MATHEMATICS

Yu. P. KRIVENKOV

REPRESENTATION OF SOLUTIONS OF THE EULER–POISSON–DARBOUX EQUATION BY ANALYTIC FUNCTIONS

(Presented by Academician M. A. Lavrent’ev on 9 IV 1957)

In the present article, for the equation

\[ \frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\partial y^{2}}+\frac{c}{y}\frac{\partial w}{\partial y}=0,\qquad c=\mathrm{const}, \tag{1} \]

for values of \(c\) belonging to the interval \(0<c<1\), the results set forth in the article \((^1)\) are strengthened. In the notation adopted there, for any domain \(T\) adjoining an interval \(L\) of the axis \(Ox\), the following theorem is proved.

Theorem 1. If a solution \(w(x,y)\) of equation (1), belonging to the class \(C_2(T)\), or the limit

\[ \lim_{y\to 0} y^c \frac{\partial w}{\partial y} \tag{2} \]

takes analytic values in \(x\) on the interval \(L\), then there exists a domain \(\sigma\), adjoining \(L\), in which the solution can be represented in the form

\[ w(x,y)=\gamma\left(\frac{c}{2}\right)\int_0^1 \frac{\varphi[x+iy(1-2\sigma)]\,d\sigma} {|\sigma(1-\sigma)|^{1-c/2}} + \]

\[ +\gamma\left(1-\frac{c}{2}\right)\left(\frac{y}{1-c}\right)^{1-c} \int_0^1 \frac{\psi[x+iy(1-2\sigma)]\,d\sigma} {[\sigma(1-\sigma)]^{c/2}} . \tag{3} \]

In this expression \(\varphi(z)\) and \(\psi(z)\) are functions analytic in \(\sigma \cup L \cup \bar{\sigma}\), satisfying on \(L\) the conditions

\[ \varphi(x)=w(x,0). \tag{4} \]

\[ \psi(x)=\lim_{y\to 0}\left(\frac{y}{1-c}\right)^c\frac{\partial w}{\partial y}. \tag{5} \]

Proof. In the case when the solution is analytic on \(L\), the function \(\varphi(x)=w(x,0)\) is analytically continued to complex values, forming in some domain \(\sigma' \cup L \cup \bar{\sigma}'\) an analytic function \(\varphi(z)\). In \(\sigma'\) consider the open semicircle \(\tau\) of radius \(\rho\), adjoining the interval \(ab\) (or \(l\)) of the axis \(Ox\) and bounded, for \(y>0\), by the semicircle \(\gamma\).

Using the values of \(w(x,y)\) on the boundary of the semicircle \(\gamma \cup l \cup \{a,b\}\), we solve problem D in \(\tau\) \((^2)\). We seek its solution in the form of the sum \(w_1(x,y)+w_2(x,y)\), where \(w_1(x,y)\) has the form of the first term of expression (3), in which \(\varphi(z)\) is just the analytic function considered.

We note the following property of solutions of equation (1). If \(w(x,y)\) satisfies equation (1) for \(c=2-2\beta\) and \(y>0\), then the expression \(y^{1-2\beta}w(x,y)\) will also satisfy (1) for \(c=2\beta\).

We seek \(w_2(x,y)\) in the form \(y^{1-2\beta}w_2^*(x,y)\), where \(w_2^*(x,y)\) is a solution of equation (1) for \(c=2-2\beta\), belonging to the class \(C_2(\tau)\), continuous on \(\gamma\), and taking on \(\gamma\) the values

\[ \left(\frac{y}{1-2\beta}\right)^{2\beta-1}[w(x,y)-w_1(x,y)]. \tag{6} \]

If on \(\gamma\) we introduce the variable \(t=\dfrac{x-x_0}{\rho}\), then expression (6) on \(\gamma\) will be a function \(F(t)\), defined on the interval \((-1,1)\). Generally speaking, on this interval \(F(t)\) is unbounded, but \((1-t^2)^{1/2-\beta}F(t)\) is always bounded. Therefore, on the basis of work \((^3)\), \(F(t)\) can be expanded in the series

\[ \sum_{n=0}^{\infty} a_n C_n^{\,1-\beta}(t), \]

converging uniformly on every segment \([-1+\delta,1-\delta]\) \((0<\delta<1)\).

To construct \(w_2^*(x,y)\) in \(\tau\), we apply the method used in work \((^1)\). We obtain an expression for \(w_2^*(x,y)\) in \(\tau\) in the form

\[ w_2^*(x,y)=\gamma\left(1-\frac{c}{2}\right) \int_0^1 \frac{\psi[x+iy(1-2\sigma)]\,d\sigma}{[\sigma(1-\sigma)]^{c/2}}, \tag{7} \]

where \(\psi(z)\) is a function analytic in \(\tau\cup l\cup \overline{\tau}\), real on \(l\). Moreover, we find that \(w_2^*(x,y)\) is continuous on \(\gamma\) and takes on \(\gamma\) the values (6), while the expression \(y^{1-2\beta}w_2^*(x,y)\) is bounded in \(\tau\). Consequently, the constructed solution \(w^*(x,y)=w_1(x,y)+w_2(x,y)\) in the form (3), where \(\varphi(z)\) and \(\psi(z)\) are functions analytic in \(\tau\cup l\cup\overline{\tau}\), belongs to the class \(C_2(T)\), is continuous on \(\gamma\), and on \(l\cup\gamma\) takes the values \(w(x,y)\). And since \(w^*(x,y)\) is bounded in \(\tau\), it is proved by the usual barrier method \((^2)\) \((v=-\ln(x^2+y^2))\) that \(w(x,y)\equiv w^*(x,y)\) in \(\tau\), i.e. \(w(x,y)\) is representable in \(\tau\) in the form (3). Since every point of the interval \(l\) is an interior point of the domain of definition of the functions \(\varphi(z),\psi(z)\), and on \(l\) the functions \(\varphi(z),\psi(z)\) satisfy the relations (4), (5), it follows, by the uniqueness property of analytic functions, that \(\varphi(z)\) and \(\psi(z)\) in the expression (3) are unique. The latter means that the representation of \(w(x,y)\) in the form (3) is unique. Considering the representations of \(w(x,y)\) in the collection of semicircles \(\tau_n\) adjacent to \(L\), we obtain the representation of \(w(x,y)\) in the form (3) in a certain domain \(\sigma\) adjacent to \(L\).

The second case of the theorem is proved analogously.

Corollary. If the solution \(w(x,y)\) is analytic in \(x\) on \(L\), then the limit (7) is analytic in \(x\) on \(L\), and conversely.

Two functions \(w(x,y)\) and \(w^*(x,y)\), defined in \(T\), will be called conjugate if they belong to the class \(C_2(T)\) and on \(L\) satisfy the relations

\[ w(x,0)=w^*(x,0),\qquad \lim_{y\to 0} y^c\frac{\partial w}{\partial y} = -\lim_{y\to 0} y^c\frac{\partial w^*}{\partial y}. \]

Theorem 2. For the representability of a solution \(w(x,y)\) of equation (1), for \(0<c<1\), belonging to the class \(C_2(T)\), where \(T\in B\), in the form (3), it is necessary and sufficient that at least one of the following conditions be fulfilled:

a) the existence in \(T\) of a conjugate solution;

b) the existence of a function analytic in \(T\) satisfying condition (4) on the interval \(L\);

c) the existence of a function analytic in \(T\) and satisfying condition (5) on the interval \(L\).

We shall first carry out the proof for case b) of the theorem. In this case, by the preceding, there exists a domain \(\sigma\) adjoining the interval \(L\) in which the solution is represented in the form (3), where \(\varphi(z)\) is a function analytic in \(T\cup L\cup \overline{T}\), and \(\psi(z)\) is analytic in \(\sigma\cup L\cup \overline{\sigma}\). From this representation it is clear that the solution \(w(x,y)\) extends to complex values of \(z\) and \(\zeta\) in the form

\[ U(z,\zeta)=w\left(\frac{z+\zeta}{2};\,\frac{z-\zeta}{2i}\right) \tag{8} \]

to the domain
\[ \{z\in \sigma\cup L\cup \overline{\sigma},\ \zeta\in \sigma\cup L\cup \overline{\sigma},\ \operatorname{Im} z>\operatorname{Im}\zeta\} \]
or to the domain \(Q(\sigma)\). Moreover, the real solution \(w(x,y)\) defined in \(T\) extends, by a theorem of I. N. Vekua (4), to complex values of \(z\) and \(\zeta\) in the form (8) in the bicylindrical domain \(\{z\in T,\ \zeta\in T\}\), or in the domain \(B(T)\). Therefore \(w(x,y)\) determines an analytic function \(U(z,\zeta)\) in the sum of the domains \(Q(\sigma)\) and \(B(T)\).

The expression (3), extended to complex values of \(z\) and \(\zeta\) in the form (8), may be written as

\[ \gamma(1-\beta)\int_0^1 \frac{\psi[z+(\zeta-z)\sigma]\,d\sigma}{[\sigma(1-\sigma)]^\beta} = \frac{ U(z,\zeta)-\gamma(\beta)\displaystyle\int_0^1 \frac{\varphi[z+(\zeta-z)\sigma]\,d\sigma}{[\sigma(1-\sigma)]^{1-\beta}} }{ \left[\dfrac{z-\zeta}{2i(1-2\beta)}\right]^{1-2\beta} }. \tag{9} \]

We choose that branch of the expression in the denominator of the right-hand side which is real and positive for real and positive \(i(\zeta-z)\).

Equality (9) is valid in the domain \(Q(\sigma)\), but its left-hand side is defined in \(B(\sigma\cup L\cup\overline{\sigma})\), and the right-hand side in the sum of \(Q(\sigma)\) and \(B(T)\). Since the right-hand and left-hand sides separately represent analytic functions of two complex variables \(z\) and \(\zeta\) in the corresponding domains and coincide in the bicylindrical domain \(B(\sigma)\), they both represent one analytic function of two complex variables \(V(z,\zeta)\), defined in the sum \(B(\sigma\cup L\cup\overline{\sigma})\) and \(B(T)\). For values \(\zeta=\overline{z}\) this function \(V(z,\overline{z})\) is a real solution of equation (1) with \(c_1=2-c>1\) of class \(C_2(T)\), and therefore, by (*), is represented in \(T\) in the form of the left-hand side of expression (9), in which \(\psi(z)\) is a function analytic in the domain \(T\cup L\cup\overline{T}\) and having real values on \(L\). Constructing with its aid the expression (3), we obtain the required representation. The proof of the theorem in case c) is carried out in a similar way.

In case a) consider the half-sum \(w_1(x,y)\) and the half-difference \(w_2(x,y)\) of the conjugate solutions. Both belong to the class \(C_2(T)\) and have the following properties: the half-sum \(w_1(x,y)\) on \(L\) has the zero boundary value (5), while the half-difference \(w_2(x,y)\) is equal to zero on \(L\), i.e. \(w_2(x,0)=0\). Therefore, in accordance with conditions c) and b) of Theorem 2, they are represented in \(T\) in the form (3), where \(\psi(z)\equiv0\) in the first case and \(\varphi(z)\equiv0\) in the second. Hence the solution itself
\[ w(x,y)=w_1(x,y)+w_2(x,y) \]
is represented in the form (3). According to the remark to Theorem 1, the representation is unique.

Corollary. Every solution of class \(C_2(T)\) of equation (1) for \(0<c<1\), representable in \(T\) in the form (3), analytically extends

in the form (8) to the following domain of two complex variables \(z\) and \(\zeta\):

\[ \left\{ \begin{array}{c} z\in D\\ \zeta\in D\\ \operatorname{Im} z>\operatorname{Im}\zeta \end{array} \right\},\qquad D=T\cup L\cup \overline{T}. \]

In the domain \(D=T\cup L\cup \overline{T}\), where \(T\in B\), consider the class of functions \(C_2(D)\). To it we assign every pair of functions belonging, respectively, to the classes \(C_2(T)\) and \(C_2(\overline{T})\) and forming in \(D\), together with the expression \(|y|^c \dfrac{\partial w}{\partial y}\), a single continuous function.

Theorem 3. Every solution \(w(x,y)\) of class \(C_2(D)\) of equation (1), for \(0<c<1\), is represented in \(D\) in the form

\[ w(x,y)=\gamma\left(\frac{c}{2}\right)\int_0^1 \frac{\varphi[x+iy(1-2\sigma)]\,d\sigma}{[\sigma(1-\sigma)]^{1-c/2}} + \]

\[ +\operatorname{sign}y\cdot \gamma\left(1-\frac{c}{2}\right) \left(\frac{|y|}{1-c}\right)^{1-c} \int_0^1 \frac{\psi[x+iy(1-2\sigma)]\,d\sigma}{[\sigma(1-\sigma)]^{c/2}}, \tag{10} \]

where \(\varphi(z)\) and \(\psi(z)\) are functions analytic in \(D\), satisfying on \(L\), respectively, condition (4) and the condition

\[ \psi(z)=\lim_{y\to 0}\left(\frac{|y|}{1-c}\right)^c \frac{\partial w}{\partial y}. \tag{11} \]

Proof. Since, according to the definition of the class \(C_2(D)\), the solution \(w(x,y)\) consists of two functions \(w^1(x,y)\) and \(w^2(x,y)\), defined, respectively, in \(T\) and \(\overline{T}\), and since equation (1) does not change under the replacement of \(y\) by \(-y\), the function \(\overline{w(x,y)}=w^2(x,-y)\), defined in \(T\), is a solution of equation (1) for \(0<c<1\) and has on \(L\) the properties

\[ w^1(x,0)=\overline{w(x,0)},\qquad \lim_{y\to 0} y^c\frac{\partial w^1}{\partial y} = -\lim_{y\to 0} y^c\frac{\partial\overline{w}}{\partial y}, \]

i.e. \(\overline{w(x,y)}\) is the solution of equation (1) in \(T\) conjugate to \(w^1(x,y)\).

Consequently, \(w(x,y)\) is represented in \(T\) in the form (10), where \(\varphi(z)\) and \(\psi(z)\) are functions analytic in \(D\) and taking real values on \(L\). Representing \(w(x,y)\) by this method also in the domain \(\overline{T}\), we obtain a representation of \(w(x,y)\) in the domain \(D=T\cup L\cup \overline{T}\), which in both cases is written in the form (10). By the preceding result the representation is unique, as was required to prove.

I express my gratitude to I. N. Vekua for posing the problem and for guidance, and also to V. I. Karabegov and L. V. Ovsyannikov for a number of valuable comments.

Moscow Physico-Technical Institute

Received
5 IV 1957

CITED LITERATURE

¹ Yu. P. Krivenkov, DAN, 116, No. 3 (1957). ² M. V. Keldysh, DAN, 77, No. 2 (1951). ³ G. Szego, Am. Math. Soc., 23, 239 (1939). ⁴ I. N. Vekua, New Methods for Solving Elliptic Equations, Moscow–Leningrad, 1948.