A. DZHURAEV

THE GENERAL LINEAR BOUNDARY-VALUE PROBLEM FOR THE EQUATION

\[ \Delta u+\lambda c(x,y)u=0 \]

(Presented by Academician I. N. Vekua, 9 X 1961)

1. Let \(D\) denote a bounded domain of class \({}^{(1)} C_\alpha,\ 0<\alpha<1\), in the plane of the variables \(x,y\); let \(\Gamma\) be its boundary. Denote by \(L_n\) an arbitrary linear differential operator of order \(n\) with variable real coefficients, Hölder-continuous on \(\Gamma\),

\[ L_n \equiv \sum_{k=1}^{n}\sum_{l=0}^{k} a^{k-l,l}(\xi,\eta)\, \frac{\partial^k}{\partial \xi^{k-l}\partial \eta^l}, \qquad \xi,\eta\in\Gamma, \]

and by \(\overset{0}{L}_n\) the principal part of the operator \(L_n\). Assuming \(c(x,y)\) to be a real-valued function of class \({}^{(1)} C_\alpha^{\,n-1}(D+\Gamma)\), in the present note we shall study the following boundary-value problem:

Problem \(C_\lambda\). Find in \(D\) real solutions of the equation

\[ \Delta u+\lambda c(x,y)u=0, \tag{1} \]

continuous together with their derivatives up to order \(n\) inclusive in the closed domain \(\overline{D}=D+\Gamma\), and satisfying on the boundary \(\Gamma\) the general linear condition

\[ \left(\overset{0}{L}_n+\lambda L_{n-1}\right)u(\xi,\eta)=h(\xi,\eta), \tag{2} \]

where \(h(\xi,\eta)\) is a prescribed real-valued function on \(\Gamma\).

Problem \(C_1\) was first formulated for a second-order equation of elliptic type, reduced to canonical form, and studied in \({}^{(2)}\) under the assumption of analyticity of the coefficients of the equation, by methods of the theory of one-dimensional singular equations. Dispensing with the analyticity requirement, we shall study problem \(C_\lambda\), reducing it to a certain Riemann—Hilbert boundary-value problem for a system of elliptic type. This method was first applied in \({}^{(3)}\) in the study of problems with oblique derivative for generalized analytic functions*. In what follows, as in \({}^{(2)}\), we shall assume that problem \(C_\lambda\) is normal, i.e., everywhere on the contour \(\Gamma\) the condition

\[ a^*(t)=\sum_{l=0}^{n} i^l a^{\,n-l,l}(\xi,\eta)\ne 0,\qquad t=\xi+i\eta\in\Gamma. \tag{3} \]

is satisfied.

Taking into account the continuity, together with the derivatives up to \(\Gamma\), of the sought function \(u(x,y)\), equation (1) and the boundary condition (2) can be brought to the form

\[ \partial^2 u/\partial z\,\partial\overline{z}+\lambda c_0(x,y)u=0, \tag{4} \]

\[ 2\partial/\partial\overline{z}=\partial/\partial x+i\partial/\partial y,\qquad 2\partial/\partial z=\partial/\partial x-i\partial/\partial y; \]

\[ \operatorname{Re}\left[ a^*(t)\frac{\partial^n u}{\partial t^n} +\sum_{k=0}^{n-1} a_k(t,\lambda)\frac{\partial^k u}{\partial t^k} \right] =h_0(t),\qquad c_0=\frac12 c,\quad h_0=\frac12 h, \tag{5} \]

where \(a_k(t,\lambda)\) are completely determined functions satisfying the condition

* In the dissertation of Teng En Cher, problem \(C_1\) was studied for \(n=1\) for a multiply connected domain \({}^{(7)}\).

Hölder in \(t\) and analytic in \(\lambda\) (polynomials in \(\lambda\)), such that \(a_k(t,0)\equiv0\), \(k=0,1,2,\ldots,n-1\). These functions are uniquely determined by the coefficients of the operator \(L_n\) and by the boundary values of the function \(c(x,y)\) and its derivatives up to order \(n-2\) inclusive. To study problem \(C_\lambda\), consider the following auxiliary problem.

Problem \(P_\lambda\). Find, continuous in the closed domain \(\overline D\), a complex-valued vector function
\[ V(z)=[v_1,v_2,\ldots,v_{2n}], \]
satisfying in \(D\) the equation
\[ \partial V/\partial \overline z+A^\lambda V+B^\lambda \overline V=0 \tag{6} \]
and the boundary condition on \(\Gamma\)
\[ \operatorname{Re}[G_\lambda(t)V(t)]=\mathscr H_0(t),\qquad \mathscr H_0=[0,h_0,0,\ldots,0], \tag{7} \]
where \(A^\lambda(z)=\|A_{kl}\|\), \(B^\lambda(z)=\|B_{kl}\|\), \(G_\lambda(t)=\|G_{kl}\|\) are square \((2n\times2n)\) matrices such that:
\[ A_{kl}=\lambda C_{k-2}^{\,l-1}\partial^{k-l-1}c_0(x,y)/\partial z^{k-l-1},\quad k=3,4,\ldots,n+1;\quad l=2,3,\ldots,n; \]
\[ \begin{aligned} &B_{1,1}=0,\quad B_{k,1}=\lambda\partial^{k-2}c_0(x,y)/\partial z^{k-2},\quad k=2,3,\ldots,n-1;\\ &B_{1,2}=B_{n+2,n+1}=B_{n+3,n}=\ldots=B_{2n,3}=-1,\\ &G_{1,1}=G_{4,n}=G_{4,n+2}=G_{6,n-1}=G_{6,n+3}=\ldots\\ &\ldots=G_{2n,2}=G_{2n,2n}=i,\quad G_{2,k}=a_{k-1}(t,\lambda),\quad G_{2,n+1}=a^*(t),\quad k=1,2,3,\ldots,n;\\ &G_{3,n+1}=G_{5,n+2}=\ldots=G_{2n-1,2n}=1,\quad G_{2k-1,n-k+2}=-1,\quad k=2,3,\ldots,n, \end{aligned} \]
and all their remaining elements are equal to zero. Here \(C_{k-2}^{\,l-1}\) denotes the binomial coefficient from \(k-2\) elements taken \(l-1\) at a time.

Theorem 1. If \(u(x,y)\) is a solution of problem \(C_\lambda\), then the vector function \(V(z)\) with components
\[ v_1=u,\quad v_k=\partial^{k-1}u/\partial z^{k-1},\quad k=2,3,\ldots,n+1; \]
\[ v_k=\partial^{2n-k+1}u/\partial z^{2n-k+1},\quad k=n+2,\ldots,2n, \]
will be a solution of problem \(P_\lambda\). Conversely, if the vector function \(V\) is a solution of problem \(P_\lambda\), and if the homogeneous Dirichlet problem for equation (1) has no nonzero solutions, then the first component \(v_1\) of the vector function \(V(z)\) will be a solution of problem \(C_\lambda\).

Since
\[ \det G_\lambda(t)=(2i)^n i a^*(t)\ne0, \]
problem \(P_\lambda\) is normal. Therefore, applying to it the results of work \((^4)\), and then comparing linearly independent solutions of the problems \(C_\lambda\) and \(P_\lambda\), we obtain the following result for an \((m+1)\)-connected domain \(D\):

Theorem 2. For the solvability of problem \(C_\lambda\) it is necessary and sufficient that the conditions
\[ \int_\Gamma h(t)v_j(t)\,dt=0,\quad j=1,2,\ldots,\hat l, \]
hold, where \(v_j(t)\) are certain linearly independent functions. The number of solvability conditions \(\hat l\) of problem \(C_\lambda\) is equal to
\[ \hat l=l_C-2[\varkappa+n(m-1)], \]
where
\[ \varkappa=\frac{1}{2\pi}\{\arg a^*(t)\}_\Gamma \]
is the index of problem \(C_\lambda\); \(l_C\) is the number of linearly independent solutions of the homogeneous problem \(C_\lambda^0\) (\(h=0\)). The number \(l_C\) is finite and bounded below:
\[ l_C\ge \max(0,2[\varkappa-n(m-1)])-q, \]
where \(q\) is a nonnegative integer \(\le l_D\), and \(l_D\) is the number of linearly independent solutions of the homogeneous Dirichlet problem for equation (1). For \(n=1\),
\[ l_C=\max(0,2[\varkappa-n(m-1)])-q,\quad \varkappa<0. \]

1. Let \(D\) be simply connected (\(m=0\)). In this case, without loss of generality, one may assume that \(D\) is the unit disk. Then the boundary matrix \(G_\lambda(t)\) can be represented in the form
   \[ G_\lambda=E_\Omega\cdot G_\chi\cdot G_\varphi, \]
   where \(E_\Omega=\|E_{kk}\|\), \(G_\chi=\|G_{kk}\|\), \(k=1,2,\ldots,2n\), are diagonal matrices, moreover such that
   \[ E_{kk}=G_{kk}=1,\quad k\ne2;\qquad E_{22}=\exp(-\Omega),\quad G_{22}=t^{-\varkappa}; \]
   \[ \Omega=\operatorname{Im}\varphi;\quad \varphi(z)=\frac{1}{2\pi i}\int_\Gamma[\arg a^*(t)+\varkappa\arg t]\frac{t+z}{t-z}\frac{dt}{t}; \]
   \[ \widehat G_\varphi=\|\widehat G_{kl}\| \]
   is a square \((2n\times2n)\) matrix all of whose elements are zero except
   \[ \widehat G_{2,k}=a_{k-1}(t,\lambda)/a^*(t),\quad k=1,2,\ldots,n; \]
   \[ \widehat G_{2k-1,n-k+2}=-1,\quad k=2,3,\ldots,n;\quad \widehat G_{3,n+1}=\widehat G_{5,n+2}=\ldots=\widehat G_{2n-1,2n}=1; \]
   \[ \widehat G_{2,n+1}=e^{i\varphi(z)};\quad \widehat G_{11}=\widehat G_{4,n}=\widehat G_{4,n+2}=\widehat G_{6,n-1}=\widehat G_{6,n+3}=\ldots \]
   \[ =\ldots=\widehat G_{2n,2}=\widehat G_{2n,2n}=i. \]

Let us now suppose that the coefficients of the operator \(L_n\) are continuously differentiable. Then, since \(\det \widehat G_\varphi(t)=(2i)^n i e^{i\varphi(t)}\ne 0\), and \(\varphi\) is a single-valued function, we have \(\{\arg\det \widehat G_\varphi(t)\}_\Gamma=0\), and the matrix can be extended into \(D\) in such a way that, if \(\widehat G_\varphi(z)\) is its extension, then \(\det \widehat G_\varphi(z)\ne 0\) and \(\widehat G_\varphi(z)\) is continuously differentiable. Making now the transformation \(\widehat G_\varphi(z)V(z)=W(z)\), we bring equation (6) and condition (7) to the form

\[ \partial W/\partial \bar z+P^\lambda W+Q^\lambda \overline W=0, \tag{8} \]

\[ \operatorname{Re}[G_\chi(t)W(t)]=\mathcal H(t), \qquad \mathcal H=E_\Omega^{-1}\mathcal H_0, \tag{9} \]

in which the matrices \(P^\lambda, Q^\lambda\) depend analytically on the parameter \(\lambda\) in such a way that \(P^0(z)\equiv 0\), while the matrix \(Q^0(z)\) has the form \(Q^0(z)=\|Q_{kl}\|\), where

\[ Q_{32}=-e^{i\overline{\varphi(z)}}, \qquad Q_{42}=-i e^{i\overline{\varphi(z)}}, \]

\[ Q_{2\bar k-1,\,2\bar k-3}=1/2; \qquad Q_{2\bar k-1,\,2\bar k-2}=1/2i, \qquad \bar k=3,4,\ldots,n; \]

\[ Q_{2\tilde k,\,2\tilde k-3}=i/2, \qquad Q_{2\tilde k,\,2\tilde k-2}=1/2, \qquad \tilde k=4,5,\ldots,n; \tag{10} \]

\[ Q_{k,l}\equiv 0, \qquad k\ne 2\bar k-1,\,2\tilde k,\quad l\ne 2\bar k-3,\,2\bar k-2,\,2\tilde k-3,\,2\tilde k-2. \]

3. Let \(\chi\ge 0\). Then the solution of equation (8) can be represented in the form

\[ W(z)=T_\lambda W+\Phi(z), \tag{11} \]

where \(T_\lambda\) is a completely continuous operator in the space of complex-valued vectors continuous in \(D\), which has the form

\[ T_\lambda W=\frac{1}{\pi}\iint_D \left\{ \frac{\chi_\lambda(t)}{t-z} + z g_\chi(z)\frac{\overline{\chi_\lambda(t)}}{1-\bar t z} \right\}\,dD_t, \qquad \chi_\lambda\equiv P^\lambda W+Q^\lambda\overline W, \tag{12} \]

and \(\Phi(z)\) is an arbitrary vector-function holomorphic in \(D\) and continuous in \(\overline D\), where \(g_\chi(z)=\|g_{kk}(z)\|\), \(k=1,2,\ldots,2n\), is a diagonal matrix such that \(g_{22}(z)=z^{2\chi}\), \(g_{kk}=1\), \(k\ne 2\). Since, obviously, \(g_\chi(t)=G_\chi^{-1}(t)\overline{G_\chi(t)}\), it is not difficult to verify that \(\operatorname{Re}[G_\chi(t)T_\lambda W]=0\). Therefore, substituting (11) into (9), we see that equation (11) is completely equivalent to problem \(P_\lambda\), provided the holomorphic vector-function \(\Phi(z)=[\Phi_1,\Phi_2,\ldots,\Phi_{2n}]\) has the form

\[ \Phi_k(z)=i\gamma_k,\qquad k=1,3,4,\ldots,2n,\qquad h_1=e^\Omega h_0, \]

\[ \Phi_2(z)=\frac{1}{2\pi i}\int_\Gamma h_1(t)\frac{t+z}{t-z}\frac{dt}{t} +i\alpha_0 z^\chi+ \tag{13} \]

\[ +\sum_{l=0}^{\chi-1}\{\alpha_l(z^l-z^{2\chi-l})+i\beta_l(z^l+z^{2\chi-l})\}. \]

Let \(\dot M\) denote the set of real numbers consisting of those \(\lambda\) for which the homogeneous equation \(W-T_\lambda W=0\) has nonzero solutions. Then, applying I. Tamarkin’s theorem \((^5)\) to equation (11) and taking (10) into account, we arrive at the following result:

Theorem 3. If the index of the problem \(C_\lambda\) is nonnegative, then the problem \(C_\lambda\) is always solvable, except, possibly, for a discrete set \(\dot M\) of values of \(\lambda\), not containing the point \(\lambda=0\). In particular, the problem \(C_\lambda\) is always solvable for sufficiently small \(\lambda\). If \(\lambda\notin \dot M\), then the homogeneous problem \(C_\lambda^0\) \((h\equiv 0)\) has exactly \(l_C=2(\chi+n)-q\) linearly independent solutions. If, however, \(\lambda\in \dot M\), then, for the solvability of the problem \(C_\lambda\), it is necessary and sufficient that \(r-s\) additional conditions be fulfilled, where \(r\) is the rank of the eigenvalue \(\lambda\), and \(s\) is a number satisfying the inequality \(0\le s\le \min[r,\,2(\chi+n)-q]\).

1. Suppose now that the index \(\varkappa\) is negative: \(\varkappa=-\varkappa_1\), \(\varkappa_1>0\). Then the boundary condition (9) can be written in the form

\[ \{\operatorname{Re}[\overline{G_{\varkappa_1}(t)}\, W(t)]\}=\mathcal H(t), \tag{14} \]

and the solution of equation (8) can be represented in the form

\[ W(z)=\frac{1}{2\pi i}\int_\Gamma \frac{W(t)}{t-z}\,dt +\frac{1}{\pi}\iint_D \frac{\chi_\lambda(t)}{t-z}\,dD_t . \tag{15} \]

Transforming the first term on the right-hand side with the use of (14) and Green’s formula, we obtain the following Fredholm integral equation, equivalent to the problem \(P_\lambda\) for \(\varkappa<0\):

\[ W-K_\lambda W =\frac{1}{\pi i}\int_\Gamma \frac{G_\varkappa(t)\mathcal H(t)}{t-z}\,dt -g_\varkappa(0)\overline{W(0)}, \tag{16} \]

where

\[ K_\lambda W=\frac{1}{\pi}\iint_D \left\{ \frac{\chi_\lambda(z)}{t-z} -\frac{g_\varkappa(t)}{t}\, \frac{\overline{\chi_\lambda(t)}}{1-tz} \right\}\,dD_t . \]

Investigating equation (16), we arrive at the following theorem:

Theorem 4. Let \(\varkappa<0\). Then the problem \(C_\lambda\) is solvable for all \(\lambda\) except, possibly, for a discrete set \(M_1\) not containing the zero point and, in particular, it is solvable for sufficiently small \(\lambda\). If \(\lambda\notin M_1\), then the homogeneous problem \(C_\lambda^0\) has no more than \(4n-2-q\) linearly independent solutions, while the nonhomogeneous problem has exactly one solution provided it is normalized by the conditions \(\partial^k u/\partial z^k|_{z=0}=0\), \(k=0,1,\ldots,n\).

If, however, \(\lambda\in M_1\), then for solvability of the problem \(C_\lambda\) it is necessary to impose \(r_1-s_1\) additional conditions, where \(r_1\) is the rank of the eigenvalue \(\lambda\), and \(s_1\) is a number determined by the inequality \(0\leq s_1\leq \min(0,\,4n-2-q)\).

1. Let \(\hat C(x,y)=\|\hat c_{kl}\|\) be a square matrix of order \(N\), whose elements are real functions of class \(C_{\alpha}^{\,n-1}(\overline D)\), \(0<\alpha<1\), and let \(a^{p,q}(\xi,\eta)=\|a_{kl}^{p,q}\|\) be square matrices of order \(N\), given on \(\Gamma\), whose elements are real functions continuous in the sense of Hölder. Consider the following problem:

Problem \(\hat C_\lambda\). It is required to determine a real vector-function
\(U=[u_1,u_2,\ldots,u_N]\), continuous in \(\overline D\) together with its derivatives up to order \(n\) inclusive, satisfying in \(D\) the equation
\(\Delta U+\lambda\hat C(x,y)U=0\) and on \(\Gamma\) the condition

\[ \left( \sum_{j=0}^{n} a^{\,n-j,j}(\xi,\eta) \frac{\partial^n}{\partial \xi^{\,n-j}\partial \eta^j} + \lambda\sum_{j=1}^{n-1}\sum_{\mu=0}^{j} a^{\,j-\mu,\mu}(\xi,\eta) \frac{\partial^j}{\partial \xi^{\,j-\mu}\partial \eta^\mu} \right)U =H(\xi,\eta), \]

where \(H=[H_1,\ldots,H_N]\) is a given real vector-function on \(\Gamma\), Hölder-continuous.

Let

\[ \hat{\mathbf a}^{\,*}(t)=\sum_{j=0}^{n} i^j a^{\,n-j,j}(\xi,\eta),\qquad \hat\varkappa=\frac{1}{2\pi}\{\arg \det \hat{\mathbf a}^{\,*}(t)\}_\Gamma . \]

Then, if \(\det \hat{\mathbf a}^{\,*}(t)\ne 0\) everywhere on \(\Gamma\), Theorems 2, 3, and 4 are valid for the problem \(\hat C_\lambda\), provided that in them \(n\) is replaced by \(n\cdot N\), and \(\varkappa\) by \(\hat\varkappa\); moreover, in the case of Theorems 3 and 4, continuous differentiability is required of the boundary matrices \(a^{p,q}\).

Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR

Received
3 X 1961

REFERENCES

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