MINIMUM PRINCIPLE OF A QUADRATIC FUNCTIONAL FOR A HYPERBOLIC EQUATION

V. M. Shalov

Introduction

In the study of equations of mathematical physics, the variational principle is widely used. Based on the physical law of conservation of energy, the variational principle makes it possible, as a rule, to pose correct problems for a given class of equations, to prove their solvability, to investigate the differential properties of solutions, and, if necessary, to construct in practice the desired solution with any degree of accuracy.

At the same time, the corresponding apparatus of investigation has been developed mainly for the case in which the variational problem is reduced to the problem of minimizing some quadratic functional.

In this case, thanks to S. L. Sobolev’s [1] introduction of the notion of a generalized derivative and the spaces \(W_p^{(l)}\), as well as to the development of embedding and extension theorems, the variational principle for boundary-value problems has received a rigorous and profound interpretation.

In the works of V. I. Kondrashov [6], L. D. Kudryavtsev [8], S. M. Nikol’skii [16], and other authors, the ideas of S. L. Sobolev received further development, and by means of the variational principle a number of the most important problems of mathematical physics were investigated.

The indicated method, however, could be applied mainly only to differential equations of elliptic type.

In [22, 23], in terms of Hilbert space, the possibility is shown of extending the variational principle of minimizing a quadratic functional to a rather broad class of non-self-adjoint equations, and the corresponding variational problems are formulated.

There it is established, in particular, that for every solvable linear equation in a Hilbert space there can be posed an equivalent variational problem of finding the point of minimum of some quadratic functional; that is, for any, for example, solvable linear differential equation, one can indicate some quadratic functional such that the original equation, together with the boundary conditions, will be the Euler equation for this functional. In this case the functional will be bounded below, and the minimum of the functional is attained only at the element that is the solution of the original equation.

In the present work, using the general approach set forth in [22, 23], a variational problem on the minimization of a quadratic functional is formulated and investigated for the equation of vibrations of a string, which, as is known, is an equation of hyperbolic type. In

in this, to a significant degree, the known methods for studying equations of elliptic type by the variational principle [1, 8, 13, 16] are used, with the only, perhaps, novelty that the minimum of the corresponding functional is sought not on the class of functions satisfying prescribed boundary conditions (as was usually done), but on the entire set of functions of the space in which the solution is sought, i.e., on the set of functions not subject to any boundary conditions.

The approach used by us is also close, in a certain sense, to the results of K. Friedrichs [2] for differential equations with a symmetrizable operator.

For the string-vibration equation under consideration, we study the existence and uniqueness of the solution of the variational problem, the existence and uniqueness of a generalized solution, and the relation of the generalized solution to the solution of the variational problem; we investigate the differential properties of the generalized solution, as well as the question of satisfying the boundary conditions and the question of the well-posedness of the problem of string vibrations. In particular, the well-posedness of a certain mixed problem for the string-vibration equation is shown in the case when the boundary data are prescribed on the entire boundary.

The main aim of the present paper is to illustrate the possibilities of the method proposed in [22, 23] in the case of studying equations of non-elliptic type, for example hyperbolic equations, although the results obtained here for the string-vibration equation apparently are also of independent interest.

§ 1. Symmetrization of the differential equation of string vibrations

Consider, in a certain two-dimensional bounded domain \(\Omega\) with a piecewise continuously differentiable boundary \(\Gamma\), the string-vibration equation

\[ \frac{\partial^{2}u}{\partial t^{2}}-\frac{\partial^{2}u}{\partial x^{2}}=g, \tag{1} \]

where \(t\) is time; \(x\) is the coordinate of points of the string; \(u\) is the unknown function corresponding to the deflection of the string; \(g\) is a certain given function depending on \(x,t\). The propagation speed of waves in the string described by equation (1) is, for simplicity, taken to be equal to 1.

Equation (1) is, as is known, an equation of hyperbolic type, and the corresponding differential operator for it is in general neither symmetric nor positive definite. However, as will be shown in this section, this operator possesses the property of symmetry and positive definiteness in a certain generalized sense; moreover, this symmetry and positivity of the operator turns out to be very useful and natural in the study of equation (1) by methods of functional analysis.

Let the domain \(\Omega\) have the property that each of the characteristics of equation (1),

\[ \xi=x+t,\qquad \eta=x-t, \tag{2} \]

intersects the boundary \(\Gamma\) in no more than two points, or coincides with the boundary \(\Gamma\) on some part of it. Denote by \(\Gamma_i=\{\xi_i,\eta_i\}\)

\((i=0, 1, 2, 3, 4)\) are the points of contact of the boundary \(\Gamma\) of the domain \(\Omega\) (Fig. 1) with the corresponding characteristics \(\xi\) or \(\eta\).

The quantities \(\xi^i, \eta^i\) here denote the coordinates of the point \(\Gamma_i\) in the coordinate system \(\xi, \eta\), indicated in Fig. 1.

Let, further, the equation of the curve \(\Gamma_3\Gamma_4\Gamma_0\) of the boundary \(\Gamma\) (see Fig. 1) in the coordinate system \(\xi, \eta\) have the form:

\[ \gamma_1=\gamma_1(\eta), \qquad \eta^0 \leqslant \eta \leqslant \eta^3, \tag{3} \]

and the curve \(\Gamma_0\Gamma_1\Gamma_2\), respectively,

\[ \gamma_2=\gamma_2(\xi), \qquad \xi^2 \leqslant \xi \leqslant \xi^0. \tag{3'} \]

Fig. 1

The portions \(\Gamma_4\Gamma_0\) and \(\Gamma_0\Gamma_1\) of the boundary \(\Gamma\) coincide respectively with the characteristics \(\eta, \xi\), passing through the points \(\Gamma_4, \Gamma_0\) and \(\Gamma_0, \Gamma_1\).

To simplify the subsequent reasoning, we transform the original equation (1) to the new coordinate system \(\xi, \eta\) (2); then we shall have

\[ \frac{\partial^2 u}{\partial \xi \partial \eta}=g, \tag{4} \]

where \(g\), as in equation (1), denotes a certain known function of \(\xi, \eta\), defined on \(\Omega\).

Along with equation (4), we shall consider the operator \(K\) defined by the relation

\[ Kv=\int_{\xi}^{\gamma_1}\frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta +\int_{\eta}^{\gamma_2}\frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau, \tag{5} \]

where \(\gamma_1\) and \(\gamma_2\) are defined by relations (3), (3′).

Now forming the product of the left-hand side of equality (4) by the expression (5) for \(Kv\), and integrating over the domain \(\Omega\), we shall have the following equality:

\[ \int_{\Omega}\frac{\partial^2 u}{\partial \xi \partial \eta} \left[ \int_{\xi}^{\gamma_1}\frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta + \int_{\eta}^{\gamma_2}\frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]d\xi d\eta = \]

\[ =- \int_{\Gamma}\frac{\partial u}{\partial \eta} \left[ \int_{\xi}^{\gamma_1}\frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta \right]d\eta + \int_{\Gamma}\frac{\partial u}{\partial \xi} \left[ \int_{\eta}^{\gamma_2}\frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]d\xi + \]

\[ +\int_{\Omega} \left[ \frac{\partial u}{\partial \xi}\frac{\partial v}{\partial \xi} + \frac{\partial u}{\partial \eta}\frac{\partial v}{\partial \eta} \right]d\xi d\eta, \tag{6} \]

obtained by integration by parts.

We transform several contour integrals in this equality. To this end, denote by \(\gamma, l_1, l_2\) (see Fig. 1) respectively the portions \(\Gamma_2\Gamma_3\), \(\Gamma_1\Gamma_2\), \(\Gamma_3\Gamma_4\) of the boundary \(\Gamma\), and use the fact that on the contour \(\Gamma\) the relations

\[ \frac{\partial}{\partial \xi} = \sin\theta\,\frac{\partial}{\partial n} + \cos\theta\,\frac{\partial}{\partial s}, \qquad \frac{\partial}{\partial \eta} = \cos\theta\,\frac{\partial}{\partial n} - \sin\theta\,\frac{\partial}{\partial s}, \tag{7} \]

and also

\[ d\xi=\cos\theta\,ds,\qquad d\eta=-\sin\theta\,ds, \tag{8} \]

hold, where \(n\) is the direction of the outward normal to \(\Gamma\), \(s\) is the positive direction along the contour \(\Gamma\), and \(\theta\) is the angle between the direction \(n\) on the contour and the \(\eta\)-axis.

Then for the contour integrals in (6) we may write:

\[ \begin{aligned} &-\int_{\Gamma}\frac{\partial u}{\partial \eta} \left[ \int_{\xi}^{\gamma_1} \frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta \right]d\eta + \int_{\Gamma}\frac{\partial u}{\partial \xi} \left[ \int_{\eta}^{\gamma_2} \frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]d\xi =\\ &= -\int_{\gamma}\frac{\partial u}{\partial \eta} \left[ \int_{\xi}^{\gamma_1} \frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta \right]d\eta - \int_{l_1}\frac{\partial u}{\partial \eta} \left[ \int_{\xi}^{\gamma_1} \frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta \right]d\eta +\\ &\quad +\int_{\gamma}\frac{\partial u}{\partial \xi} \left[ \int_{\eta}^{\gamma_2} \frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]d\xi + \int_{l_2}\frac{\partial u}{\partial \xi} \left[ \int_{\eta}^{\gamma_2} \frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]d\xi =\\ &= \int_{\gamma}\frac{\partial u}{\partial s} \left[ -\sin^2\theta \int_{\xi}^{\gamma_1} \frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta + \cos^2\theta \int_{\eta}^{\gamma_2} \frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]ds +\\ &\quad +\int_{\gamma}\frac{\partial u}{\partial n}\sin\theta\cdot\cos\theta \left[ \int_{\xi}^{\gamma_1} \frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta + \int_{\eta}^{\gamma_2} \frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]ds -\\ &\quad -\int_{l_1}\frac{\partial u}{\partial \eta} \left[ \int_{\xi}^{\gamma_1} \frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta \right]d\eta + \int_{l_2}\frac{\partial u}{\partial \xi} \left[ \int_{\eta}^{\gamma_2} \frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]d\xi . \tag{9} \end{aligned} \]

Since on the portions of the boundary \(\gamma_1\) and \(\gamma_2\) the corresponding integrals vanish.

Taking relation (9) into account, equality (6) can finally be written in the following form:

\[ \begin{aligned} &\int_{\Omega} \frac{\partial^2 u}{\partial \xi\,\partial \eta} \left[ \int_{\xi}^{\gamma_1} \frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta + \int_{\eta}^{\gamma_2} \frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]d\xi\,d\eta +\\ &\quad +\int_{\gamma}uv\,ds + \int_{\gamma}\frac{\partial u}{\partial s} \left[ -\sin^2\theta \int_{\xi}^{\gamma_1} \frac{\partial v}{\partial \eta}(\zeta,\eta)\,d\zeta + \cos^2\theta \int_{\eta}^{\gamma_2} \frac{\partial v}{\partial \xi}(\xi,\tau)\,d\tau \right]ds + \end{aligned} \]

\[ +\int_{\gamma}\frac{\partial u}{\partial n}\sin\theta\cdot\cos\theta \left[ \int_{\xi}^{\gamma_1}\frac{\partial v}{\partial\eta}(\zeta,\eta)\,d\zeta + \int_{\eta}^{\gamma_2}\frac{\partial v}{\partial\xi}(\xi,\tau)\,d\tau \right]ds+ \]

\[ +\int_{l_1}\frac{\partial u}{\partial\eta} \left[ \int_{\xi}^{\gamma_1}\frac{\partial v}{\partial\eta}(\zeta,\eta)\,d\zeta \right]d\eta - \int_{l_2}\frac{\partial u}{\partial\xi} \left[ \int_{\eta}^{\gamma_2}\frac{\partial v}{\partial\xi}(\xi,\tau)\,d\tau \right]d\xi = \]

\[ =\int_{\Omega}\left( \frac{\partial u}{\partial\xi}\frac{\partial v}{\partial\xi} + \frac{\partial u}{\partial\eta}\frac{\partial v}{\partial\eta} \right)d\xi d\eta + \int_{\gamma}uv\,ds, \tag{10} \]

where the quantity \(\displaystyle \int_{\gamma}uv\,ds\) has been added to both sides of the equality. Relation (10) is valid for any smooth functions \(u, v\) and is, in fact, the fundamental equality from which follows the formulation of the boundary-value problem for the equation of vibrations of a string and the possibility of solving this problem by the variational method.

It is not difficult to see that the first factors under the integrals on the left-hand side of equality (10) represent a certain operator \(A\), which assigns to the function \(u\) six functions in another space, while the second terms define, respectively, an auxiliary operator \(B\), which assigns to the element \(v\) likewise six functions. On the right-hand side of equality (10) there is a quadratic positive-definite form; therefore the operator \(A\), corresponding to equation (4), will be \(B\)-symmetric and \(B\)-positive. In order to define the operators \(A\) and \(B\) more precisely, we introduce the following notation:

\[ Kv= \int_{\xi}^{\gamma_1}\frac{\partial v}{\partial\eta}(\zeta,\eta)\,d\zeta + \int_{\eta}^{\gamma_2}\frac{\partial v}{\partial\xi}(\xi,\tau)\,d\tau, \]

\[ Tv= -\sin^2\theta\int_{\xi}^{\gamma_1}\frac{\partial v}{\partial\eta}(\zeta,\eta)\,d\zeta + \cos^2\theta\int_{\eta}^{\gamma_2}\frac{\partial v}{\partial\xi}(\xi,\tau)\,d\tau, \]

\[ K_{\Theta}v= \sin\theta\cdot\cos\theta \left[ \int_{\xi}^{\gamma_1}\frac{\partial v}{\partial\eta}(\zeta,\eta)\,d\zeta + \int_{\eta}^{\gamma_2}\frac{\partial v}{\partial\xi}(\xi,\tau)\,d\tau \right], \]

\[ L_1v=-\int_{\xi}^{\gamma_1}\frac{\partial v}{\partial\eta}(\zeta,\eta)\,d\zeta,\qquad L_2v=\int_{\eta}^{\gamma_2}\frac{\partial v}{\partial\xi}(\xi,\tau)\,d\tau. \tag{11} \]

Then the operators \(A\) and \(B\) can be defined by the following relations:

\[ Au= \left\{ \frac{\partial^2 u}{\partial\xi\,\partial\eta},\; u,\; \frac{\partial u}{\partial s},\; \frac{\partial u}{\partial n},\; \frac{\partial u}{\partial\xi},\; \frac{\partial u}{\partial\eta} \right\}, \]

\[ Bv=\{Kv,\;v,\;Tv,\;K_{\Theta}v,\;L_1v,\;L_2v\}, \tag{12} \]

where the vector-functions on the right are defined, respectively, on \(\Omega, \gamma, \gamma, \gamma, l_1, l_2\).

Let us now introduce the space \(\widetilde W_2^1(\Omega)\) with scalar product

\[ [u,v]_{\widetilde W_2^1(\Omega)} = \int_{\Omega}\left( \frac{\partial u}{\partial \xi}\frac{\partial v}{\partial \xi} + \frac{\partial u}{\partial \eta}\frac{\partial v}{\partial \eta} \right)d\xi d\eta + \int_{\gamma}uv\,ds \]

and norm

\[ \|u\|_{\widetilde W_2^1(\Omega)}^2 = \int_{\Omega}\left[ \left(\frac{\partial u}{\partial \xi}\right)^2 + \left(\frac{\partial u}{\partial \eta}\right)^2 \right]d\xi d\eta + \int_{\gamma}u^2\,ds, \tag{13} \]

as well as the space \(L_2(\Omega_\gamma^l)\) of vector-functions of the form

\[ f=\left\{g,\varphi,\frac{\partial\varphi}{\partial s},\psi,\chi,\omega\right\}, \tag{14} \]

defined, respectively, on \(\Omega, \gamma, \gamma, \gamma, l_1, l_2\).

In the space \(L_2(\Omega_\gamma^l)\) we define the scalar product and norm by the relations:

\[ [f_1,f_2]_{L_2(\Omega_\gamma^l)} = \int_{\Omega}g_1g_2\,d\xi d\eta + \int_{\gamma}\varphi_1\varphi_2\,ds + \]

\[ + \int_{\gamma}\frac{\partial\varphi_1}{\partial s} \frac{\partial\varphi_2}{\partial s}\,ds + \int_{\gamma}\psi_1\psi_2\,ds + \int_{l_1}\chi_1\chi_2\,d\eta + \int_{l_2}\omega_1\omega_2\,d\xi, \tag{15} \]

\[ \|f\|_{L_2(\Omega_\gamma^l)}^2 = \int_{\Omega}g^2\,d\xi d\eta + \int_{\gamma}\varphi^2\,ds + \int_{\gamma}\left(\frac{\partial\varphi}{\partial s}\right)^2 ds + \]

\[ + \int_{\gamma}\psi^2\,ds + \int_{l_1}\chi^2\,d\eta + \int_{l_2}\omega^2\,d\xi. \]

It is clear that the space \(\widetilde W_2^1(\Omega)\) is the space of S. L. Sobolev and is equivalent to the space \(W_2^1(\Omega)\), in which, as usual, the norm is taken to be

\[ \|u\|_{W_2^1(\Omega)}^2 = \int_{\Omega}\left[ \left(\frac{\partial u}{\partial \xi}\right)^2 + \left(\frac{\partial u}{\partial \eta}\right)^2 + u^2 \right]d\xi d\eta. \tag{16} \]

We shall regard the spaces \(\widetilde W_2^1(\Omega)\), \(W_2^1(\Omega)\), and \(L_2(\Omega_\gamma^l)\) as complete. The operators \(A\) and \(B\), defined by relations (12), may be regarded as acting from \(\widetilde W_2^1(\Omega)\) into the space \(L_2(\Omega_\gamma^l)\). Moreover, it is clear that \(\widetilde W_2^1(\Omega)\subset L_2(\Omega_\gamma^l)\). Using the notation adopted in (12), (13), equality (10) can be written in the following form:

\[ (Au,Bv)=[u,v]_{\widetilde W_2^1(\Omega)}, \tag{17} \]

where the scalar product \((Au,Bv)\) denotes the left-hand side of equality (10).

Let us now return to the equation of oscillations of a string. Suppose that in the domain \(\Omega\) for equation (1) the following boundary-value problem is prescribed:

\[ \frac{\partial^{2}u}{\partial t^{2}}-\frac{\partial^{2}u}{\partial x^{2}}=g(x,t),\qquad u\big|_{\gamma}=\varphi(s),\qquad \frac{\partial}{\partial s}\left(u\big|_{\gamma}\right)=\frac{\partial\varphi}{\partial s}(s), \]

\[ \left.\frac{\partial u}{\partial n}\right|_{\gamma}=\psi(s),\qquad \left.\frac{\partial u}{\partial \xi}\right|_{l_{1}}=\chi(s),\qquad \left.\frac{\partial u}{\partial \eta}\right|_{l_{2}}=\omega(s), \tag{18} \]

where the right-hand side \(\left\{g,\varphi,\dfrac{\partial\varphi}{\partial s},\psi,\chi,\omega\right\}\) is some known element of the space \(L_{2}(\Omega_{l}^{t})\). The formulation of such a problem, as we shall see below, follows directly from relation (10).

In the present paragraph we have established, in fact, that if the left-hand side of the indicated boundary-value problem (18) is regarded as an operator \(A\) acting from the space \(\widetilde W_{2}^{1}(\Omega)\) into the space of vector-functions \(L_{2}(\Omega_{l}^{t})\), then, according to relations (10), (12), (17), the operator \(A\) is \(B\)-symmetric and \(B\)-positive [22], i.e., for any smooth elements \(u,v\) the equality

\[ (Au,Bv)=(Bu,Av)\quad (B\text{-symmetry}) \]

holds, and, moreover,

\[ (Au,Au)>0\quad \text{for } u\ne 0, \]

and from the condition \((Au,Bu)\to 0\) it follows that

\[ \|u\|_{\widetilde W_{2}^{1}(\Omega)}\to 0\quad (B\text{-positivity}). \]

The presence of the indicated properties of the operator \(A\) makes it possible to study easily the variational problem and the differential equation of string vibrations.

§ 2. The variational problem for the equation of string vibrations

Let us agree that by the variational problem for the equation of string vibrations we shall understand the problem of finding the minimum of a certain quadratic functional possessing the property that the equation of string vibrations is the Euler equation for this functional, and the solution of the equation realizes a strict minimum of the indicated functional.

We note that the quadratic functional for the string usually used in mathematical physics,

\[ J(u)=\int_{0}^{T}\int_{0}^{l}\left[\left(\frac{\partial u}{\partial x}\right)^{2}-\left(\frac{\partial u}{\partial t}\right)^{2}-2gu\right]\,dxdt, \tag{19} \]

constructed in accordance with Hamilton’s principle, does not belong to the given class of functionals, since it is bounded neither above nor below. Therefore, for the mathematical investigation of the equation of string vibrations in this case, one cannot directly use the variational principle, well developed for equations of elliptic type.

To verify the unboundedness of the functional (19), consider the simplest case, when the boundary conditions for the string are given in the following form:

\[ u(0,t)=u(l,t)=0,\qquad u(x,0)=\frac{\partial u}{\partial t}(x,0)=0. \tag{20} \]

Let us compute the value of the functional (19) on the element

\[ \tilde u(x,t)=at^n(x^m-xl^{m-1}), \tag{21} \]

where \(a\) is a certain coefficient; \(m,n\) are positive integers.

It is obvious that an element of the form (21) satisfies the boundary conditions (20) and, moreover, in a finite domain, the condition

\[ \int_0^T\int_0^l \left[ \left(\frac{\partial \tilde u}{\partial t}\right)^2+ \left(\frac{\partial \tilde u}{\partial x}\right)^2 \right]\,dxdt<\infty, \]

which expresses the finiteness of the total energy of the moving string. Substituting the element (21) into expression (19), we obtain

\[ \begin{aligned} J(\tilde u) &= \frac{a^2T^{2n-1}l^{2m+1}(m-1)^2} {3(4n^2-1)(4m^2-1)(m+2)} \left[ 2n^2(2n+1)(2m+1) \right.\\ &\qquad\left. -3\left(\frac{T}{l}\right)^2(2n-1)(2m+1)(m+2) \right]\\ &\quad -2a\int_0^T\int_0^l g(x,t)t^n(x^m-xl^{m-1})\,dxdt = a^2C(m,n)-2aD(m,n), \end{aligned} \]

whence it is seen that the coefficient \(C(m,n)\) of \(a^2\) may take both positive and negative values under a corresponding choice of the numbers \(m\) and \(n\). Since the coefficient \(a\) can then be taken arbitrarily, we conclude from this that the functional \(J(\tilde u)\) may take any value from \(-\infty\) to \(+\infty\), i.e., for the functional (21) the relation

\[ -\infty \le J(u)\le +\infty \]

holds.

Thus, we have verified that the functional (21) is bounded neither from above nor from below.

Using the results obtained in § 1, as well as the method set forth in the note [23], we shall construct for the equation of string vibrations (18) a quadratic functional bounded from below and allowing the methods of functional analysis to be used for the investigation of the corresponding problems.

According to [23], for the equation of string vibrations (18), written in the form

\[ Au=f, \tag{22} \]

the corresponding functional for the variational problem has the form

\[ D(u)=(Au,Bu)-2(f,Bu), \tag{23} \]

where the operators \(A,B\) are defined by the relations (11), (12), and the element \(f\) by the relation (14), with \(f\in L_2(\Omega_T^l)\).

By virtue of conditions (17), (13), the functional (23) can be written in the form

\[ D(u)=\|u\|_{W_2^1(\Omega)}^2-2(f,Bu), \tag{24} \]

or, taking into account the notation (11), (13), (14), in a more expanded form

\[ \begin{aligned} D(u)={}& \int_{\Omega}\left[\left(\frac{\partial u}{\partial \xi}\right)^2+ \left(\frac{\partial u}{\partial \eta}\right)^2\right]\,d\xi\,d\eta +\int_{\gamma}u^2\,ds \\ &-2\int_{\Omega}gKu\,d\xi\,d\eta -2\int_{\gamma}\varphi u\,ds -2\int_{\gamma}\frac{\partial\varphi}{\partial s}Tu\,ds -\int_{\gamma}\psi K_{\theta}u\,ds \\ &-2\int_{l_1}\chi L_1u\,d\eta -2\int_{l_2}\omega L_2u\,d\xi , \end{aligned} \tag{25} \]

where the linear part of the functional \(D(u)\) denotes the functional \(-2(f,Bu)\).

It is not difficult to see that the functional \(D(u)\) (24), (25) is defined on the whole space \(\widetilde W_2^1(\Omega)\), since its quadratic part is simply the norm of the element \(u\) in the space \(\widetilde W_2^1(\Omega)\), and the functional \((f,Bu)\) is also defined on all elements \(u\in \widetilde W_2^1(\Omega)\), as is directly seen from relations (11), (12), (14). We note that, in the case where the functional \((f,Bu)\) were defined only on a set of functions everywhere dense in the space \(\widetilde W_2^1(\Omega)\), it could be extended by continuity to the entire space, since, as we shall verify below, the functional \((f,Bu)\) is linear in the space \(\widetilde W_2^1(\Omega)\), and all the subsequent arguments would remain valid.

Before considering the variational problem (24), we prove that the following holds.

Lemma 1. For any element \(f\in L_2(\Omega_\gamma^l)\), the functional \((f,Bu)\) is linear in the space \(\widetilde W_2^1(\Omega)\), and the estimate

\[ |(f,Bu)|\le c\|f\|_{L_2(\Omega_\gamma^l)}\cdot \|u\|_{\widetilde W_2^1(\Omega)}, \tag{26} \]

holds for all \(u\in \widetilde W_2^1(\Omega)\).

Proof. By virtue of the notation (11), (12), (14), one may write

\[ (f,Bu)= \int_{\Omega}gKu\,d\xi\,d\eta +\int_{\gamma}\varphi u\,ds +\int_{\gamma}\frac{\partial\varphi}{\partial s}Tu\,ds + \]

\[ +\int_{\gamma}\psi K_{\theta}u\,ds +\int_{l_1}\chi L_1u\,d\eta +\int_{l_2}\omega L_2u\,d\xi . \tag{26'} \]

Let us estimate in turn each term on the right-hand side of this equality. By the Cauchy–Bunyakovsky inequality, for the first term we shall have

\[ \left|\int_{\Omega}gKu\,d\xi\,d\eta\right| = \left|\int_{\Omega}g\left[ \int_{\xi}^{\gamma_1}\frac{\partial u}{\partial\eta}(\zeta,\eta)\,d\zeta + \int_{\eta}^{\gamma_2}\frac{\partial u}{\partial\xi}(\xi,\tau)\,d\tau \right]d\xi\,d\eta\right| \le \]

\[ \le \int_{\Omega}|g|\left[ \left(\int_{\xi}^{\gamma_1}d\zeta\right)^{1/2} \left(\int_{\xi}^{\gamma_1}\frac{\partial u^2}{\partial\eta}(\zeta,\eta)\,d\zeta\right)^{1/2} + \right. \]

\[ +\left(\int_{\eta}^{\gamma_2} d\tau\right)^{1/2} \left(\int_{\eta}^{\gamma_2}\frac{\partial u^2}{\partial \xi}(\xi,\tau)d\tau\right)^{1/2} \Bigg]\,d\xi d\eta . \tag{27} \]

Putting further

\[ c=\sup_{\xi,\eta\subset \Omega}2\{|\gamma_1-\xi|,\ |\gamma_2-\eta|\} \]

and increasing the limits of integration in the right-hand side of (27), we obtain

\[ \left|\int_{\Omega} gKu\,d\xi d\eta\right| \leq c\left(\int_{\Omega} g^2\,d\xi d\eta\right)^{1/2} \left\{\int_{\Omega}\left[\left(\frac{\partial u}{\partial \xi}\right)^2+ \left(\frac{\partial u}{\partial \eta}\right)^2\right]d\xi d\eta\right\}^{1/2}. \]

Hence, by the definition of the norm in the space \(\widetilde W_2^1(\Omega)\) (13), the estimate follows all the more:

\[ \left|\int_{\Omega} gKu\,d\xi d\eta\right| \leq c\left(\int_{\Omega} g^2\,d\xi d\eta\right)^{1/2} \|u\|_{\widetilde W_2^1(\Omega)} . \tag{28} \]

For the second term in relation (27) the estimate is quite elementary:

\[ \left|\int_{\gamma}\varphi u\,ds\right| \leq \left(\int_{\gamma}\varphi^2 ds\right)^{1/2} \left(\int_{\gamma}u^2 ds\right)^{1/2} \leq \left(\int_{\gamma}\varphi^2 ds\right)^{1/2} \|u\|_{\widetilde W_2^1(\Omega)} . \tag{29} \]

Let us now estimate the third term. By virtue of the notation (11) we have

\[ \int_{\gamma}\frac{\partial \varphi}{\partial s}Tu\,ds = \int_{\gamma}\frac{\partial \varphi}{\partial s} \left[ -\sin^2\theta\int_{\xi}^{\gamma_1}\frac{\partial u}{\partial \eta}(\zeta,\eta)d\zeta + \cos^2\theta\int_{\eta}^{\gamma_2}\frac{\partial u}{\partial \xi}(\xi,\tau)d\tau \right]ds . \tag{30} \]

For the first integral in the right-hand side of this equality we may write the estimate

\[ \left| \int_{\gamma}\frac{\partial \varphi}{\partial s}\sin^2\theta \left[\int_{\xi}^{\gamma_1}\frac{\partial u}{\partial \eta}(\zeta,\eta)d\zeta\right]ds \right| \leq \]

\[ \leq \int_{\gamma}\left|\frac{\partial \varphi}{\partial s}\right| \left(\int_{\xi}^{\gamma_1}d\zeta\right)^{1/2} \left(\int_{\xi}^{\gamma_1}\frac{\partial u^2}{\partial \eta}(\zeta,\eta)d\zeta\right)^{1/2} d\eta, \tag{31} \]

obtained taking into account that

\[ d\eta=-\sin\theta\,ds \quad\text{and}\quad |\sin\theta|\leq 1. \]

Here \(\xi\) in relation (31) should properly be replaced by the quantity \(\gamma(\eta)\), if by \(\gamma(\eta)\), \(\eta^2\leq \eta\leq \eta^3\), we denote the equation of the segment \(\Gamma_2\Gamma_3\) of the boundary \(\Gamma\) (Fig. 1).

Increasing the limits of integration in the right-hand side of (31) and applying the Cauchy—Bunyakovsky inequality, we obtain

\[ \left| \int_{\gamma}\frac{\partial \varphi}{\partial s}\sin^2\theta \left[\int_{\xi}^{\gamma_1}\frac{\partial u}{\partial \eta}(\zeta,\eta)d\zeta\right]ds \right| \leq \]

\[ \leqslant c\left[\int_\gamma \left(\frac{\partial\varphi}{\partial s}\right)^2 ds\right]^{1/2} \cdot \left[\int_\Omega \frac{\partial u^2}{\partial \eta}(\xi,\eta)\,d\xi\,d\eta\right]^{1/2}. \tag{32} \]

Quite analogously, for the second integral in the right-hand side of expression (30) one obtains the estimate

\[ \left|\int_\gamma \frac{\partial\varphi}{\partial s}\cos^2\theta \left[\int_\eta^{\gamma_2}\frac{\partial u}{\partial \xi}(\xi,\tau)\,d\tau\right]ds\right| \leqslant \]

\[ \leqslant \left[\int_\gamma \left(\frac{\partial\varphi}{\partial s}\right)^2 ds\right]^{1/2} \left[\int_\Omega \frac{\partial u^2}{\partial \xi}(\xi,\tau)\,d\xi\,d\tau\right]^{1/2}. \tag{33} \]

Taking now into account relations (32), (33), (30), we obtain for the third term the final estimate

\[ \left|\int_\gamma \frac{\partial\varphi}{\partial s}T u\,ds\right| \leqslant c\left[\int_\gamma \left(\frac{\partial\varphi}{\partial s}\right)^2 ds\right]^{1/2} \left\{\int_\Omega\left[\left(\frac{\partial u}{\partial \xi}\right)^2+ \left(\frac{\partial u}{\partial \eta}\right)^2\right]d\xi\,d\eta\right\}^{1/2} \leqslant \]

\[ \leqslant c\left[\int_\gamma \left(\frac{\partial\varphi}{\partial s}\right)^2 ds\right]^{1/2} \cdot \|u\|_{\widetilde W_2^1(\Omega)} . \tag{34} \]

The fourth term on the right-hand side of the equality is estimated quite analogously to the third term, namely

\[ \left|\int_\gamma \psi K_\theta u\,ds\right| = \left|\int_\gamma \psi\sin\theta\cos\theta \left[ \int_\xi^{\gamma_1}\frac{\partial u}{\partial \eta}(\zeta,\eta)\,d\zeta + \int_\eta^{\gamma_2}\frac{\partial u}{\partial \xi}(\xi,\tau)\,d\tau \right]ds\right| \leqslant c\left[\int_\gamma \psi^2\,ds\right]^{1/2} \|u\|_{\widetilde W_2^1(\Omega)} . \tag{35} \]

In exactly the same way, for the last two terms in (26) we obtain the estimates

\[ \left|\int_{l_1}\chi L_1 u\,d\eta\right| = \left|\int_{l_1}\chi \left[\int_\xi^{\gamma_1}\frac{\partial u}{\partial \eta}(\zeta,\eta)\,d\zeta\right]d\eta\right| \leqslant c\left[\int_{l_1}\chi^2\,d\eta\right]^{1/2} \cdot \|u\|_{\widetilde W_2^1(\Omega)}, \tag{36} \]

\[ \left|\int_{l_2}\omega L_2 u\,d\xi\right| = \left|\int_{l_2}\omega \left[\int_\eta^{\gamma_2}\frac{\partial u}{\partial \xi}(\xi,\tau)\,d\tau\right]d\xi\right| \leqslant c\left[\int_{l_2}\omega^2\,d\xi\right]^{1/2} \cdot \|u\|_{\widetilde W_2^1(\Omega)} . \tag{37} \]

Using now estimates (28), (29), (34)—(37), for the functional \((f,Bu)\) (26) one can write the general estimate

\[ |(f,Bu)| \leqslant c\left\{ \left[\int_\Omega g^2\,d\xi\,d\eta\right]^{1/2} + \left[\int_\gamma \varphi^2\,ds\right]^{1/2} + \left[\int_\gamma \left(\frac{\partial\varphi}{\partial s}\right)^2 ds\right]^{1/2} + \right. \]

\[ \left. + \left[\int_\gamma \psi^2\,ds\right]^{1/2} + \left[\int_{l_1}\chi^2\,d\eta\right]^{1/2} + \left[\int_{l_2}\omega^2\,d\xi\right]^{1/2} \right\} \cdot \|u\|_{\widetilde W_2^1(\Omega)} \leqslant \]

\[ \leqslant c\|f\|_{L_2(\Omega_\gamma^1)}\cdot \|u\|_{\widetilde W_2^1(\Omega)}, \]

where \(c\) is a certain constant depending only on the domain \(\Omega\).

In the last passage here we used the inequality

\[ |a_1+a_2+\ldots+a_n|\leq \sqrt{n}\,(a_1^2+a_2^2+\ldots+a_n^2)^{1/2} \]

and the definition of the norm (15) in the space \(L_2(\Omega_\gamma^l)\). Lemma 1 is thereby proved.

Thus we have established that the functional \((f,Bu)\) is indeed bounded in the space \(\widetilde W_2^1(\Omega)\).

We now turn to the study of the variational problem (24) for the equation of vibrations of a string. We shall seek the minimum of the functional \(D(u)\) on the class of functions \(u\in \widetilde W_2^1(\Omega)\) that are not subject to any boundary conditions, since the corresponding boundary conditions are included in the functional (24).

We shall show that the following is true.

Theorem 1. For any element \(f\in L_2(\Omega_\gamma^l)\), in the space \(\widetilde W_2^1(\Omega)\) there exists a unique solution of the variational problem corresponding to the functional \(D(u)\).

Moreover, the variational problem is correctly posed in the sense that a small change of the element \(f\) in the space \(L_2(\Omega_\gamma^l)\) corresponds to a small change of the sought solution in the space \(\widetilde W_2^1(\Omega)\).

The proof of the theorem is carried out according to the usual scheme (see, for example, [1]) used for the study of elliptic equations by the variational method.

First of all, let us establish that the functional \(D(u)\) is bounded below. For this we shall use the results of Lemma 1. Taking into account estimate (26) for the functional \((f,Bu)\), we can write

\[ D(u)=\|u\|_{\widetilde W_2^1(\Omega)}^2-2(f,Bu)\geq \|u\|_{\widetilde W_2^1(\Omega)}^2-2(f,Bu)\geq \]

\[ \geq \|u\|_{\widetilde W_2^1(\Omega)}^2 -2c\|f\|_{L_2(\Omega_\gamma^l)}\|u\|_{\widetilde W_2^1(\Omega)}\geq \]

\[ \geq \left[\|u\|_{\widetilde W_2^1(\Omega)}-c\|f\|_{L_2(\Omega_\gamma^l)}\right]^2 -c^2\|f\|_{L_2(\Omega_\gamma^l)}^2 \geq -c^2\|f\|_{L_2(\Omega_\gamma^l)}^2 . \]

Hence we conclude that for any elements \(u\in \widetilde W_2^1(\Omega)\)

\[ d\leq D(u)<\infty, \tag{38} \]

where \(d\) is the exact lower bound of the functional \(D(u)\).

By the definition of the exact lower bound in \(\widetilde W_2^1(\Omega)\), one can indicate a minimizing sequence \(\{u_n\}\), for which

\[ \lim_{n\to\infty} D(u_n)=d. \tag{39} \]

We shall show that this sequence converges in the space \(\widetilde W_2^1(\Omega)\) to some element \(u_0\), which is a solution of the variational problem. For this it is sufficient to establish that the sequence \(\{u_n\}\) is fundamental in the space \(\widetilde W_2^1(\Omega)\), i.e.,

\[ \|u_m-u_n\|_{\widetilde W_2^1(\Omega)}\to 0, \qquad m,n\to\infty . \]

Obviously, the numbers \(m\) and \(n\) can be chosen so large that for any arbitrarily small \(\varepsilon>0\) one has

\[ D(u_n) \leq d+\frac{\varepsilon^2}{4}, \qquad D(u_m) \leq d+\frac{\varepsilon^2}{4}. \tag{40} \]

For the element \(\dfrac{u_m+u_n}{2}\), which also belongs to \(\widetilde W_2^1(\Omega)\), by virtue of condition (38) we shall have

\[ D\left(\frac{u_m+u_n}{2}\right)\geq d. \tag{41} \]

Considering now the obvious identity

\[ \|u_m-u_n\|_{\widetilde W_2^1(\Omega)}^2 = 2D(u_m)+2D(u_n)-4D\left(\frac{u_m+u_n}{2}\right), \tag{42} \]

on the basis of relations (40), (41) we obtain

\[ \|u_m-u_n\|_{\widetilde W_2^1(\Omega)}\leq \varepsilon, \]

whence, in view of the arbitrariness of \(\varepsilon\), it follows that the sequence \(\{u_n\}\) will indeed be fundamental in the space \(\widetilde W_2^1(\Omega)\).

But since \(\widetilde W_2^1(\Omega)\) is a complete space, it follows that for the sequence \(\{u_n\}\) there exists a limiting element \(u_0\in \widetilde W_2^1(\Omega)\), for which

\[ \|u_n-u_0\|_{\widetilde W_2^1(\Omega)} \xrightarrow[n\to\infty]{} 0. \tag{43} \]

We now show that the lower bound of the functional \(D(u)\) is attained precisely at the element \(u_0\). Indeed, using inequality (26), we may write:

\[ \begin{aligned} \bigl|D(u_0)-D(u_n)\bigr| &\leq \left|\|u_0\|_{\widetilde W_2^1(\Omega)}^2 - \|u_n\|_{\widetilde W_2^1(\Omega)}^2\right| \\ &\quad +2\bigl|(Bu_0-Bu_n,f)\bigr| \\ &\leq \left|\|u_0\|_{\widetilde W_2^1(\Omega)}^2 - \|u_n\|_{\widetilde W_2^1(\Omega)}^2\right| \\ &\quad +2c\|f\|_{L_2(\Omega_2^1)} \|u_0-u_n\|_{\widetilde W_2^1(\Omega)}. \end{aligned} \tag{44} \]

On the basis of relation (43), the right-hand side of this inequality tends to zero as \(n\to\infty\); therefore we obtain

\[ \lim_{n\to\infty}\bigl|D(u_0)-D(u_n)\bigr|=0, \]

whence, by virtue of condition (39), it follows that

\[ D(u_0)=d, \]

i.e., the element \(u_0\) indeed realizes the minimum of the functional \(D(u)\).

The spaces \(\widetilde W_2^1(\Omega)\) and \(W_2^1(\Omega)\) (see (13), (16)) are equivalent; therefore \(u_0\in W_2^1(\Omega)\) and the condition

\[ \|u_n-u_0\|_{W_2^1(\Omega)} \xrightarrow[n\to\infty]{} 0, \]

holds, following directly from (43).

We now show that the solution \(u_0\) is unique in the space \(\widetilde W_2^1(\Omega)\) and, accordingly, in the space \(W_2^1(\Omega)\). Suppose that, besides \(u_0\), there exists another solution \(u_0'\) of the variational problem; then, obviously,

\[ D(u_0)=d \quad \text{and} \quad D(u_0')=d, \]

and, moreover,

\[ D\left(\frac{u_0+u_0'}{2}\right)\geq d. \]

Using relation (42), where in place of \(u_m, u_n\) we put \(u_0, u_0'\), we obtain

\[ \|u_0-u_0'\|_{\widetilde W_2^1(\Omega)}^2 \leq 2d+2d-4d=0, \]

whence it follows that

\[ \|u_0-u_0'\|_{\widetilde W_2^1(\Omega)}=0 \]

or

\[ u_0=u_0'. \]

Thus, indeed, in the space \(\widetilde W_2^1(\Omega)\), and by virtue of the equivalence also in the space \(W_2^1(\Omega)\), the solution \(u_0\) of the variational problem is unique. The theorem is thereby proved.

We now turn to the consideration of the differential equation of string vibrations (18).

§ 3. Generalized solution of the equation of string vibrations

The solution of the variational problem (24) is closely connected with the concept of a generalized solution for the equation of string vibrations (18), which we rewrite here in the variables \(\xi,\eta\):

\[ \frac{\partial^2 u}{\partial \xi \partial \eta}=g,\qquad u|_\gamma=\varphi,\qquad \frac{\partial}{\partial s}(u|_\gamma)=\frac{\partial \varphi}{\partial s}, \]

\[ \left.\frac{\partial u}{\partial \eta}\right|_\gamma=\psi,\qquad \left.\frac{\partial u}{\partial \eta}\right|_{l_1}=\chi,\qquad \left.\frac{\partial u}{\partial \xi}\right|_{l_2}=\omega, \tag{45} \]

where the element
\[ f=\left\{g,\varphi,\frac{\partial \varphi}{\partial s},\psi,\chi,\omega\right\}\in L_2(\Omega_\gamma^t). \]

In this paragraph we shall define in two ways a generalized solution of problem (49) (by means of an integral relation and by means of a limiting relation), and show that the generalized solution of problem (45) and the solution of the variational problem (24) coincide in the space \(\widetilde W_2^1(\Omega)\).

Definition 1. An element \(u\in \widetilde W_2^1(\Omega)\) will be called a generalized solution of problem (45) if, for every element \(v\in \widetilde W_2^1(\Omega)\), the equality

\[ (f,Bv)= \int_\Omega \left( \frac{\partial u}{\partial \xi}\frac{\partial v}{\partial \xi} + \frac{\partial u}{\partial \eta}\frac{\partial v}{\partial \eta} \right)\,d\xi\,d\eta + \int_\gamma uv\,ds, \tag{46} \]

holds, where the functional \((f,Bv)\) has the form (26′).

Let us consider in somewhat more detail the definition of a generalized solution introduced above.

The following is true.

Lemma 2. In order that an element \(u \in \widetilde{W}^{1}_{2}(\Omega)\) be a generalized solution of problem (45), it is necessary and sufficient that in the space \(\widetilde{W}^{1}_{2}(\Omega)\) there exist a sequence of smooth functions \(\{u_k\}\) such that the relations

\[ \left\|u_k-u\right\|_{\widetilde{W}^{1}_{2}(\Omega)} \to 0, \qquad k\to\infty, \tag{47} \]

\[ (Au_k-f,Bv)\to 0, \qquad k\to\infty, \tag{48} \]

hold simultaneously, where the last condition is satisfied for any element \(v\in \widetilde{W}^{1}_{2}(\Omega)\).

Let \(u\) be a generalized solution of problem (45). By definition, relation (46) is valid for the element \(u\). By virtue of the property of the space \(\widetilde{W}^{1}_{2}(\Omega)\), one can choose in it a sequence of smooth functions \(\{u_k\}\) which converges in \(\widetilde{W}^{1}_{2}(\Omega)\) to the element \(u\), i.e. condition (47) will hold.

Using the notation (13), we write equality (46) in the form

\[ (f,Bv)=[u,v]_{\widetilde{W}^{1}_{2}(\Omega)}, \]

and for smooth elements \(u_k\), \(v\), on the basis of (17), we shall also have the equality

\[ (Au_k,Bv)=[u_k,v]_{\widetilde{W}^{1}_{2}(\Omega)}. \tag{49} \]

Subtracting these equalities one from the other, we obtain

\[ \left|(Au_k-f,Bv)\right| = \left|[u_k-u,v]_{\widetilde{W}^{1}_{2}(\Omega)}\right| \le \left\|v\right\|_{\widetilde{W}^{1}_{2}(\Omega)} \cdot \left\|u_k-u\right\|_{\widetilde{W}^{1}_{2}(\Omega)}, \]

whence, by virtue of the convergence of \(\{u_k\}\) to \(u\) in the space \(\widetilde{W}^{1}_{2}(\Omega)\), condition (48) follows. Thus, the necessity of conditions (47), (48) is established.

We now prove sufficiency. Suppose it is given that, for some element \(u\in\widetilde{W}^{1}_{2}(\Omega)\), there exists a sequence of smooth functions \(\{u_k\}\) such that relations (47), (48) hold. It is necessary to show that in this case the element \(u\) will certainly be a generalized solution of problem (45).

For this purpose consider the following relation:

\[ \left|(f,Bv)-[u,v]_{\widetilde{W}^{1}_{2}(\Omega)}\right| \le \left|(-Au_k+f,Bv)\right|+ \]

\[ +\left|(Au_k,Bv)-[u_k,v]_{\widetilde{W}^{1}_{2}(\Omega)}\right| + \left|[u_k-u,v]_{\widetilde{W}^{1}_{2}(\Omega)}\right|. \tag{50} \]

Taking into account equality (49), as well as conditions (47), (48), we obtain that the right-hand side of inequality (50) can be made arbitrarily small, whence it follows that

\[ (f,Bv)-[u,v]_{\widetilde{W}^{1}_{2}(\Omega)}=0. \]

But this equality is nothing other than relation (46), which defines a generalized solution. Thus, it has been established that the fulfillment of conditions (47), (48) is sufficient for the element \(u\) to be a generalized solution of problem (45). The lemma is completely proved.

The relations (47), (48) can be regarded, just as the definition of a generalized solution of problem (45), as equivalent to Definition 1.

Let us now consider the question of the existence of a generalized solution of problem (45).

The following theorem holds.

Theorem 2. For any right-hand side \(f \in L_2(\Omega_\gamma^1)\), in the space \(\widetilde W_2^1(\Omega)\) there exists a unique generalized solution of problem (45) on the vibrations of a string.

Moreover, the indicated generalized solution and the solution of the corresponding variational problem coincide.

For the proof, let us verify that the solution \(u_0\) of the variational problem (24), whose existence and uniqueness were established by Theorem 1, will be a generalized solution of problem (45) in the sense of Definition 1. In addition, we shall show that any generalized solution will also be a solution of the variational problem (24). Let \(u_0\) be a solution of the variational problem. Then, by definition, for any \(u \in \widetilde W_2^1(\Omega)\),

\[ D(u_0) \leq D(u). \]

Putting \(u = u_0 + tv\), where \(v\) is an arbitrary element of \(\widetilde W_2^1(\Omega)\), and \(t\) is some parameter, we obtain

\[ D(u_0 + tv) - D(u_0) = \|u_0 + tv\|_{\widetilde W_2^1(\Omega)}^2 \]

\[ -\|u_0\|_{\widetilde W_2^1(\Omega)}^2 - 2(f, Bu_0 + tBv - Bu_0) \geq 0, \]

whence follows the relation

\[ t^2\|v\|_{\widetilde W_2^1(\Omega)}^2 - 2t\bigl[(f,Bv)-[u_0,v]_{\widetilde W_2^1(\Omega)}\bigr] \geq 0. \]

But this is possible only in the case when

\[ (f,Bv)-[u_0,v]_{\widetilde W_2^1(\Omega)}=0. \tag{51} \]

Since the element \(v\) may here be arbitrary in \(\widetilde W_2^1(\Omega)\), it follows from equality (51), taking into account (13), (46), that \(u_0\) is a generalized solution of problem (45). Let us now show that if some element \(u_0' \in \widetilde W_2^1(\Omega)\) is a generalized solution of problem (45), then \(u_0'\) will also be a solution of the variational problem (24).

Indeed, equality (46) for the element \(u_0'\) is valid for any \(v \in \widetilde W_2^1(\Omega)\). Then one may write the equality

\[ (f,Bu) = \int_\Omega \left( \frac{\partial u_0'}{\partial \xi}\,\frac{\partial u}{\partial \xi} + \frac{\partial u_0'}{\partial \eta}\,\frac{\partial u}{\partial \eta} \right) \,d\xi\,d\eta + \int_\gamma u_0' u\,ds = [u_0',u]_{\widetilde W_2^1(\Omega)} \]

and the functional \(D(u)\) (24) can therefore be represented in the form

\[ D(u) = \|u\|_{\widetilde W_2^1(\Omega)}^2 - 2(f,Bu) = \|u\|_{\widetilde W_2^1(\Omega)}^2 - 2[u_0',u]_{\widetilde W_2^1(\Omega)} = \]

\[ = \|u-u_0'\|_{\widetilde W_2^1(\Omega)}^2 - \|u_0'\|_{\widetilde W_2^1(\Omega)}^2. \tag{52} \]

From this it is immediately clear that the minimum of the functional \(D(u)\) is attained at the element \(u_0'\), i.e. \(u_0'\) is indeed a solution of the variational problem. The uniqueness of the generalized solution directly-

follows directly from the preceding arguments. Indeed, we have shown that every generalized solution will also be a solution of the variational problem; but since, by Theorem 1, the solution of the variational problem is unique, the generalized solution of problem (45) will also be unique. To prove uniqueness of the generalized solution one can also make direct use of definition (46). Let \(u_0\) and \(u'_0\) be two generalized solutions of problem (45). Then, obviously, for them the relations (46), (13) must hold,

\[ (f,Bv)=[u_0,v]_{\widetilde W_2^1(\Omega)},\qquad (f,Bv)=[u'_0,v]_{\widetilde W_2^1(\Omega)}, \]

whence, by subtraction, we obtain the equality

\[ [u_0-u'_0,v]_{\widetilde W_2^1(\Omega)}=0, \]

valid for all \(v\in \widetilde W_2^1(\Omega)\). From this equality it follows that

\[ u_0=u'_0 . \]

Thus, we have established that for any right-hand side \(f\in L_2(\Omega_\gamma)\) the generalized solution of problem (45) exists and is unique and, moreover, coincides in the space \(\widetilde W_2^1(\Omega)\) with the solution of the variational problem. The theorem is thereby proved.

It follows from Theorem 2 that

Corollary. The construction of the generalized solution of problem (45) and the solution of the variational problem (24) are equivalent problems in the sense that every solution of the variational problem is a generalized solution of problem (45) and, conversely, every generalized solution of problem (45) is also a solution of the variational problem (24).

§ 4. Differential properties of the generalized solution

We now turn to the consideration of the differential properties of the generalized solution of problem (45) and investigate the question of the satisfaction of the differential equation and the boundary conditions. Recall that all the arguments carried out above, and in particular Lemma 1, referred to the case when the boundary \(\Gamma\) of the domain \(\Omega\) (Fig. 1) may contain portions coinciding with the characteristics \(\xi\) or \(\eta\). In the general case the domain \(\Omega\), consequently, has the form shown in Fig. 2, where the following notation is adopted: \(\Gamma_2,\Gamma_3\) are the points where the domain \(\Omega\) rests on the characteristics \(\xi,\eta\); \(l_i^j\) \((i,j=1,2)\) and \(\gamma',\gamma''\) are parts of the contour \(\Gamma\) intersected by each of the characteristics \(\xi,\eta\) in no more than one point; \(\xi_i,\eta_i\) \((i=0,1,2)\) are parts of the contour \(\Gamma\) coinciding respectively with the characteristics \(\xi,\eta\).

In what follows we shall also use the notation

\[ \gamma_\eta=\gamma''+\gamma',\qquad \gamma=\gamma''+\xi_0+\eta_0+\gamma', \]

\[ l_1=l_1^2+l_1^1,\qquad l_2=l_2^1+l_2^2, \tag{53} \]

and, as before (see Figs. 1–2), we assume that

\[ \gamma_1=l_2^1+\eta_1+l_2^2+\eta_2,\qquad \gamma_2=\xi_2+l_2^2+\xi_1+l_1^1 . \tag{54} \]

The formulation of boundary-value problems for a domain of the type indicated in Fig. 2 will differ, generally speaking, from the formulation of analogous problems for the domain \(\Omega\) (Fig. 1). The formulation of the named problems differs because on the segments \(\xi_i,\eta_i\) (see Fig. 2) the quantities \(\sin\theta,d\eta\) and \(\cos\theta,d\xi\), respectively, vanish. Therefore, as is seen from relations (10)—(12), on the segments \(\xi_i,\eta_i\ (i=0,1,2)\) the corresponding boundary conditions need not be prescribed.

The boundary-value problem for the equation of vibrations of a string (45) in the case of a domain of general form (Fig. 2) corresponds to the situation where the length of the string is, during the vibrations, a variable quantity (i.e. the points \(\Gamma_2\) and \(\Gamma_3\), depending on the time \(t\), move along the curves \(l_1,l_2\) (see Fig. 2)), while the “initial deflection” and the “initial velocity” of the string are prescribed on the segment \(\Gamma_2\Gamma_3\) not at the zero instant of time, but, for different points of the string, at different times (Fig. 2).

Here, at any instant of time \(t\), the ends of the string may be “released” and, after some time, again “fixed” at a definite place.

Further, on the segments \(\xi_0,\eta_0\) (Fig. 2) it is necessary to prescribe only the values of the “initial deflection” of the string, and it is not required also to prescribe the “initial velocity” of the string.

Thus, in the case of the general domain indicated in Fig. 2, for the boundary-value problem of string vibrations the values of the required function are prescribed on the segment \(\gamma''+\xi_0+\eta_0+\gamma'\), the values of the normal derivative on the segment \(\gamma'+\gamma''\), and the values of \(\dfrac{\partial u}{\partial\eta}\), \(\dfrac{\partial u}{\partial\xi}\), respectively, on the segments \(l_1^1+l_1^2\) and \(l_2^2+l_2^1\). On the segments \(\xi_i\eta_i\ (i=1,2)\) no boundary conditions are prescribed. Consequently, the boundary-value problem for the equation of string vibrations in the case of a domain of general form (Fig. 2) is written as follows:

\[ -\frac{\partial^2 u}{\partial \xi \partial \eta}=g,\quad u|_\gamma=\varphi,\quad \frac{\partial}{\partial s}(u|_\gamma)=\frac{\partial\varphi}{\partial s},\quad \left.\frac{\partial u}{\partial n}\right|_{\gamma'+\gamma''}=\psi \]

\[ \left.\frac{\partial u}{\partial\eta}\right|_{l_1^1+l_1^2}=\chi,\quad \left.\frac{\partial u}{\partial\xi}\right|_{l_2^2+l_2^1}=\omega . \tag{55} \]

It is obvious that the boundary-value problem (45) for the domain \(\Omega\) indicated in Fig. 1 will be a particular case of the general boundary-value problem (55) for the domain indicated in Fig. 2.

The right-hand side of equation (55)

\[ f=\left\{g,\varphi,\frac{\partial\varphi}{\partial s},\psi,\chi,\omega\right\} \]

still belongs to the space \(L_2(\Omega_l^1)\). We note that in the definition of the norm (15) in the space \(L_2(\Omega_l^1)\), in the case of a domain of general form (Fig. 2), the integral of \(\psi_1\psi_2\) and of \(\psi^2\) is taken not over the whole segment \(\gamma\), but only over \(\gamma'+\gamma''\), since in all other relations (see (10)—(12)) the integrals of the term with the quantity \(\dfrac{\partial u}{\partial n}\) over the segment \(\xi_0+\eta_0\) vanish.

The generalized solution of problem (55), analogously to (46), is defined by the relation

\[ (f,Bv)=\int_{\Omega} gKvd\xi d\eta+\int_{\gamma}\varphi vds+ \int_{\gamma}\frac{\partial\varphi}{\partial s}Tvds+\int_{\gamma_\eta}\psi K_\theta vds+ \]

\[ + \int_{l_1} \chi L_1 v\, d\eta + \int_{l_2} \omega L_2 v\, d\xi = \int_{\Omega} \left(-\frac{\partial u_0}{\partial \xi}\frac{\partial v}{\partial \xi} + \frac{\partial u_0}{\partial \eta}\frac{\partial v}{\partial \eta}\right)\, d\xi d\eta + \int_{\gamma} u_0 v\, ds, \tag{56} \]

valid for every \(v \in \widetilde W_2^1(\Omega)\).

The existence and uniqueness of the generalized solution (56) for the boundary-value problem (55) follows from Theorem 2 proved earlier, since the domain \(\Omega\) for which Theorem 2 was proved could have portions of the boundary coinciding with the characteristics \(\xi,\eta\), and consequently could have the form of the domain indicated in Fig. 2. Therefore, considering relation (56) established, we shall show that the generalized solution \(u_0\) of the boundary-value problem (55) will also be a solution of problem (55) in a certain strong sense, namely a solution almost everywhere. For this purpose we introduce the following

Definition 2. The element \(u_0\) will be called a strong solution almost everywhere of the boundary-value problem (55), if \(u_0 \in \widetilde W_2^1(\Omega)\) has, in addition, a mixed generalized derivative with respect to \(\xi,\eta\), and the equalities

\[ \frac{\partial^2 u_0}{\partial \xi \partial \eta} - g = 0, \qquad u_0|_{\gamma} - \varphi = 0 \tag{57} \]

hold almost everywhere respectively on \(\Omega\) and \(\gamma\), while the remaining boundary conditions in (55) are satisfied in the weak sense, namely, for every smooth sequence of elements \(\{u_k\}\) satisfying the condition

\[ \|u_k-u_0\|_{\widetilde W_2^1(\Omega)} \to 0, \qquad k \to \infty, \tag{58} \]

the relations

\[ \int_{\gamma_n}\left(\frac{\partial u_k}{\partial n}-\psi\right)\sin\theta\cdot \cos\theta\cdot u\, ds \underset{k\to\infty}{\longrightarrow}0, \qquad \int_{l_1}\left(\frac{\partial u_k}{\partial \eta}-\chi\right)v\, d\eta \underset{k\to\infty}{\longrightarrow}0, \]

\[ \int_{l_2}\left(\frac{\partial u_k}{\partial \xi}-\omega\right)w\, d\xi \underset{k\to\infty}{\longrightarrow}0 \tag{59} \]

hold for any smooth functions \(u,v,w\) defined on \(\Omega\).

Let us also introduce the space \(W_2^{(1,1)}(\Omega)\) with norm

\[ \|u\|_{W_2^{(1,1)}(\Omega)}^2 = \int_{\Omega} \left[ u^2 + \left(\frac{\partial u}{\partial \xi}\right)^2 + \left(\frac{\partial u}{\partial \eta}\right)^2 + \left(\frac{\partial^2 u}{\partial \xi \partial \eta}\right)^2 \right]\, d\xi d\eta, \tag{60} \]

which is a space of S. L. Sobolev.

Having introduced these notions, one can now consider the question of existence and uniqueness of a strong solution satisfying almost everywhere the equation of vibrations of a string. We shall first show that the following holds.

Lemma 3. For smooth functions \(\mathring g\) and \(\mathring\varphi\), given respectively on \(\Omega\) and \(\gamma\), there exists a unique element \(v \in \widetilde W_2^1(\Omega)\) such that, for any \(u \in \widetilde W_2^1(\Omega)\), the relation is satisfied

\[ \int_\Omega u\,\frac{\partial^2 \mathring g}{\partial \xi\,\partial \eta}\,d\xi\,d\eta + \int_\gamma u\,\mathring\varphi\,ds = [u,\mathring v]_{\widetilde W_2^1(\Omega)} \tag{61} \]

for almost all points in \(\Omega\) the equality

\[ K\mathring v=\mathring g . \tag{62} \]

Let concrete functions \(\mathring g\) and \(\mathring\varphi\), defined respectively on \(\Omega\) and \(\gamma\), be given. It is not hard to see that, for the functional standing on the left-hand side of (61), the following estimate is valid:

\[ \left| \int_\Omega u\,\frac{\partial^2 \mathring g}{\partial \xi\,\partial \eta}\,d\xi\,d\eta + \int_\gamma u\,\mathring\varphi\,ds \right| \le C\|u\|_{W_2^1(\Omega)} \le C_1\|u\|_{\widetilde W_2^1(\Omega)}, \]

since the norms in the spaces \(W_2^1(\Omega)\) and \(\widetilde W_2^1(\Omega)\) are equivalent. The indicated functional is therefore a bounded linear functional in the space \(\widetilde W_2^1(\Omega)\), and, according to the Riesz theorem, is representable in the space \(\widetilde W_2^1(\Omega)\) uniquely in the form of a scalar product; i.e., relation (61) indeed holds for every \(u\in\widetilde W_2^1(\Omega)\).

To prove equality (62), we transform relation (61) somewhat. For the right-hand side of (61) we shall have

\[ \int_\Omega \left( \frac{\partial u}{\partial \xi}\frac{\partial \mathring v}{\partial \xi} + \frac{\partial u}{\partial \eta}\frac{\partial \mathring v}{\partial \eta} \right)d\xi\,d\eta + \int_\gamma u\mathring v\,ds = \]

\[ = \int_\Gamma \frac{\partial u}{\partial \xi} \left[ \int_\eta^{\gamma_2} \frac{\partial \mathring v}{\partial \xi}(\xi,\tau)\,d\tau \right]d\xi - \int_\Gamma \frac{\partial u}{\partial \eta} \left[ \int_\xi^{\gamma_1} \frac{\partial \mathring v}{\partial \eta}(\zeta,\eta)\,d\zeta \right]d\eta + \]

\[ + \int_\Omega \frac{\partial^2 u}{\partial \xi\,\partial \eta} \left[ \int_\eta^{\gamma_2} \frac{\partial \mathring v}{\partial \xi}(\xi,\tau)\,d\tau + \int_\xi^{\gamma_1} \frac{\partial \mathring v}{\partial \eta}(\zeta,\eta)\,d\zeta \right]d\xi\,d\eta + \int_\gamma u\mathring v\,ds \]

and correspondingly for the left-hand side

\[ \int_\Omega u\,\frac{\partial^2\mathring g}{\partial \xi\,\partial \eta}\,d\xi\,d\eta + \int_\gamma u\mathring\varphi\,ds = \]

\[ = \int_\Omega \frac{\partial^2u}{\partial \xi\,\partial \eta}\,\mathring g\,d\xi\,d\eta + \int_\gamma u\mathring\varphi\,ds + \int_\Gamma u\,\frac{\partial\mathring g}{\partial \xi}\,d\xi + \int_\Gamma \mathring g\,\frac{\partial u}{\partial \eta}\,d\eta, \]

whence we obtain the following equality:

\[ \int_\Omega \frac{\partial^2u}{\partial \xi\,\partial \eta}\,\mathring g\,d\xi\,d\eta + \int_\gamma u\mathring\varphi\,ds + \int_\Gamma u\,\frac{\partial\mathring g}{\partial \xi}\,d\xi + \int_\Gamma \mathring g\,\frac{\partial u}{\partial \eta}\,d\eta = \]

\[ = \int_{\Gamma} \frac{\partial u}{\partial \xi} \left[\int_{\eta}^{\gamma_2} \frac{\partial \dot v}{\partial \xi}(\xi,\tau)\,d\tau \right] d\xi - \int_{\Gamma} \frac{\partial u}{\partial \eta} \left[\int_{\xi}^{\gamma_1} \frac{\partial \dot v}{\partial \eta}(\zeta,\eta)\,d\zeta \right] d\eta + \]
\[ + \int_{\Omega} \frac{\partial^2 u}{\partial \xi \partial \eta} K \dot v\,d\xi d\eta + \int_{\gamma} u \dot v\,ds, \tag{63} \]

valid for any smooth function \(u\).

We choose the function \(u\) in what follows in the form

\[ u=\int_{\gamma_1}^{\xi}\int_{\gamma_2}^{\eta} W(\zeta,\tau)\,d\zeta d\tau + C(\xi)+D(\eta), \tag{63'} \]

where \(W(\zeta,\tau)\) is some smooth function (generally speaking, arbitrary), and \(C(\xi)\), \(D(\eta)\) are chosen so that, for fixed \(W(\zeta,\tau)\), the contour integrals in equality (63) vanish identically. Then, substituting the element (63′) into equality (63), we obtain the condition

\[ \int_{\Omega} (K\dot v-\dot g) w\,d\xi d\eta = 0, \]

whence, by virtue of the arbitrariness of the choice of the element \(W(\zeta,\tau)\), equality (62) follows.

Lemma 3 is thereby proved. We now prove that the following holds.

Theorem 3. For any right-hand side \(f \in L_2(\Omega_\gamma^1)\) and for a domain \(\Omega\) of the form indicated in Fig. 2, with piecewise differentiable boundary \(\Gamma\), there exists a unique strong solution \(u_0\) of the boundary-value problem (55), which is also a solution of the variational problem (24). Moreover, the solution \(u_0\) has generalized derivatives \(\partial u_0/\partial \xi\), \(\partial u_0/\partial \eta\), \(\partial^2 u_0/\partial \xi \partial \eta\); belongs to the space \(W_2^{(1,1)}(\Omega)\), and satisfies equation (55) almost everywhere.

In addition, the boundary-value problem (55) is a correctly posed problem in the sense that a small change of the right-hand side in the space \(L_2(\Omega_\gamma^1)\) corresponds to a small change of the solution \(u_0\) in the space \(W_2^{(1,1)}(\Omega)\).

To prove the theorem, we show that the element \(u_0\), which is a solution of the variational problem (24) and a generalized solution of problem (55) in the sense of definition (56), will also be a strong solution and satisfies equation (55) almost everywhere.

First let us show that almost everywhere on \(\gamma\) the equality

\[ u_0\big|_{\gamma} - \varphi = 0. \tag{64} \]

holds. For this purpose consider the functional

\[ l(v) = (f,Bv) - \int_{\gamma} \varphi v\,ds + \int_{\gamma} u_0 v\,ds, \tag{65} \]

which, just like the functional \((f,Bv)\) (see Lemma 1), will be linear in the space \(\widetilde W_2^1(\Omega)\). Therefore, on the basis of the Riesz theorem, the functional (65) can be represented in the form of a scalar product, i.e.

\[ (f,Bv) + \int_{\gamma} (u_0-\varphi)v\,ds = [u_1,v]_{\widetilde W_2^1(\Omega)}, \tag{66} \]

where \(u_1\) is some fixed element of \(\widetilde W_2^1(\Omega)\).

Along with this, consider equality (56), which we write in the form

\[ (f, Bv)=[u_0,v]_{\widetilde W_2^1(\Omega)} . \tag{67} \]

Subtracting equalities (66) and (67), we obtain the following relation:

\[ \int_\gamma (u_0-\varphi)v\,ds=[u_1-u_0,v]_{\widetilde W_2^1(\Omega)}, \tag{68} \]

which is valid for any \(v\in \widetilde W_2^1(\Omega)\).

Expanding this relation and integrating by parts, we obtain

\[ \begin{aligned} \int_\gamma (u_0-\varphi)v\,ds &-\int_\Omega\left(\frac{\partial u_1}{\partial\eta}-\frac{\partial u_0}{\partial\eta}\right) \frac{\partial v}{\partial\eta}\,d\xi d\eta -\int_\gamma (u_1-u_0)v\,ds \\ &+\int_\Gamma (u_1-u_0)\frac{\partial v}{\partial\xi}\,d\eta =\int_\Omega (u_1-u_0)\frac{\partial^2 v}{\partial\xi^2}\,d\xi d\eta, \end{aligned} \tag{69} \]

where \(v\) is a smooth function.

But since equality (69) holds for any smooth function \(v\), let us choose \(v\) in the following form:

\[ v=\int_{\xi_1}^{\xi}\int_{\eta_1}^{\rho} w(\zeta,\eta)\,d\zeta d\rho+\xi C(\eta)+D(\eta), \]

where \(w\) is a prescribed function, and \(C(\eta), D(\eta)\) are chosen so that, for fixed \(w\), the left-hand side of equality (69) vanishes identically. Substituting the indicated element \(v\) into equality (69), we obtain

\[ \int_\Omega (u_1-u_0)w\,d\xi d\eta=0, \]

whence, by the arbitrariness in the choice of the element \(w\), we conclude that almost everywhere in \(\Omega\) the equality

\[ u_1-u_0=0 \]

holds. But then, on the basis of equality (69), it follows that for any \(v\in \widetilde W_2^1(\Omega)\) the relation

\[ \int_\gamma (u_0-\varphi)v\,ds=0 \]

holds.

From this relation there follows, in fact, assertion (65), since for any smooth function prescribed on the segment \(\gamma\) one can, in our case, construct an extension into the domain \(\Omega\) such that this extension belongs to the space \(\widetilde W_2^1(\Omega)\) (see, for example, (25)). Thus, the validity of assertion (64) has been established.

We now show that the element \(u_0\), which is the solution of the variational problem (24) and the generalized solution of the boundary-value problem (55), has a square-summable mixed generalized derivative

\[ \frac{\partial^2 u_0}{\partial\xi\,\partial\eta}. \]

Let there be given some smooth finite function \(\overset{\circ}{g}\) in \(\Omega\) and a function \(\overset{\circ}{\varphi}=0\) defined on \(\gamma\), for which, according to Lemma 13, there exists an element such that the equality holds

\[ K\overset{\circ}{v}= \int_{\xi}^{\gamma_1}\frac{\partial \overset{\circ}{v}}{\partial \eta}(\zeta,\eta)\,d\zeta+ \int_{\eta}^{\gamma_2}\frac{\partial \overset{\circ}{v}}{\partial \xi}(\xi,\tau)\,d\tau =\overset{\circ}{g}. \tag{70} \]

From this equality, by virtue of the finiteness of the function \(\overset{\circ}{g}\), the relations

\[ K_{\theta}\overset{\circ}{v}\big|_{\gamma_n}=0,\qquad L_1\overset{\circ}{v}\big|_{l_1}=0,\qquad L_2\overset{\circ}{v}\big|_{l_2}=0, \tag{71} \]

follow, where the notation (11) has been used. Taking into account equalities (70), (71), relation (56) for \(v=\overset{\circ}{v}\) takes the following form:

\[ \int_{\Omega}\overset{\circ}{g}\,\overset{\circ}{g}\,d\xi\,d\eta+ \int_{\gamma}\varphi\,\overset{\circ}{v}\,ds+ \int_{\gamma}\frac{\partial\varphi}{\partial s}\,T\overset{\circ}{v}\,ds = [u_0,v]_{\overset{\circ}{W}{}^{1}_{2}(\Omega)}, \tag{72} \]

where the elements \(\overset{\circ}{g}\) and \(\overset{\circ}{v}\) are connected by equality (70). Consider, in addition, relation (10), which will also be valid for \(v=\overset{\circ}{v}\). Taking into account equalities (70), (71) and the notation (11), (13), relation (10) can be rewritten in the following form:

\[ \int_{\Omega}\frac{\partial^2 u}{\partial \xi\,\partial \eta}\, \overset{\circ}{g}\,d\xi\,d\eta+ \int_{\gamma}u\,\overset{\circ}{v}\,ds+ \int_{\gamma}\frac{\partial u}{\partial s}\,T\overset{\circ}{v}\,ds = [u,\overset{\circ}{v}]_{\overset{\circ}{W}{}^{1}_{2}(\Omega)}, \tag{73} \]

where the function \(u\) is smooth, while \(\overset{\circ}{g}\), \(\overset{\circ}{v}\) are still connected by equality (70). Performing integration by parts of the first term in (73), taking into account the finiteness of \(\overset{\circ}{g}\), we obtain the relation

\[ \int_{\Omega}u\,\frac{\partial^2\overset{\circ}{g}}{\partial \xi\,\partial \eta}\,d\xi\,d\eta+ \int_{\gamma}u\,\overset{\circ}{v}\,ds+ \int_{\gamma}\frac{\partial u}{\partial s}\,T\overset{\circ}{v}\,ds = [u,\overset{\circ}{v}]_{\overset{\circ}{W}{}^{1}_{2}(\Omega)}, \tag{74} \]

valid for smooth functions \(u\). But, on the other hand, by Lemma 3 equality (61) has been established, valid for any element \(u\in \overset{\circ}{W}{}^{1}_{2}(\Omega)\). Comparing equalities (74) and (61), for smooth functions \(u\) we obtain the following condition:

\[ \int_{\gamma}u\,\overset{\circ}{v}\,ds+ \int_{\gamma}\frac{\partial u}{\partial s}\,T\overset{\circ}{v}\,ds = \int_{\gamma}u\,\varphi\,ds \]

or, rewriting,

\[ \int_{\gamma}u(\overset{\circ}{v}-\varphi)\,ds = -\int_{\gamma}\frac{\partial u}{\partial s}\,T\overset{\circ}{v}\,ds. \tag{75} \]

We shall show that from this condition there follows the existence of a generalized derivative with respect to \(s\) of the element \(T\overset{\circ}{v}\big|_{\gamma}\). Indeed, the element \(\overset{\circ}{v}\) is chosen so that for smooth finite \(\overset{\circ}{g}\) equality (70) is satisfied. But since the right-hand side of equality (70) certainly belongs to the space \(\overset{\circ}{W}{}^{1}_{2}(\Omega)\), each of the terms on the left-hand side of (70) has a square-summable generalized derivative respectively with respect to \(\xi\) and with respect to \(\eta\); consequently, each of the terms on the left-hand side also belongs to the space \(\overset{\circ}{W}{}^{1}_{2}(\Omega)\). And then, using the expression for \(Tv\) (11), we conclude that the element \(T\overset{\circ}{v}\in\overset{\circ}{W}{}^{1}_{2}(\Omega)\), and therefore

one may speak of the trace of the element \(T\overset{\circ}{v}\) on \(\gamma\). By virtue of relation (75), we obtain that the indicated trace has, in addition, a generalized derivative with respect to \(s\), and in this case the equality

\[ \frac{\partial}{\partial s}\left(T\overset{\circ}{v}\bigm|_{\gamma}\right) = \overset{\circ}{v}\bigm|_{\gamma}-\varphi , \tag{76} \]

is satisfied, whence we conclude that the element \(T\overset{\circ}{v}\bigm|_{\gamma}\) is an absolutely continuous function of \(s\), and therefore we may speak of the value of \(T\overset{\circ}{v}\bigm|_{\gamma}\) at any point on \(\gamma\). From relations (70) and (11), in view of the finiteness of \(\overset{\circ}{g}\) and the vanishing at \(0\) on \(\Gamma_2,\Gamma_3\) of the corresponding integrals, it follows that the element \(T\overset{\circ}{v}\bigm|_{\gamma}\) vanishes at the endpoints of the interval \(\gamma\).

Taking into account condition (70) and the equality to zero of the element \(T\overset{\circ}{v}\bigm|_{\gamma}\) at the points \(\Gamma_2,\Gamma_3\), the contour integrals in relation (72) are transformed as follows:

\[ \int_{\gamma}\varphi\,\overset{\circ}{v}\,ds + \int_{\gamma}\frac{\partial\varphi}{\partial s}\,T\overset{\circ}{v}\,ds = \int_{\gamma}\varphi\,\overset{\circ}{v}\,ds - \int_{\gamma}\varphi\,\frac{\partial}{\partial s}T\overset{\circ}{v}\,ds = \int_{\gamma}\varphi\varphi\,ds . \]

Then relation (72) can be written in the form

\[ \int_{\Omega}\overset{\circ}{g}\,g\,d\xi\,d\eta + \int_{\gamma}\varphi\varphi\,ds = [u_0,\overset{\circ}{v}]_{\widetilde{W}_2^1(\Omega)}, \]

and, moreover, we write equality (61) in the case \(u=u_0\) (since equality (61) is valid for any \(u\in \widetilde{W}_2^1(\Omega)\)):

\[ \int_{\Omega}u_0\,\frac{\partial^2\overset{\circ}{g}}{\partial \xi\,\partial \eta}\,d\xi d\eta + \int_{\gamma}u_0\,\overset{\circ}{\varphi}\,ds = [u_0,\overset{\circ}{v}]_{\widetilde{W}_2^1(\Omega)} . \]

From the written relations, by virtue of condition (64), there follows the final equality

\[ \int_{\Omega}\overset{\circ}{g}\,g\,d\xi d\eta = \int_{\Omega}u_0\,\frac{\partial^2\overset{\circ}{g}}{\partial \xi\,\partial \eta}\,d\xi d\eta . \]

But since as \(\overset{\circ}{g}\) we could take any smooth finite function in \(\Omega\), it follows from this (see, for example, (1)) that the element \(u_0\) has a generalized mixed derivative with respect to \(\xi,\eta\), and, moreover, almost everywhere in \(\Omega\) the equality

\[ \frac{\partial^2 u_0}{\partial \xi\,\partial \eta}-g=0 \tag{77} \]

is satisfied.

By virtue of the square summability of the function \(g\) (see (15)), from condition (77) there follows the square summability of the mixed generalized derivative with respect to \(\xi,\eta\) of the function \(u_0\).

Thus, we have established that the element \(u_0\) indeed has the generalized mixed derivative \(\dfrac{\partial^2 u_0}{\partial \xi\,\partial \eta}\) and belongs to the space \(W_2^{(1,1)}(\Omega)\).

Let us now consider the question of satisfying the boundary conditions in problem (55).

It was shown earlier (see (64)) that the sought solution \(u_0\) of problem (55) coincides almost everywhere on the segment \(\gamma\) with the prescribed values \(\varphi\).

We shall show that the remaining boundary conditions of problem (55) are satisfied in the weak sense (58)—(59). To this end choose a sequence of smooth functions \(\{u_k\}\) such that the condition

\[ \left\|u_k-u_0\right\|_{\widetilde W_2^{(1,1)}(\Omega)} \underset{k\to\infty}{\longrightarrow}0 \tag{78} \]

is fulfilled, and, all the more,

\[ \left\|u_k-u_0\right\|_{\widetilde W_2^1(\Omega)} \underset{k\to\infty}{\longrightarrow}0. \tag{79} \]

It is not difficult to see (cf. Lemma 2) that then for any \(v\in \widetilde W_2^1(\Omega)\) the relation

\[ (Au_k-f,Bv)=[u_k-u_0,v]_{\widetilde W_2^1(\Omega)} \underset{k\to\infty}{\longrightarrow}0, \tag{80} \]

holds, where \((Au_k,Bv)\) is defined by equalities (17), (10). Using the notation (11), (56), and also equalities (64) and (77), we write relation (80) in the following form:

\[ \begin{aligned} &\int\limits_{\Omega}\left( \frac{\partial^2 u_k}{\partial \xi\,\partial \eta} - \frac{\partial^2 u_0}{\partial \xi\,\partial \eta} \right)Kv\,d\xi\,d\eta + \int\limits_{\gamma}(u_k-u_0)v\,ds \\ &\quad + \int\limits_{\gamma}\left( \frac{\partial u_k}{\partial s} - \frac{\partial u_0}{\partial s} \right)Tv\,ds + \int\limits_{\gamma_n}\left( \frac{\partial u_k}{\partial n}-\psi \right)K_\theta v\,ds \\ &\quad + \int\limits_{l_1}\left( \frac{\partial u_k}{\partial \eta}-\chi \right)L_1v\,d\eta + \int\limits_{l_2}\left( \frac{\partial u_k}{\partial \xi}-\omega \right)L_2v\,d\xi = [u_k-u_0,v]_{\widetilde W_2^1(\Omega)} \underset{k\to\infty}{\longrightarrow}0, \end{aligned} \]

whence, on the basis of conditions (78), (79), it follows that for any \(v\in \widetilde W_2^1(\Omega)\) the relation

\[ \int\limits_{\gamma}\left( \frac{\partial u_k}{\partial n}-\psi \right)K_\theta v\,ds + \int\limits_{l_1}\left( \frac{\partial u_k}{\partial \eta}-\chi \right)L_1v\,d\eta + \int\limits_{l_2}\left( \frac{\partial u_k}{\partial \xi}-\omega \right)L_2v\,d\xi \underset{k\to\infty}{\longrightarrow}0 \tag{81} \]

is also valid.

Assertions (58), (59) follow directly from condition (81), if one uses Lemma 3.

Indeed, according to Lemma 3, for any smooth element \(\overset{\circ}{g}\), defined on \(\Omega\), one can indicate an element \(\overset{\circ}{v}\in \overset{\circ}{\widetilde W}_2^1(\Omega)\) such that the equality

\[ \overset{\circ}{K}\overset{\circ}{v}=\overset{\circ}{g} \]

holds.

But, on the other hand, on the basis of relations (11) we have

\[ K_\theta\overset{\circ}{v}\big|_{\gamma} = \sin\theta\cdot\cos\theta\, \overset{\circ}{K}\overset{\circ}{v}\big|_{\gamma}, \qquad L_1\overset{\circ}{v}\big|_{l_1} = \overset{\circ}{K}\overset{\circ}{v}\big|_{l_1}, \tag{82} \]

\[ -\,L_2\overset{\circ}{v}\big|_{l_2} = \overset{\circ}{K}\overset{\circ}{v}\big|_{l_2}. \]

Suppose now that smooth elements \(u, v, w\) are given on \(\Omega\). We modify them so that \(u\) vanishes on the boundary portions \(l_1, l_2\), while the elements \(v, w\) vanish respectively on \(\gamma_n, l_2\) and \(\gamma_n, l_1\). Now setting \(g\) successively equal to \(u, v, w\), and taking into account the equalities (82), from condition (81) we obtain

\[ \int_{\gamma_n}\left(\frac{\partial u_k}{\partial n}-\psi\right)\sin\theta\cdot\cos\theta\,u\,ds \underset{k\to\infty}{\longrightarrow}0, \]

\[ \int_{l_1}\left(\frac{\partial u_k}{\partial \eta}-\chi\right)v\,d\eta \underset{k\to\infty}{\longrightarrow}0,\qquad \int_{l_2}\left(\frac{\partial u_k}{\partial \xi}-\omega\right)w\,d\xi \underset{k\to\infty}{\longrightarrow}0, \tag{83} \]

where \(u, v, w\) are the elements modified in \(\Omega\) in the indicated way.

In the relations (83), only the values of the elements \(u, v, w\) on the boundary portions \(\gamma_n, l_1, l_2\), respectively, are essential, and it is immaterial what these elements are equal to on the remaining part of the domain \(\Omega\). Therefore we may modify \(u, v, w\) back to their original values; hence we obtain that the relations (83) hold for any smooth elements \(u, v, w\).

We note that, in the case of a smooth boundary of the domain \(\Gamma\), the relations (83) simply denote the weak convergence of the corresponding sequences \(\{u_k\}\).

Thus, having proved the validity of the relations (77), (64), and (83), we have thereby established that, for the boundary-value problem (55), there exists a unique strong solution
\(u_0\in W_2^{(1,1)}(\Omega)\), satisfying almost everywhere the original equation (55).

Let us now show that problem (55) for the vibrations of a string is a well-posed problem.

For this we use the result of Lemma 1 and relation (56) with \(v=u_0\). Then, on the basis of (26) and (56), we may write

\[ \|u_0\|_{W_2^1(\Omega)}^2 =(f,Bu_0)\leq c\|f\|_{L_2(\Omega_\gamma^l)}\cdot \|u_0\|_{\widetilde W_2^1(\Omega)}, \]

whence the inequality follows:

\[ \|u_0\|_{\widetilde W_2^1(\Omega)} \leq c\|f\|_{L_2(\Omega_\gamma^l)} . \tag{84} \]

Further, by virtue of equality (77) and the square-summability of the function \(g\) (see (15)), from (84) we obtain the final estimate

\[ \|u_0\|_{W_2^{(1,1)}(\Omega)} \leq c\|f\|_{L_2(\Omega_\gamma^l)}, \tag{85} \]

where the norm in the space \(W_2^{(1,1)}(\Omega)\) is defined by relation (60), and the norm in the space \(L_2(\Omega_\gamma^l)\) by relation (15). From the linearity of the boundary-value problem (55) and estimate (85), the well-posedness of the problem follows immediately. Theorem 3 is thereby proved.

In conclusion, let us note that from the general boundary-value problem (55) there follow the previously known particular problems for the equation of vibrations of a string, namely: the Cauchy problem (the case when \(\xi_0=\eta_0=0;\ l_i^j=0;\ i,j=1,2\), see Fig. 2), the Goursat problem (all portions of the boundary \(\Gamma\) are equal to zero, except for \(l_1,\xi_0,\eta_2\), which are different from zero and form a closed domain in the form of a right triangle with a curved diago-

that is, the Darboux problem (if all portions of the boundary \(\Gamma\) are equal to zero, except for \(\xi_0, \eta_0, \xi_2, \eta_2\), which are nonzero and form a rectangular region).

Let us also note a new problem for the hyperbolic equation (1), which follows from problem (55) and from the form of the domain \(\Omega\) (Fig. 2). This problem corresponds to the case when the boundary portions \(\xi_1=\xi_2=\eta_1=\eta_2=0\), and the domain \(\Omega\) is situated strictly inside the angle formed by the characteristics \(\xi,\eta\) and “suspended” from above over the domain \(\Omega\) (Fig. 2), i.e. \(\Omega\) is situated inside the characteristics \(\xi,\eta\), touching the domain \(\Omega\) at only one point, namely at the point \(\Gamma_0\), where \(\xi,\eta\) intersect.

Fig. 2

For such a problem, boundary conditions \(u,\dfrac{\partial u}{\partial n},\dfrac{\partial u}{\partial \xi},\dfrac{\partial u}{\partial \eta}\) are prescribed along the entire boundary of the domain. According to Theorem 3, such a problem will be well posed.

From other points of view, the mixed problem for hyperbolic equations was studied in the works of O. A. Ladyzhenskaya [10], V. A. Il’in [5], and other authors. For the case of two variables, estimate (85) is sharper than the analogous estimate of O. A. Ladyzhenskaya (see [10], p. 77). The first general functional approach to the study of hyperbolic equations was given by S. L. Sobolev [1]. The well-posed problem for a hyperbolic equation with data on the entire boundary was also first considered by S. L. Sobolev [19].

In conclusion, the author takes the opportunity to sincerely thank Academician S. L. Sobolev for drawing attention to this problem, to thank his scientific adviser Professor L. D. Kudryavtsev for his constant attention and assistance in the work, and also Candidate of Physical and Mathematical Sciences G. N. Yakovlev, who made a number of valuable comments to the author.

References

1. Sobolev S. L. Some applications of functional analysis in mathematical physics. LSU, 1950.
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Received by the editors
April 29, 1965

Central Aerohydrodynamic Institute
named after N. E. Zhukovsky