y′_{1}(x) + a_{2}(x)y_{1}(x))y_{2}(x)
= ∫_{x_{0}}^{x} (b_{1}(t) − a_{2}(t))W (t) dt + 1, a.e. x ∈ (0, 1),

from which follows W (x) ≠ 0 for a.e. x ∈ (0, 1) (cf. Corollay 3). Because of Lemma 5, the two functions y_{1} and y_{2} are linearly independent.

2. Let now y_{3} ∈ H^{1}(0, 1) be another solution to (22) and x_{0} ∈ (0, 1) be taken such that y′_{i}(x_{0}) ∈ ℝ, i = 1, 2, 3, and a_{i}(x_{0}) ∈ ℝ, i = 1, 2. With a nontrivial solution c_{1}^{0}, c_{2}^{0}, c_{3}^{0} ∈ ℝ to the algebraic equation system

∑_{i=1}^{3} c_{i}y_{i}(x_{0}) = 0, ∑_{i=1}^{3} c_{i}y′_{i}(x_{0}) = 0

we define the funtion y = ∑_{i=1}^{3} c_{i}^{0}y_{i}, which is a solution to the homogeneous Cauchy problem (25) and, hence, vanishes on (0, 1):

c_{1}^{0}y_{1}(x) + c_{2}^{0}y_{2}(x) + c_{3}^{0}y_{3}(x) = 0, a.e. x ∈ (0, 1).

Since y_{1} and y_{2} are linearly independent, we have c_{3}^{0} ≠ 0. This means, y_{3} is a linear combination of y_{1} and y_{2}.

4. Dirichlet Boundary Value Problem

4.1. Fredholm Alternative

Now we are ready for considering our Dirichlet boundary value problem (P) described at the beginning of the introduction. So we are going to look for solutions of

𝔏y = f − g′ in (0, 1), y(0) = 0, y(1) = 0,

under the same assumptions as before. In accordance with the above considerations we will base on the following definition.

Definition 3 A Function y ∈ H_{0}^{1}(0, 1) is said to be a (generalized) solution to the Dirichlet boundary value problem (P) if

Π(y, z) = < φ, z > ∀ z ∈ H_{0}^{1}(0, 1),

where Π(·, ·) : H^{1}(0, 1) × H^{1}(0, 1) ⟶ ℝ and < φ, · >: H^{1}(0, 1) ⟶ ℝ are defined in (13) and (14) (with α = 0 and β = 1), respectively.

We know there exist a unique operator A ∈ L(H_{0}^{1}(0, 1)) and a unique element h ∈ H_{0}^{1}(0, 1) such that

\{ \begin{matrix}Π(y, z) & = & (Ay, z)_{0} ∀ y, z ∈ H_{0}^{1}(0, 1), \\ < φ, z > & = & (h, z)_{0} ∀ z ∈ H_{0}^{1}(0, 1). \end{matrix} (26)

Thus, y ∈ H_{0}^{1}(0, 1) is a solution to (P) if an only if it solves the operator equation

Ay = h. (27)

Thereby, the homogeneous boundary value problem (hP), that means

𝔏y = 0 in (0, 1), y(0) = 0, y(1) = 0,

corresponds to the homogeneous operator equation

Ay = 0. (28)

The set of all solutions to (hP) (that means the kernel of A) will be denoted by S_{0}. Below we show dim S_{0} ≤ 1, a result which we have used already in [8]. Theorem 5 describes completely the solvability of (P).

Theorem 5 For the Dirichlet boundary value problem (P) the Fredholm Alternative holds. Furthermore, the solution set S_{0} of (hP) is at most one-dimensional.

P r o o f. 1. Let A_{1} ∈ L(H_{0}^{1}(0, 1)) be dfined by

∫_{0}^{1} a_{1}(x)y′(x)z′(x) dx = (A_{1}y, z)_{0} ∀ y, z ∈ H_{0}^{1}(0, 1)

and A_{2} ∈ L(H_{0}^{1}(0, 1)) by

∫_{0}^{1} {a_{2}(x)y(x)z′(x) + [b_{1}(x)y′(x)+b_{2}(x)y(x)]z(x)} dx = (A_{2}y, z)_{0} ∀ y, z ∈ H_{0}^{1}(0, 1).

Then A = A_{1} + A_{2}, A_{1} is continuously invertible and A_{2} is compact. Hence, to A the Riesz-Schauder-Theory can be applied, which results in our first statement. (The reader is referred to K. Yosida [22, Chap. X, § 5], H. Triebel [20, Sect. 2.4.] or S. Mizohata [15, Chap.3, § 10].)

2. Due to the Riesz-Schauder-Theory, it is dim S_{0} < ∞. We want to show dim S_{0} ≤ 1. Let y_{1}, y_{2} ∈ H_{0}^{1}(0, 1) be two arbitrary nontrivial solutions to (hP), that is y_{1}, y_{2} ∈ S_{0}. For their Wronskian W = W (y_{1}, y_{2}) we claim

W (x) = 0, a.e. x ∈ (0, 1). (29)

Clearly, the function

z(x) = ∫_{0}^{x} W (t) exp \{∫_{t}^{x} \frac{a_{2}(τ) − b_{1}(τ)}{a_{1}(τ)} dτ\} dt, x ∈ [0, 1],

belongs to H^{1}(0, 1) and it holds

z′(x) = W (x) + \frac{a_{2}(x) − b_{1}(x)}{a_{1}(x)}z(x), a.e. x ∈ (0, 1).

So, by Lemma 1, we have y_{1}z, y_{2}z ∈ H_{0}^{1}(0, 1) and identity (23) yields

∫_{0}^{1} a_{1}(x)W^{2}(x) dx = 0,

from which follows (29). Using Lemma 5 we conclude dim S_{0} ≤ 1.

The next theorem is a consequence of Theorem 5.

Theorem 6 For any f ∈ L^{1}(0, 1) and g ∈ L^{2}(0, 1) the inhomogeneous boundary value problem (P) has exactly one solution y ∈ H_{0}^{1}(0, 1) if and only if the related homogeneous boundary value problem (hP) has the trivial solution only.

In the last, but most important section of this paper we consider the case characterized by this theorem in more details. An analogue to the Green formula for the solution to (P) will be derived.

4.2. Green Function

Throughout the rest of the paper we assume

S_{0} = {0}. (30)

Hence, in virtue of Theorem 6, for any f ∈ L^{1}(0, 1) and g ∈ L^{2}(0, 1) the boundary value problem (P) has a unique solution y ∈ H_{0}^{1}(0, 1). The main purpose of this section is to derive for it a explicit integral representation in terms of a Green function and to establish some basic properties of this Green function. In the next theorem the occuring operator A ∈ L(H_{0}^{1}(0, 1)) is defined in (26).

Theorem 7 Let S_{0} = {0}. Then for any x ∈ [0, 1] there exists a unique function G(x, ·) ∈ H_{0}^{1}(0, 1) with ∥G(x, ·)∥_{0} ≤ ∥A^{−1}∥, such that

(z, G(x, ·))_{0} = (A^{−1}z)(x) ∀ z ∈ H_{0}^{1}(0, 1). (31)

In particular, it holds G(0, ξ) = 0 and G(1, ξ) = 0 for a.e. ξ ∈ (0, 1).

P r o o f. Because of the results in the foregoing section and the assumption (30) the operator A ∈ L(H_{0}^{1}(0, 1)) defined in (26) is continuously invertible, i. e., A^{−1} ∈ L(H_{0}^{1}(0, 1)). Let x ∈ [0, 1] be arbitrary, but fixed. Consider the map

F_{x} : H_{0}^{1}(0, 1) ⟶ ℝ,

which is defined by

< F_{x}, z > = (A^{−1}z)(x), z ∈ H_{0}^{1}(0, 1).

Obviously, F_{x} is linear and, in virtue of (5), it holds

| < F_{x}, z > | ≤ ∥A^{−1}z∥_{0} ≤ ∥A^{−1}∥ ∥z∥_{0} ∀ z ∈ H_{0}^{1}(0, 1).

So, we have F_{x} ∈ H^{−1}(0, 1) with ∥F_{x}∥ ≤ ∥A^{−1}∥. The Riesz Representation Theorem applied to F_{x} gives a unique element G(x, ·) ∈ H_{0}^{1}(0, 1) having the properties

< F_{x}, z > = (z, G(x, ·))_{0} ∀ z ∈ H_{0}^{1}(0, 1) and ∥F_{x}∥ = ∥G(x, ·)∥_{0},

which proves (31) and the estimation ∥G(x, ·)∥_{0} ≤ ∥A^{−1}∥. Since A^{−1}z ∈ H_{0}^{1}(0, 1) for each z ∈ H_{0}^{1}(0, 1), this implies also the statements on G(x, ·) in case of x = 0 and x = 1.

Definition 4 The function G = G(x, ξ), x, ξ ∈ [0, 1], defined in Theorem 7 is called (generalized) Green function to the differential operator 𝔏.

The name will be justified by the subsequent considerations.

Theorem 8 Let S_{0} = {0} and G be the Green function to the differential operator 𝔏. Then for any f ∈ L^{1}(0, 1) and g ∈ L^{2}(0, 1) the unique solution y ∈ H_{0}^{1}(0, 1) to the boundary value problem (P) is given by the formula

y(x) = ∫_{0}^{1} G(x, ξ)f(ξ) dξ + ∫_{0}^{1} G_{ξ}(x, ξ)g(ξ) dξ, a.e. x ∈ (0, 1).

P r o o f. The unique solution to (P) is given by y = A^{−1}h, where A ∈ L(H_{0}^{1}(0, 1)) and h ∈ H_{0}^{1}(0, 1) are defined in (26). Therefore, due to Theorem 7, we have

y(x) = (A^{−1}h)(x) = (h, G(x, ·))_{0} = < φ, G(x, ·) >, a.e. x ∈ (0, 1).

The definition of φ yields the wanted formula (cf. (14) with α = 0 and β = 1).

To establish some basic properties of the Green function we introduce now the (formal) adjoint differential operator 𝔏^{∗} to 𝔏 by

(𝔏^{∗}y)(x) = −\frac{d}{dx}[a_{1}(x)y′(x) + b_{1}(x)y(x)] + [a_{2}(x)y′(x) + b_{2}(x)y(x)].

For 0 ≤ α < β ≤ 1 the corresponding bilinear form Π_{αβ}^{∗}(·, ·) : H^{1}(0, 1) × H^{1}(0, 1) ⟶ ℝ is given by

Π_{αβ}^{∗}(y, z)
= ∫_{α}^{β} {[a_{1}(x)y′(x) + b_{1}(x)y(x)]z′(x) + [a_{2}x)y′(x) + b_{2}(x)y(x)]z(x)} dx,

that means

Π_{αβ}^{∗}(y, z) = Π_{αβ}(z, y) ∀ y, z ∈ H_{0}^{1}(0, 1)

(cf. (13)). In analogy to our Definition 1 we say y ∈ H_{0}^{1}(0, 1) is a (generalized) solution to the homogeneous adjoint equation

𝔏^{∗}y = 0 in (α, β),

if

Π_{αβ}(z, y) = 0 ∀ z ∈ H_{0}^{1}(α, β).

We prove the following important theorem.

Theorem 9 For each x ∈ (0, 1) the Green function G(x, ·) to 𝔏 solves the homogeneous differential equation 𝔏^{∗}y = 0 both in the interval (0, x) and in (x, 1).

P r o o f. Let x_{0} ∈ (0, 1) be fixed. By definition, G(x, ·) ∈ H_{0}^{1}(0, 1) and (26) yields

Π(z, G(x_{0}, ·)) = (Az, G(x_{0}, ·))_{0} ∀ z ∈ H_{0}^{1}(0, 1),

from which, in virtue of (31), we obtain

Π(z, G(x_{0}, ·)) = z(x_{0}) ∀ z ∈ H_{0}^{1}(0, 1). (32)

Taking an arbitrary z ∈ H_{0}^{1}(0, x_{0}) and setting

z̃(x) = \{ \begin{matrix}z(x) if & x ∈ (0, x_{0}] \\ 0 if & x ∈ (x_{0}, 1) \end{matrix} ,

we get z̃ ∈ H_{0}^{1}(0, 1) with z̃(x_{0}) = 0, so that (32) implies

Π_{0x_{0}}(z, G(x_{0}, ·)) = Π(z̃, G(x_{0}, ·)) = 0 ∀ z ∈ H_{0}^{1}(0, x_{0}).

This means G(x, ·) solves 𝔏^{∗}y = 0 in (0, x_{0}). Since the other statement follows in the same way, the theorem is proved.

Let us shortly consider the equation (32). With the characteristic function χ_{[x_{0},1]} of the interval [x_{0}, 1] we have

z(x_{0}) = − ∫_{x_{0}}^{1} z′(x) dx = − ∫_{0}^{1} χ_{[x_{0},1]}(x)z′(x) dx ∀ z ∈ H_{0}^{1}(0, 1)

and relation (32) shows that for each x_{0} ∈ (0, 1) the Green function is the unique solution to the boundary value problem

𝔏^{∗}y = χ′_{[x_{0},1]} in (0, 1), y(0) = 0, y(1) = 0. (33)

Introducing h_{0} ∈ H_{0}^{1}(0, 1) by

h_{0}(x) = \{ \begin{matrix}(1 − x_{0})x if & x ∈ [0, x_{0}] \\ x_{0}(1 − x) if & x ∈ [x_{0}, 1] \end{matrix} ,

we see, furthermore, G(x, ·) solves uniquely the operator equation

A^{∗}y = h_{0}

or, what the same is, the operator equation

Ã^{∗}y = δ_{x_{0}}.

Here Ã ∈ L(H_{0}^{1}(0, 1), H^{−1}(0, 1)) is defined by

Π(y, z) = < Ãy, z > ∀ y, z ∈ H_{0}^{1}(0, 1),

and δ_{x_{0}} ∈ H^{−1}(0, 1) denotes the Dirac functional given by

< δ_{x_{0}}, z > = (h_{0}, z)_{0} = z(x_{0}) ∀ z ∈ H_{0}^{1}(0, 1).

The boundary value problem (33) (or one of the both operator equations above mentioned) could have been taken also to define the Green function.

Now, taking advantage of Theorem 2 and Theorem 9, in the next Corollary 4 we get identities, which involve G(x, ·) and G_{ξ}(x, ·). These relations will afterwards allow us to prove a jumping condition at ξ = x.

Corollary 4 For any fixed x ∈ (0, 1) it holds

a_{1}(ξ)G_{ξ}(x, ξ) + b_{1}(ξ)G(x, ξ) (34)

= \{ \begin{matrix}∫_{0}^{ξ} [a_{2}(t)G_{ξ}(x, t) + b_{2}(t)G(x, t)] dt + c_{1} a.e. ξ ∈ & (0, x) \\ − ∫_{ξ}^{1} [a_{2}(t)G_{ξ}(x, t) + b_{2}(t)G(x, t)] dt + c_{2} a.e. ξ ∈ & (x, 1) \end{matrix} ,

with the constants (depending on x)

c_{1} = \frac{1}{x} ∫_{0}^{x} \{a_{1}(ξ)G_{ξ}(x, ξ) + b_{1}(ξ)G(x, ξ) − ∫_{0}^{ξ} [a_{2}(t)G_{ξ}(x, t) + b_{2}(t)G(x, t)] dt\} dξ,
c_{2} = \frac{1}{1 − x} ∫_{x}^{1} \{a_{1}(ξ)G_{ξ}(x, ξ) + b_{1}(ξ)G(x, ξ) + ∫_{ξ}^{1} [a_{2}(t)G_{ξ}(x, t) + b_{2}(t)G(x, t)] dt\} dξ.

Theorem 10 Let x ∈ (0, 1) be fixed and the functions u, u_{−}, and u_{+} be defined by

u(ξ) = a_{1}(ξ)G_{ξ}(x, ξ) + b_{1}(ξ)G(x, ξ), a.e. ξ ∈ (0, 1),
u_{−}(ξ) = u(ξ), a.e. ξ ∈ (0, x), and u_{+}(ξ) = u(ξ), a.e. ξ ∈ (x, 1),

respectively. Then it holds

u_{−} ∈ W^{1,1}(0, x), u_{+} ∈ W^{1,1}(x, 1)

and u_{−}(x) − u_{+}(x) = 1. (35)

P r o o f. Besides u we define

v(ξ) = a_{2}(ξ)G_{ξ}(x, ξ) + b_{2}(ξ)G(x, ξ), a.e. ξ ∈ (0, 1).

Then, relation (34) in Corollary 4 reads as

u_{−}(ξ) = ∫_{0}^{ξ} v(t) dt + c_{1}, a.e. ξ ∈ (0, x),
u_{+}(ξ) = − ∫_{ξ}^{1} v(t) dt + c_{2}, a.e. ξ ∈ (x, 1).

Since u ∈ L^{2}(0, 1) and v ∈ L^{1}(0, 1), we have u_{−} ∈ W^{1,1}(0, x) and u_{+} ∈ W^{1,1}(x, 1). From (32) it follows that

Π_{0x}(z, G(x, ·)) + Π_{x1}(z, G(x, ·)) = z(x) ∀ z ∈ H_{0}^{1}(0, 1).

For Π_{0x}(z, G(x, ·) we find

Π_{0x}(z, G(x, ·) = ∫_{0}^{x} (u(ξ)z′(ξ) + v(ξ)z(ξ)) dξ
= ∫_{0}^{x} (u_{−}(ξ)z′(ξ) + u′_{−}(ξ)z(ξ)) dξ
= u_{−}(x)z(x)  ∀ z ∈ H_{0}^{1}(0, 1)

and analogously

Π_{x1}(z, G(x, ·) = −u_{+}(x)z(x) ∀ z ∈ H_{0}^{1}(0, 1).

Consequently, it holds

(u_{−}(x) − u_{+}(x))z(x) = z(x) ∀ z ∈ H_{0}^{1}(0, 1),

from which assertion (35) follows.

If, for a moment, we assume a_{1}, b_{1} ∈ C[0, 1], then for any x ∈ (0, 1) the one-sided limits G_{ξ}(x, x ± 0) are given by

G_{ξ}(x, x ± 0) = \frac{1}{a_{1}(x)}(u_{±}(x) − b_{1}(x)G_{ξ}(x, x))

and (35) results in

G_{ξ}(x, x − 0) − G_{ξ}(x, x + 0) = \frac{1}{a_{1}(x)},

which means G_{ξ}(x, ·) itself has a jump discontinuity at ξ = x. Equation (35) will be referred to as the jump equation.

Our next aim is to derive upper bounds for |G(x, ξ)| and |G_{ξ}(x, ξ)| needed, for example, in [9]. Already in Theorem 7 it was stated that ∥G(x, ·)∥_{0} ≤ ∥A^{−1}∥ for any x ∈ [0, 1]. Therefore, in virtue of (5), we have

|G(x, ξ)| ≤ ∥A^{−1}∥ ∀ x, ξ ∈ [0, 1]. (36)

If the bilinear form Π(·, ·) (which defines A) is coercive on H_{0}^{1}(0, 1) with a coercitvity constant μ > 0, then, here and also in the following Lemma 6, the number ∥A^{−1}∥ can be estimated above by μ^{−1}. To get a similar estimation for |G_{ξ}(x, ξ)| is not so easy. First of all, G_{ξ}(x, ·) ∈ L^{∞}(0, 1) provided b_{1} ∈ L^{∞}(0, 1) (see Corollary 2). To find an upper bound for ∥G_{ξ}(x, ·)∥_{∞} independent on x certain difficulties are caused mainly by the first terms in c_{1} and c_{2} (see Corollary 4).

Lemma 6 If a_{2} ∈ L^{∞}(0, 1), then for the Green function G to the differential operator 𝔏 it holds

\frac{1}{x} ∫_{0}^{x} |G_{ξ}(x, ξ)| dξ ≤ c_{0} ∀ x ∈ (0, 1)

and

\frac{1}{1 − x} ∫_{x}^{1} |G_{ξ}(x, ξ)| dξ ≤ c_{0} ∀ x ∈ (0, 1),

where the constant c_{0} is given by

c_{0} = 2γ^{−1}[1 + (∥a_{1}∥_{∞} + ∥a_{2}∥_{∞} + ∥b_{1}∥ + ∥b_{2}∥_{1})∥A^{−1}∥].

P r o o f. Let t ∈ (0, 1) be fixed. We define two functions g_{i}(t, ·) ∈ L^{∞}(0, 1), i = 1, 2, by

g_{1}(t, ξ) = χ_{(0,t)}(ξ)sign G_{ξ}(t, ξ), ξ ∈ (0, 1),
g_{2}(t, ξ) = χ_{(t,1)}(ξ)sign G_{ξ}(t, ξ), ξ ∈ (0, 1),

and consider the two boundary value problems

𝔏y = −g_{i}(t, ·)′ in (0, 1),  y(0) = 0, y(1) = 0,  i = 1, 2, (37)

In virtue of S_{0} = {0} each of them has a unique solution y_{i}(t, ·) ∈ H_{0}^{1}(0, 1) given by

y_{i}(t, x) = ∫_{0}^{1} G_{ξ}(x, ξ)g_{i}(t, ξ) dξ, a.e. x ∈ (0, 1),

(cf. Theorem 6 and Theorem 8). In particular we have

y_{1}(t, t) = ∫_{0}^{t} |G_{ξ}(t, ξ)| dξ, y_{2}(t, t) = ∫_{t}^{1} |G_{ξ}(t, ξ)| dξ.

Now, because of

|y(x)| ≤ ∫_{0}^{x} |y′(ξ)| dξ, |y(x)| ≤ ∫_{x}^{1} |y′(ξ)| dξ, ∀ x ∈ [0, 1], ∀ y ∈ H_{0}^{1}(0, 1),

and since, due to Corollary 2, it holds y_{i}(t, ·)′ ∈ L^{∞}(0, 1), we get

∫_{0}^{t} |G_{ξ}(t, ξ)| dξ ≤ t ∥y_{1}(t, ·)′∥_{∞}, ∫_{t}^{1} |G_{ξ}(t, ξ)| dξ ≤ (1 − t)∥y_{2}(t, ·)′∥_{∞}. (38)

So it remains to verify that ∥y_{i}(t, ·)′∥_{∞} can be estimated above by the constant c_{0} given in the formulation of the lemma. Using again Corollary 2 and also estimation (15) (with α = 0 and β = 1) we find

∥y_{i}(t, ·)′∥_{∞} ≤ 2γ^{−1}[1 + (∥a_{1}∥_{∞} + ∥a_{2}∥_{∞} + ∥b_{1}∥ + ∥b_{2}∥_{1})∥y_{i}(t, ·)∥_{0}]. (39)

By Riesz’ Representation Theorem, for i = 1, 2, there exists a unique element h_{i}(t, ·) ∈ H_{0}^{1}(0, 1) with ∥h_{i}(t, ·)∥_{0} ≤ 1, such that

∫_{0}^{1} g_{i}(t, ξ)z′(ξ) dξ = (h_{i}(t, ·), z)_{0} ∀ zH_{0}^{1}(0, 1).

Since, on the other hand, for the solutions y_{i}(t, ·) to (37) we have

y_{i}(t, ·) = A^{−1}h_{i}(t, ·),

it follows ∥y_{i}(t, ·)∥_{0} ≤ ∥A^{−1}∥, for i = 1, 2. (40)

Combining (38,39) and (40) we obtain the wanted estimations.

Theorem 11 If a_{2}, b_{1} ∈ L^{∞}(0, 1), then for any x ∈ [0, 1] it holds G_{ξ}(x, ·) ∈ L^{∞}(0, 1) with

|G_{ξ}(x, ξ)| ≤ (1 + γ^{−1}∥a_{1}∥_{∞})c_{0}, a.e. ξ ∈ (0, 1),

where the constant c_{0} is given by

c_{0} = 2γ^{−1}[1 + (∥a_{1}∥_{∞} + ∥a_{2}∥_{∞} + ∥b_{1}∥_{∞} + ∥b_{2}∥_{1})∥A^{−1}∥]. (41)

P r o o f. First we note that, by (36),

|G(x, ξ)| ≤ ∥A^{−1}∥ ∀ x, ξ ∈ [0, 1]

and, by Theorem 7,

∫_{0}^{1} |G_{ξ}(x, ξ)| dξ ≤ ∥G_{ξ}(x, ·)∥ ≤ ∥A^{−1}∥ ∀ x ∈ [0, 1].

Since for x = 0 and x = 1 there is nothing to be proved, we assume x ∈ (0, 1) be arbitrary, but fixed. We consider G_{ξ}(x, ·) over the subinterval (0, x); concerning the other subinterval (x, 1) the arguments are completely the same. Taking advantage of the above estimations, formula (34) in Corollary 4 and Lemma 6 with the constant c_{0} given in (41), we have

|G_{ξ}(x, ξ)| ≤ γ^{−1}[∥a_{2}∥_{∞} + ∥b_{1}∥_{∞} + ∥b_{2}∥_{1})∥A^{−1}∥ + |c_{1}|], a.e. ξ ∈ (0, x),
|c_{1}| ≤ \frac{1}{x} ∫_{0}^{x} (|a_{1}(ξ)||G_{ξ}(x, ξ)| + |b_{1}(ξ)||G(x, ξ)|) dξ

+ \frac{1}{x} ∫_{0}^{x} [∫_{0}^{ξ} (|a_{2}(t)||G_{ξ}(x, t)| + |b_{2}(t)||G(x, t)|) dt] dξ

≤ c_{0}∥a_{1}∥_{∞} + (∥a_{2}∥_{∞} + ∥b_{1}∥_{∞} + ∥b_{2}∥_{1})∥A^{−1}∥,

from which we get

|G_{ξ}(x, ξ)| ≤ c_{0} + γ^{−1}∥a_{1}∥_{∞}c_{0} a.e. ξ ∈ (0, x).

So far for the Green function G = G(x, ξ) we have proved only properties with respect to the second variable ξ. In the following theorem we give a formula for G, which allows us to consider G as a function of x and to establish some more of its properties. Furthermore, the procedure of finding this formula may enable us to calculate the Green function for specific differential operators 𝔏.

Consider the two Cauchy problems

𝔏^{∗}y = 0 in (0, 1), y(0) = 0, (a_{1}y′ + b_{1}y)(0) = 1, (42)

and 𝔏^{∗}y = 0 in (0, 1), y(1) = 1, (a_{1}y′ + b_{1}y)(1) = 0. (43)

By Theorem 3 each of them has a unique solution in H^{1}(0, 1) denoted by y_{1} and y_{2}, respectively. Since we still assume S_{0} = {0}, the homogeneous boundary value problem

𝔏^{∗}y = 0 in (0, 1), y(0) = 0, y(0) = 0,

has only the trivial solution (cf. Section 4.1.) and, therefore, it holds

y_{1}(1) ≠ 0, y_{2}(0) ≠ 0. (44)

Let now x ∈ (0, 1) be fixed. Particularly in view of Theorem 9, it makes sense to look for α_{i}(x), β_{i}(x) ∈ ℝ, i = 1, 2, such that

G(x, ξ) = \{ \begin{matrix}∑_{i=1}^{2} (α_{i}(x) + β_{i}(x))y_{i}(ξ) if & 0 ≤ ξ ≤ x \\ ∑_{i=1}^{2} (α_{i}(x) − β_{i}(x))y_{i}(ξ) if & x ≤ ξ ≤ 1. \end{matrix}

Like in the classical theory of linear differential equations the unknowns α_{i}(x) and β_{i}(x) are uniquely determined by the continuity of G(x, ·) at ξ = x, the jump equation (35) and the both boundary conditions G(x, 0) = G(x, 1) = 0 (together with (44)). Thus a straightforward calculation leads to the following last theorem.

Theorem 12 Let S_{0} = {0}. Then for any x ∈ [0, 1] the Green function G = G(x, ξ) to the differential operator 𝔏 is given by

G(x, ξ) = \{ \begin{matrix}−\frac{y_{2}(x)y_{1}(ξ)}{(a_{1}W)(x)} if & 0 ≤ ξ ≤ x \\ −\frac{y_{1}(x)y_{2}(ξ)}{(a_{1}W)(x)} if & x ≤ ξ ≤ 1, \end{matrix}

where y_{1}, y_{2} ∈ H^{1}(0, 1) are defined by (42) and (43), respectively, and W = W (y_{1}, y_{2}) denotes their Wronskian.

Now it is evident that G(·, ξ), G_{ξ}(·, ξ) ∈ H^{1}(0, 1) for all ξ ∈ [0, 1]. Using

(a_{1}W)′(x) = (b_{2}(x) − a_{1}(x))W (x), a.e. x ∈ (0, 1),

(cf. Lemma 4, but note that here y_{1} and y_{2} are solutions to 𝔏^{∗}y = 0 and not to 𝔏y = 0) for the derivative of the unique solution y ∈ H_{0}^{1}(0, 1) to (P) the formula

y′(x) = ∫_{0}^{1} G_{x}(x, ξ)f(ξ) dξ + ∫_{0}^{1} G_{ξx}(x, ξ)g(ξ) dξ + \frac{g(x)}{a_{1}(x)}, a.e. x ∈ (0, 1),

is readily verified.

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