V. S. ROGOZHIN

A GENERAL SCHEME FOR SOLVING BOUNDARY-VALUE PROBLEMS IN THE SPACE OF GENERALIZED FUNCTIONS

(Presented by Academician V. I. Smirnov on 6 II 1965)

The Riemann boundary-value problem in the space of generalized functions has been studied by many authors. The formulation of the problem apparently belongs to O. S. Parasyuk (\(^{1}\)), who considered the problem in the case when the contour is the real axis. Earlier, Köthe (\(^{2}\)) considered a special case of the Riemann problem—the problem of decomposing a generalized function into the difference of boundary values of functions analytic inside and outside an analytic curve (for bibliography, see (\(^{3}\))). In the present note a general scheme is set forth for solving boundary-value problems in the space of generalized functions, applicable both to the Riemann problem and to the problem of linear conjugation with shift, the Carleman problem (\(^{4}\)) and its various generalizations (\(^{5}\)), as well as to the Hilbert problem ((\(^{6}\), p. 236)).

1. Let \(L\) be a closed simple curve dividing the plane into two domains \(D^+\) (the interior of \(L\)) and \(D^-\) (the exterior of \(L\)). We shall assume that the function
   \(z=\Psi(w)\) \((\Psi(\infty)=\infty,\ \Psi'(\infty)>0)\), mapping the exterior of the unit circle \(|w|>1\) onto the exterior of \(L\), is continuous in the closed domain \(|w|\geq 1\) together with derivatives of arbitrary order, and moreover \(\Psi'(e^{i\theta})\ne 0\). Consider on \(L\) the basic space \(S\), consisting of infinitely differentiable functions \(\varphi(t)\) of the points \(t=x+iy\) of the contour \(L\). We introduce convergence in \(S\) as follows: the sequence \(\{\varphi_k\}_1^\infty\) is called convergent to zero, and we write \(\varphi_k\to 0\) as \(k\to\infty\), if for every fixed \(s\) the sequences \(d^s\varphi_k(t)/dt^s\) converge uniformly. Linear functionals \((f,\varphi)\) that are continuous in the sense of the convergence introduced in \(S\) will be called generalized functions (g.f.). In (\(^{3}\)) the subspaces \(S^+\) and \(S^-\) of the space \(S\) were introduced, and g.f. of plus type \(f^+\) and minus type \(f^-\) were defined. It was proved that g.f. of plus type are analytically continued into \(D^+\), and g.f. of minus type into \(D^-\). Recall that \((f^+,\varphi^+)=0\) for all \(\varphi^+\in S^+\) and \((f^-,\varphi^-)=0\) for all \(\varphi^-\in S^-\).

2. Consider the following problems:

a) the Riemann problem
\[ f^+ = G(t)f^-+g,\quad t\in L; \]

b) the problem of linear conjugation with shift
\[ f^+[\alpha] = G(t)f^-+g,\quad t\in L; \tag{1} \]

c) the Carleman problem
\[ f^+ = G(t)f^+[\alpha]+g,\quad t\in L; \]

d) the Hilbert problem*
\[ \operatorname{Re}[f^+/G(t)] = g,\quad t=e^{i\theta},\ 0\leq \theta\leq 2\pi. \]

Here \(G(t)\in S\) \((G(t)\ne 0)\)**, \(\alpha(t)\in S\) \((\alpha'(t)\ne 0)\), and the g.f. \(g\) are given; \(f^+, f^-\) are the unknown g.f. The shift of a generalized function is defined by the equality
\[ (f[\alpha],\varphi(t))=(f,\varphi[\beta(t)]\cdot \beta'(t)), \]
where \(\alpha[\beta(t)]\equiv t\), and it is assumed that

* For an explanation of the notation of boundary condition d), see item 4.

** If in problem a)
\[ G(t)=\prod_{k=1}^{M}(t-\alpha_k)^{m_k}\prod_{k=1}^{N}(t-\beta_k)^{-n_k}G_0(t); \]
\((m_k,n_k\) are positive integers, \(G_0(t)\in S,\ G_0(t)\ne 0)\), then we obtain the exceptional case of the Riemann problem, requiring special consideration. The proposed scheme, however, remains valid.

\(\alpha(t)\) maps \(L\) onto itself one-to-one with preservation of the direction of traversal in problem b) and reversal of it in problem c).

Together with problems (1), let us consider in the space \(S\) the auxiliary problems

\[ \begin{aligned} \text{a}^{*})\quad & \varphi^{+}(t)=[G(t)]^{-1}\varphi^{-}(t)+\varphi(t),\quad t\in L;\\ \text{b}^{*})\quad & \varphi^{+}[\alpha(t)]\alpha'(t)=[G(t)]^{-1}\varphi^{-}(t)+\varphi(t),\quad t\in L;\\ \text{c}^{*})\quad & \varphi^{+}(t)=[G(t)]^{+1}\alpha'(t)\varphi^{+}(t)+\varphi(t),\quad t\in L;\\ \text{d}_{1}^{*})\quad & \operatorname{Re}\left[G(t)\frac{dt}{d\theta}\varphi^{+}(t)\right]=\varphi(t),\\ & \qquad\qquad t=e^{i\theta}\\ \text{d}_{2}^{*})\quad & \operatorname{Re}\left[G(t)i\frac{dt}{d\theta}\psi^{+}(t)\right]=\varphi(t). \end{aligned} \tag{2} \]

Each of the problems \(\text{a}^{*}, \text{b}^{*}, \text{c}^{*}\) is called the adjoint with respect to the corresponding problem (1). With problem d) two problems \(\text{d}_{1}^{*}\) and \(\text{d}_{2}^{*}\) are adjoint. We shall assume that everywhere in (2) \(\varphi(t)\in S\), and in the problems \(\text{d}_{1}^{*}\) and \(\text{d}_{2}^{*}\), \(\operatorname{Im}\varphi(t)=0\).

In solving each of the problems (2), three different cases may occur: 1) the problem is solvable uniquely for any right-hand side \(\varphi(t)\in S\); 2) the problem is solvable if and only if \(\varphi(t)\) is orthogonal to a certain number of linearly independent functions; in this case the solution will be unique; 3) the problem is solvable for any right-hand side; the solution depends linearly on a certain number of arbitrary constants.

In cases 1) and 2), the solution will be represented in the form of a Cauchy-type integral whose density contains the multiplier function \(\varphi(t)\). Using this fact, one can show that the solution belongs to the space \(S\) and depends continuously on \(\varphi(t)\). A particular solution possessing these properties can also be singled out in case 3).

1. We shall solve the Riemann problem a), assuming that \(\chi=\operatorname{ind}G(t)=0\). Let us express the functionals \((f^{+},\varphi)\) and \((f^{-},\varphi)\) through \(G(t)\) and \((g,\varphi)\), using the fact that any fundamental function \(\varphi(t)\) is uniquely representable in the form \(\varphi(t)=\varphi^{+}(t)-[G(t)]^{-1}\varphi^{-}(t)\), since the problem \(\text{a}^{*})\), adjoint to a), for \(\chi=0\), is solvable uniquely. We have
   \[ (f^{+},\varphi(t))=(f^{+},\varphi^{+}(t)-[G(t)]^{-1}\varphi^{-}(t)) =-(f^{+},[G(t)]^{-1}\varphi^{-}(t)) =-(G(t)f^{-}+g,[G(t)]^{-1}\varphi^{-}(t)) =-(g,[G(t)]^{-1}\varphi^{-}(t)) =(g,\varphi^{+}(t)-\varphi(t)). \]
   Arguing analogously, we obtain
   \[ (f^{-},\varphi(t))=-(g,[\varphi(t)/G(t)]^{+}), \]
   where \([\varphi(t)/G(t)]^{+}=\psi^{+}(t)\) is the solution of the problem
   \[ \psi^{+}(t)=[G(t)]^{-1}(\psi^{-}(t)+\varphi(t)). \]
   It is easy to verify that the constructed functionals satisfy the boundary condition a). Their continuity follows from the fact that \(\varphi^{\pm}(t)\to0\) when \(\varphi(t)\to0\), and from the continuity of the functional defining the generalized function \(g\).

In an analogous way one can obtain the solution of problem b) for \(\chi=0\) and of problem c) for \(\chi=-1\). For the Hilbert problem the cases \(\chi=0\) and \(\chi<0\) are essentially the same, and we shall carry out the corresponding investigation in the next item.

1. Let \(\chi<0\). Arguing as above, we consider together with a) the problem \(\text{a}^{*})\). Since the index of \(\text{a}^{*})\), equal to \(-\chi\), will be positive, a linear combination with arbitrary coefficients of the solutions of the corresponding homogeneous problem will enter into its solution. From the relation
   \[ (f^{+},\varphi(t))=(g,\varphi^{+}(t)-\varphi(t)) \]
   it is seen that the functional \((f^{+},\varphi(t))\) will be defined and continuous if and only if the conditions
   \[ (g,\varphi_k^{+}(t))=0,\quad k=1,2,\ldots,|\chi|, \]
   are satisfied, where \(\varphi_k(t)\) are the solutions of the adjoint homogeneous problem
   \[ \varphi^{+}(t)=[G(t)]^{-1}\varphi^{-}(t). \]
   These same conditions ensure the definiteness and continuity of the functional \((f^{-},\varphi(t))\). Thus, they will be the solvability conditions for problem a). Analogous arguments are applicable also to problem b) for \(\chi<0\) and to problem c) for \(\chi<-1\).

To investigate the Hilbert boundary-value problem d) in the space of generalized functions, let us rewrite its boundary condition in the form \(f^{+}=(g+i\rho)G(t)\), where \(\rho\) is the required real generalized function, i.e., one such that \(\operatorname{Im}\left(\rho,\varphi(t)/\dfrac{dt}{d\theta}\right)=0\) for all real test functions \(\varphi(t)\). If \(\operatorname{ind} G(t)=\varkappa\leq 0\), then each of the problems \(\Gamma_1^*\) and \(\Gamma_2^*\) has \(1-2\varkappa\) linearly independent solutions. It can be shown that

\[ (f^{+},\varphi(t))=(g,G(t)[2\varphi(t)-\varphi^{+}(t)-\psi^{+}(t)]) . \tag{3} \]

This functional is continuous if, for any solution \(\varphi_k^{+}(t)\), \(k=1,2,\ldots,1-2\varkappa\), of the homogeneous problem \(\operatorname{Re}\left[G(t)\dfrac{dt}{d\theta}\varphi^{+}(t)\right]=0\), the conditions \((g,G(t)\varphi_k^{+}(t))=0\) are satisfied. When these conditions are fulfilled, formula (3) gives a solution of problem d).

1. The adjoint problem may turn out to be solvable only when conditions of the form

\[ \int_L \varphi(t)h_j(t)\,dt=0,\qquad j=1,2,\ldots,N, \tag{4} \]

are satisfied, where \(h_j(t)\) are certain linearly independent test functions. Such a situation, for example, occurs for \(\varkappa>0\) in problems a) and b). Thus, in this case we can define the required functionals only on the subspace \(S_h\) of the test-function space \(S\), consisting of test functions \(\varphi(t)\) satisfying conditions (4).

To each test function \(\Phi(t)\) let us assign a function \(\varphi(t)\in S_h\) by setting

\[ \varphi(t)=\Phi(t)-\sum_{k=1}^{N}\Phi_k(t)\int_L \Phi(t)h_k(t)\,dt . \tag{5} \]

Here the test functions \(\Phi_k(t)\) are chosen so that \(\int_L \Phi_k(t)h_m(t)\,dt=\delta_{km}\), \(k,m=1,2,\ldots,N\) (\(\delta_{km}=0\) for \(k\ne m\), \(\delta_{kk}=1\)). The construction of such functions is easy to carry out by slightly modifying the method described in (7), pp. 567–569.

Theorem. A functional \((f,\varphi)\), defined and continuous on the subspace \(S_h\), extends, with preservation of continuity, to the whole space \(S\) by the formula

\[ (f,\Phi)=(f,\varphi)+\int_L \sum_{k=1}^{N} C_k h_k(t)\Phi(t)\,dt, \tag{6} \]

where \(C_k\) are arbitrary constants; \(\varphi(t)\) is determined from \(\Phi(t)\) by relation (5). The extension possessing the indicated properties is unique up to the choice of the constants \(C_k\).

The proof is carried out by direct verification of all assertions of the theorem.

The theorem makes it possible to complete the solution in the case \(\varkappa>0\). For example, for the Riemann problem \(h_k(t)=X^{+}(t)t^{k-1}\), \(k=1,2,\ldots,\varkappa\). Thus we arrive at the expression

\[ (f^{+},\Phi(t))=(f^{+},\varphi(t))+\int_L \sum_{k=1}^{\varkappa} C_k t^{k-1}X^{+}(t)\Phi(t)\,dt = \]

\[ = -\left(g,\frac{\varphi^{-}(t)}{G(t)}\right) +\int_L \sum_{k=1}^{\varkappa} C_k t^{k-1}X^{+}(t)\Phi(t)\,dt . \tag{7} \]

Taking into account that \(\varphi^{\pm}(t)\) is a solution of the problem

\[ \varphi^{+}(t)-\frac{\varphi^{-}(t)}{G(t)} =\Phi(t)-\sum_{k=1}^{\varkappa}\Phi_k(t)(h_k(t),\Phi(t)), \]

and using the representation of \(G(t)\) in terms of the limiting values of the canonical function \(X(z)\) of problem \(a^*\), we give formula (7) the form

\[ (f^{+},\Phi(t))=(X^{+}(t)\{(g/X^{+}(t))+P_{\varkappa-1}(t)\},\Phi(t)), \tag{8} \]

where \(P_{\varkappa-1}(z)\) is a polynomial of degree \(\varkappa-1\) with arbitrary coefficients. This form of the solution is completely analogous to that which it has in the classical case.

1. In conclusion, let us consider the linear equation

\[ Kf=g, \tag{9} \]

regarding \(g\) as a generalized function on the basic space \(S\). We define the operator \(K\) by means of the adjoint operator \(K^*\) through the relation \((Kf,\varphi)=(f,K^*\varphi)\), which must reduce to an identity if \(f\in S\). Suppose the equation \(K^*\psi=\varphi\), \(\varphi\in S\), \(\psi\in S\), is solvable under the fulfillment of \(\alpha\) (\(\alpha\geqslant0\)) conditions of the form

\[ \int_L \varphi(t)h_j(t)\,dt=0,\quad j=1,2,\ldots,\alpha, \]

where \(h_j(t)\) are certain linearly independent basic functions, and moreover the solution has the form

\[ \psi(t)=\psi_0(t)+\sum_{k=1}^{\beta}A_k\psi_k(t), \]

where \(\psi_0(t)\to0\) if \(\varphi(t)\to0\); \(A_k\) are arbitrary constants; \(\beta\geqslant0\). Under these conditions the functional \((f,\varphi)\) on the subspace \(S_h\) is defined as follows:

\[ (f,\varphi)=(f,K^*\psi)=(Kf,\psi)=(g,\psi) =\left(g,\psi_0+\sum_{k=1}^{\beta}A_k\psi_k(t)\right). \]

For the continuity of the functional \((f,\varphi)\) on \(S_h\) it is necessary and sufficient that the conditions

\[ (g,\psi_k(t))=0,\quad k=1,2,\ldots,\beta \]

be satisfied. According to the theorem of item 5, the functional on the whole space \(S\) is obtained by the formula

\[ (f,\Phi(t))=(f,\varphi(t))+ \left(\sum_{k=1}^{\alpha}C_kh_k(t),\Phi(t)\right) =(g,K^{*-1}\varphi)+ \left(\sum_{k=1}^{\alpha}C_kh_k(t),\Phi(t)\right), \]

where \(C_k\) are arbitrary constants. Thus, the functions \(h_k(t)\) will be solutions of the homogeneous equation \(Kf=0\).

The indicated scheme can be applied to the study, in the class of generalized functions, of singular integral equations with Cauchy kernel and of paired integral equations of convolution type.

Received
4 II 1965

REFERENCES

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