MATHEMATICS

CHZHAO CHZHEN

SOLUTION OF THE DIRICHLET PROBLEM IN THE PLANE FOR A SECOND-ORDER EQUATION OF ELLIPTIC TYPE BY THE METHOD OF EXPANSION IN A SERIES

(Presented by Academician I. N. Vekua, 10 II 1960)

1. In the paper \((^{1})\) a method was indicated for solving the generalized Riemann—Hilbert problem, using a certain system of particular solutions of the adjoint equation. In the present paper, by the same method, the Dirichlet problem (problem D) in the plane will be studied for a second-order elliptic equation with (real) analytic coefficients \((^{2})\).

2. For simplicity we shall restrict ourselves to the consideration of a simply connected domain, since without particular difficulty the propositions given below can be extended to the case of a multiply connected domain.

Problem D. It is required to find a solution \(u(x,y)\), regular in \(T\) and continuous in \(T+\Gamma\), of the equation

\[ \mathfrak{M}u=\Delta u+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}+c(x,y)u=0, \tag{1} \]

satisfying the boundary condition

\[ u(t)\big|_{\Gamma}=\gamma(t),\qquad t\in\Gamma, \tag{2} \]

where the coefficients \(a,b,c\) are real analytic functions in some fundamental domain \(\mathfrak{D}\) of equation (1) \((^{2})\). Suppose that the domain \(T\) lies inside the fundamental domain \(\mathfrak{D}\), and that its boundary \(\Gamma\) is a closed \(H\)-continuous curve.

If \(u(x,y)\) is a regular solution of equation (1), then \((^{2})\)

\[ u(x,y)=\int_{\Gamma}\left[u(t)N\omega(x,y,\xi,\eta)-\frac{du(t)}{d\nu}\omega(x,y,\xi,\eta)\right]ds, \tag{3} \]

where \(N\omega=d\omega/d\nu-[a\cos(\nu,x)+b\cos(\nu,y)]\omega\); \(\nu\) is the inward normal at the point \((\xi,\eta)\); \(\omega\) is the normalized standard elementary solution of equation (1).

It is obvious that formula (3) will allow us to obtain the solution of problem D, if the boundary values \(du(t)/d\nu\) on the boundary \(\Gamma\) are known (\(u(t)\) is already prescribed). It remains to determine the boundary values \(du(t)/d\nu\). For this purpose we use a certain complete system \(\{v_k(x,y)\}\) of particular solutions of the adjoint equation*

\[ \mathfrak{N}v\equiv\Delta v-\frac{\partial av}{\partial x}-\frac{\partial bv}{\partial y}+cv=0. \tag{4} \]

* Completeness here is understood in the sense that every solution of equation (4) regular in \(T\) can be uniformly approximated inside \(T\) by means of linear combinations of the form \(\xi_1v_1+\xi_2v_2+\cdots+\xi_nv_n\) \((^{2})\).

Every solution \(v\) of this equation can be represented in the form \({}^{(2)}\)

\[ v(x,y)=\operatorname{Re}\left[H_0(z)\varphi(z)+\int_0^z H(z,\tau)\varphi(\tau)\,d\tau\right], \tag{5} \]

where \(\varphi(z)\) is an arbitrary holomorphic function. Substituting into (5), in place of \(\varphi(z)\), the system \(z^p, iz^p\) \((p=0,1,2,\ldots)\), we obtain one complete (infinite) system \(\{v_k(x,y)\}\) of particular solutions of equation (4) with respect to the domain \(T\).

Moreover, for any solution \(v\) of equation (4) and solution \(u\) of problem D, the equality

\[ \iint_T [v\mathfrak{M}u-u\mathfrak{M}v]\,dx\,dy = \int_\Gamma\left[\gamma(t)Nv(t)-\frac{du(t)}{d\nu}v(t)\right]\,ds =0 \tag{6} \]

holds. This is, obviously, equivalent to

\[ \int_\Gamma \frac{du(t)}{d\nu}v_k(t)\,ds = \int_\Gamma \gamma(t)Nv_k(t)\,ds \equiv c_k,\qquad k=1,2,\ldots, \tag{7} \]

where \(c_k\) are known constants.

1. We begin the construction of the solution of problem D with the case when the homogeneous problem \(\mathrm{D}_0\) has no solution. It is not difficult to prove the following theorem:

Theorem 1. If the homogeneous problem \(\mathrm{D}_0\) has no solution, then on the contour \(\Gamma\) the system \(v_k(t)\), \(k=1,2,\ldots\), is linearly independent; moreover, as is known \({}^{(2)}\), it is closed in the space \(L^2(\Gamma)\), i.e., for any \(\varepsilon>0\) and any function \(f(t)\) belonging to the space \(L^2(\Gamma)\), there is a system of constants \(\xi_1,\xi_2,\ldots,\xi_n\) such that

\[ \left\|f(t)-\sum_{k=1}^{n}\xi_k v_k(t)\right\|_{L^2}<\varepsilon. \]

Thus, in this case one may assume that on \(\Gamma\) the system \(v_k(t)\), \(k=1,2,\ldots\), is complete and orthonormalized in the space \(L^2(\Gamma)\). Consequently, according to formula (7), we obtain that the Fourier series \(\sum_{k=1}^{\infty} c_k v_k(t)\) on the contour \(\Gamma\) converges in mean to the function \(du(t)/d\nu\). Hence the following theorem follows:

Theorem 2. If the homogeneous problem \(\mathrm{D}_0\) has no solution, then in the domain \(T\), for the solution of the Dirichlet problem, the equality

\[ u(x,y)=\int_\Gamma\left[\gamma(t)N\omega(x,y,\xi,\eta) -\sum_{k=1}^{\infty}c_k v_k(t)\omega(x,y,\xi,\eta)\right]\,ds, \tag{8} \]

holds, and the right-hand side of (8) converges uniformly to the solution \(u(x,y)\) in the closed domain \(T+\Gamma\).

1. We proceed to consider the case when the homogeneous problem \(\mathrm{D}_0\) has nontrivial solutions. As is known \({}^{(2)}\), the number of linearly independent solutions of problem \(\mathrm{D}_0\) is always finite, and the adjoint problem \(\mathrm{D}_0^*\) has as many linearly independent solutions as problem \(\mathrm{D}_0\). Suppose that \(u_1,u_2,\ldots,u_l\) is some complete system of solutions of problem \(\mathrm{D}_0\). In this case, by a suitable choice of the system \(\{v_k(x,y)\}\), \(k=1,2,\ldots\), it is always possible to ensure that on the contour \(\Gamma\), \(v_1(t)=\cdots=v_l(t)=0\), and that the system \(\{v_{l+k}(t)\}\), \(k=1,2,\ldots\), is orthonormal.

According to formula (7),

\[ \int_{\Gamma}\gamma(t)\frac{dv_k(t)}{dv}\,ds=0,\qquad k=1,2,\ldots,l. \tag{9} \]

These are the solvability conditions for problem \(D\).

Consider the space of all solutions of problem \(D_0\) (an \(l\)-dimensional space). For each element \(\tilde u(x,y)\not\equiv 0\) of this space we have \(d\tilde u(t)/dv\not\equiv 0\), \(t\in\Gamma\), since otherwise, according to formula (3), it would follow that \(\tilde u(x,y)\equiv 0\) in \(T\). All boundary values \(d\tilde u(t)/dv\), \(t\in\Gamma\), form an \(l\)-dimensional subspace \(R\) of the space \(C(\Gamma)\). It is easy to see that the linearly independent system \(du_k(t)/dv\), \(k=1,2,\ldots,l\), may be taken as a basis of the subspace \(R\).

If \(f(t)\in L^2(\Gamma)\), then for any \(\varepsilon>0\) there exists a continuous function \(g(t)\) such that \(\|f(t)-g(t)\|_{L^2}<\varepsilon/2\); on the other hand, the continuous function \(g(t)\) can be uniquely represented in the form \(g(t)=g_1(t)+g_2(t)\), where \(g_1(t)\) belongs to the subspace \(R\), i.e.

\[ g_1(t)=\sum_{k=1}^{l}\eta_k\frac{du_k(t)}{dv}, \]

and \(g_2(t)\) belongs to the orthogonal complement to \(R\), i.e. for \(g_2(t)\) we have

\[ \int_{\Gamma}g_2(t)\frac{du_k(t)}{dv}\,ds=0,\qquad k=1,2,\ldots,l. \tag{10} \]

This is nothing other than the solvability condition for the adjoint problem \(D^*\), so that one can find a solution \(v(x,y)\) of problem \(D^*\), and \(v(t)|_{\Gamma}=g_2(t)\). It is easy to see that in this case the holomorphic function \(\varphi(z)\), by means of which \(v(x,y)\) is expressed in the form (5), belongs (at least) to the space \(L^2(T+\Gamma)\) (³), therefore \(\varphi(t)\), \(t\in\Gamma\), can be approximated in the mean by a polynomial on \(\Gamma\); consequently, there exists a system of constants \(\xi_1,\xi_2,\ldots,\xi_n\) such that on \(\Gamma\) the inequality

\[ \left\|g_2(t)-\sum_{k=1}^{n}\xi_k v_k(t)\right\|_{L^2}<\frac{\varepsilon}{2}, \]

holds; whence it follows that

\[ \left\|f(t)-\sum_{k=1}^{l}\eta_k\frac{du_k(t)}{dv}-\sum_{k=1}^{n}\xi_k v_k(t)\right\|_{L^2}\leq \]

\[ \leq \|f(t)-g(t)\|_{L^2}+ \left\|g(t)-\sum_{k=1}^{l}\eta_k\frac{du_k(t)}{dv}-\sum_{k=1}^{n}\xi_k v_k(t)\right\|_{L^2}<\varepsilon. \]

Theorem 3. If the homogeneous problem \(D_0\) has \(l\) linearly independent solutions \(u_1,u_2,\ldots,u_l\), then the system

\[ \frac{du_1(t)}{dv},\ \frac{du_2(t)}{dv},\ldots,\frac{du_l(t)}{dv},\ v_{l+1}(t),\ v_{l+2}(t),\ldots, \tag{11} \]

will be closed in the space \(L^2(\Gamma)\).

Thus, one may assume that the system (11) is a complete orthonormal system in the space \(L^2(\Gamma)\). Under conditions (9), if we put

\[ a_k=\int_{\Gamma}\frac{du(t)}{dv}\frac{du_k(t)}{dv}\,ds,\qquad k=1,2,\ldots,l, \]

then the Fourier series \(\displaystyle \sum_{k=1}^{l}a_k\frac{du_k(t)}{dv}+\)

\[ +\sum_{k=1}^{\infty} c_{l+k}v_{l+k}(t) \]
converges in the mean to the function \(\dfrac{du(t)}{d\nu}\). In particular, the series

\[ \sum_{k=1}^{\infty} c_{l+k}v_{l+k}(t) \]
converges in the mean to the function
\[ \frac{d}{d\nu}\left(u(t)-\sum_{k=1}^{l}a_k u_k(t)\right). \]

If we denote
\[ u^*(x,y)\equiv u(x,y)-\sum_{k=1}^{l}a_k u_k(x,y), \]
then, evidently, \(u^*(x,y)\) is a certain particular solution of problem D. Hence it follows that

\[ \int_{\Gamma}\left[\gamma(t)N\omega(x,y,\xi,\eta)-\sum_{k=1}^{\infty}c_{l+k}v_{l+k}(t)\,\omega(x,y,\xi,\eta)\right]\,ds \]
converges uniformly to the particular solution \(u^*(x,y)\) in the closed domain \(T+\Gamma\). After this, the general solution of problem D in the domain \(T\) can be obtained from the formula

\[ u(x,y)=u^*(x,y)+\sum_{k=1}^{l}\lambda_k u_k(x,y), \]

where \(\lambda_k\) are arbitrary constants.

In conclusion I express my sincere gratitude to my supervisor, Academician I. N. Vekua, for posing the problem and for valuable comments in carrying out this work.

Moscow State University
named after M. V. Lomonosov

Received
8 II 1960

CITED LITERATURE

1. Chzhao Chzhen, DAN, 128, No. 2 (1959).
2. I. N. Vekua, New Methods for Solving Elliptic Equations, Moscow–Leningrad, 1948.
3. B. V. Khvedelidze, Tr. Tbilisi Math. Inst., 23 (1956).