On indirect variational formulations for operator equations 1

A scheme for the construction of indirect variational formulations for a wide class of equations is suggested.


Introduction
An important problem in applications of variational methods is a representation of a given system of equations in the form of the Euler -Lagrange equations. It means the construction of a functional F N such that its extremals are solutions of the given system of equations. This is known as the classical inverse problem of the calculus of variations [1,2,4,7].
In spite of the remarkable number of papers on the subject different approaches for constructing of integral variational principles for equations with nonpotential operators should be developed. They will allow to obtain so-called indirect variational formulations of given problems.
The main aim of the present paper is a constructive determination of indirect variational formulations for any operator equation with the Gâteaux differentiable operator N .

Bilinear forms and variational principles
Let N be an operator such that its domain of definition D(N ) ⊆ U ⊆ V and range of values R(N ) ⊆ V , where U and V are linear normed spaces over R, i. e.
If there exists a limit then it is called the Gâteaux variation of the operator N at the point u or the first variation of the operator N at the point u . δN (u, h) is homogeneous with respect to h : δN (u, λh) = λδN (u, h), but the operator δN (u, ·) : U → V is not always additive with respect to h.
If δN (u, h) is a linear operator with respect to h, when u is a fixed element of D(N ), then we say that the operator N is Gâteaux differentiable at the point u . The expression δN (u, h) is called the Gâteaux differential and denoted by DN (u, h). In this case we will also write DN (u, h) = N u h and say that N u is the Gâteaux derivative of operator N at the point u .
If N is a linear operator then N u h = N h, i. e. the Gâteaux derivative of the linear operator coincides with it.
Further assume that for any given operator N : for all ε sufficiently small. In this case h ∈ D(N u ) is called an admissible element.
If the Gâteaux derivative of the operator N exists, then the following equality holds [3] (2) Note that for any linear operatorÑ u which may depend on u in a nonlinear way, the Gâteaux derivative is defined by The second Gâteaux derivative N u of the operator N is given by In the most general applications N u satisfies the symmetry condition Now we need some notations and notions about bilinear forms and potential operators.

Definition 1.
A mapping Φ(·, ·) : V × U → R is said to be a nonlocal bilinear form if it is linear with respect to every argument.
Definition 3. A bilinear mapping Φ(u; ·, ·) : V × U → R is said to be a local bilinear form if it depends on u . In particular Φ(u; ·, ·) : A mapping Φ(·; v, g) : U → R is a particular kind of operator. Its Gâteaux derivative Φ u is given by Definition 4. The operator N : D(N ) ⊂ U → V is said to be potential on the set D(N ) with respect to the local bilinear form Φ(u; ·, ·) : The functional F N is called the potential of the operator N , and in turn the operator N is called the gradient of the functional F N . In this case we The following theorem is needed for the sequel.

Theorem 1. [4]
Consider the operator N : 1]. For N to be potential on the convex open set D(N ) with respect to Φ it is necessary and sufficient to have Under this condition the potential F N is given by where u 0 is a fixed element of D(N ).

Remark 1.
In the case of a nonlocal bilinear form the condition (6) takes the form

Indirect variational formulations
Consider the problem where the operator N is not potential with respect to the local bilinear form Φ(u; ., .) : V × U → R. Now we shall construct an indirect variational formulation for the above equation (9).

Definition 5. An invertible linear operator
such that the operatorÑ = M u N is potential on D(N ) with respect to the same bilinear form Φ is called a variational multiplier for N .
be a twice continuously Gâteaux differentiable operator. Let C u be any linear operator such that the following conditions take place: Then the operator N is potential on D(N ) with respect to the local bilinear form and the potential F N is given by Proof. Using (12) one gets Thus for the left-hand side of (6) we have On the other hand Now, bearing in mind the symmetry of C u and the conditions (10), (11) we conclude that According to Theorem 1 the given operator N is potential on D(N ) with respect to the local bilinear form (12) and the potential is given by To prove (13) we find Taking into consideration the symmetry of Cũ (λ) we have (20) Hence, from (19) one obtains The use of (18), (21) shows that the sought potential F N can be represented in the form (13).

Remark 3.
When C u = C is a constant operator and bilinear form Φ 1 is nonlocal then Hence, in this case we can take CN(u)).
This formula was obtained by Tonti [7] as a solution of the inverse problem of the calculus of variations in an extended sense.
By using Theorem 1, it is easy to prove the following theorems.  Proof. The criteria of potentiality (6) for N 1 = M u N (u) on the set D(N 1 ) = D(N ) with respect to the bilinear form Φ can be written as

Theorem 3. A Gâteaux differentiable invertible operator M u is a variational multiplier for N ⇔ the operator N is potential on D(N ) with respect to the bilinear form
for all u ∈ D(N 1 ) and g, h ∈ D(N 1u ). By using the equality N 1u = M u N u h + M u (N (u); h) and (22),(24), we obtain (23).

Examples
1. Consider the following partial differential equation: We set where ϕ i ∈ C[0, l] and ψ i ∈ C[0, T ] (i = 1, 2). We denote V = C(Q T ) and determine the local bilinear form by setting Let us prove that the operator N in (25)  To this end using (25) and (26), we get From (30) we obtain the following equality: Hence, in the given case the condition (6) can be written in the form for all u ∈ D(N ) and g, h ∈ D(N u ). By integrating (29) by parts and taking into consideration the conditions g| t=0 = h| t=0 = g| t=T = h| t=T = 0, If we take (28) into account, it follows that for all u ∈ D(N ) and g, h ∈ D(N u ). Thus, under the conditions (28) the given operator N is potential on the set D(N ) (26) with respect to the local bilinear form (27). The corresponding functional is Taking into account Theorem 3, we obtain that M u = e u is a variational multiplier for N (25).
It follows from the equation 2. Consider the following system of partial differential equations: where a, b, c, d are constants, u(x, t) = (u 1 (x, t), u 2 (x, t)) T is an unknown vector function. We denote by D (N ) the domain of definition of the operator N = (N 1 , N 2 ) T in (31): : 2) are given functions. Let us consider the nonlocal bilinear form If a = 0, c = 0 then the equations (31) can not be represented in the form of the Euler-Lagrange equations on the set D(N ) (32) with respect to (33) because the operator [6] is not symmetric on the set : By using Theorem 4, we shall find a variational multiplier for the given system of equations. For that we obtain In this case The operator A is invertible and The operator K u is symmetric with respect to the bilinear form (33) under the conditions a = c, b = d. Indeed, i.e.
Therefore, if a = c, b = d then K u = K * u . By using Theorem 4, we get that the operator A −1 in (34) is a variational multiplier for (31).

Remark 4.
Construction of an indirect variational formulation for the boundary value problem for the general Navier -Stokes equations and the equation of continuity was done in [5].