Corresponding Member of the USSR Academy of Sciences A. V. BITSADZE

ON ONE PARTICULAR CASE OF THE OBLIQUE DERIVATIVE PROBLEM FOR HARMONIC FUNCTIONS IN THREE-DIMENSIONAL DOMAINS

In a simply connected finite domain \(D\) of the variables \(x, y, z\), bounded by a Lyapunov surface \(S\), one seeks a regular harmonic function \(U(x,y,z)\), continuous together with its first partial derivatives in \(D+S\) and satisfying the boundary condition

\[ \mathbf P(p,q,r)\cdot \operatorname{grad} U=f,\qquad (x,y,z)\in S, \tag{1} \]

where \(p=x-a,\ q=y,\ r=z\), and \(a=\mathrm{const}\).

In view of the fact that, along with \(U(x,y,z)\), the expression \((x-a)U_x+yU_y+zU_z\) is a regular harmonic function in the domain \(D\), from (1) we have

\[ (x-a)U_x+yU_y+zU_z=V(x,y,z),\qquad (x,y,z)\in D, \tag{2} \]

where \(V(x,y,z)\) is a known regular harmonic function in \(D\), coinciding with \(f\) on \(S\).

Thus, the problem posed above (which in what follows we shall call problem (1)) is reduced to finding regular harmonic solutions in the domain \(D\) of the linear first-order equation (2).

For simplicity, below we shall restrict ourselves to considering the case of the ball \(D\) with boundary \(S: x^2+y^2+z^2=1\). In this case the function \(V(x,y,z)\) is given by the well-known Poisson formula.

First suppose that \(|a|<1\). From the form of equation (2) it is clear that, for the existence of a solution of problem (1), it is necessary that the function \(f\) satisfy the integral condition

\[ V(a,0,0)=\frac{1}{4\pi}\iint_S \frac{1-a^2}{(1+a^2-2a\cos\theta)^{3/2}}\, f\, dS=0. \tag{3} \]

If condition (3) is satisfied, then a regular harmonic solution of equation (2) exists and is given by the formula

\[ U(x,y,z)=\int_0^1 V[a+t(x-a),ty,tz]\,t^{-1}dt+C, \]

where \(C\) is an arbitrary constant. In the case under consideration, problem (1) has no other solutions \(({}^1,{}^2)\).

Thus, for \(|a|<1\), condition (3) is necessary and sufficient for the existence of a solution of problem (1).

Let now \(|a|>1\). In this case all solutions of equation (2) that are twice continuously differentiable in the ball \(D\) are given by the formula

\[ U(x,y,z)= \int_{\frac{a}{a-x}}^1 V[a+t(x-a),ty,tz]\,t^{-1}dt+\varphi(\xi,\eta), \tag{4} \]

where \(\varphi(\xi,\eta)\) is a twice continuously differentiable function of the variables

\[ \xi=\frac{y}{x-a},\qquad \eta=\frac{z}{x-a}. \]

In order that the function \(U(x,y,z)\) represented by formula (4) be a regular solution of Laplace’s equation in the ball \(D\), it is necessary and sufficient that the function \(\varphi(\xi,\eta)\) satisfy the linear elliptic equation of the second order

\[ (1+\xi^2)\varphi_{\xi\xi}+(1+\eta^2)\varphi_{\eta\eta} +2\xi\eta\varphi_{\xi\eta}+2\xi\varphi_\xi+2\eta\varphi_\eta = -\left.\frac{\partial}{\partial \alpha}(1-\alpha) V[\alpha a,(\alpha-1)a\xi,(\alpha-1)a\eta]\right|_{\alpha=0}. \tag{5} \]

Let us note that the direction \(\mathbf P(x-a,y,z)\) goes out into the plane tangent to the sphere \(S\) along the circle
\(x=\frac1a,\ x^2+y^2+z^2=1\). At the same time, when the point \((\xi,\eta)\) runs over the disk
\(\xi^2+\eta^2\le \frac{1}{a^2-1}\), the point \((x,y,z)\) runs over the ball
\(x^2+y^2+z^2\le 1\).

Thus, for \(|a|>1\), problem (1) always has solutions, which can be represented in the form \(U=U_0+\Psi\), where \(U_0\) is a particular solution of the nonhomogeneous equation (5), and \(\Psi(\xi,\eta)\) is the general regular solution of the homogeneous equation

\[ (1+\xi^2)\Psi_{\xi\xi}+(1+\eta^2)\Psi_{\eta\eta} +2\xi\eta\Psi_{\xi\eta}+2\xi\Psi_\xi+2\eta\Psi_\eta=0. \tag{6} \]

The formula \(U=U_0+\Psi\) gives all solutions of problem (1). This follows from the fact that equation (6) has a unique solution, regular in the disk
\(\xi^2+\eta^2<\frac{1}{a^2-1}\), assuming prescribed continuous values on the circle
\(\xi^2+\eta^2=\frac{1}{a^2-1}\), or, what is the same, on the circle
\(x^2+y^2+z^2=1,\ x=\frac1a\) (see \((1^2)\)).

Consequently, for \(|a|>1\) a solution of problem (1) always exists, and it is determined uniquely if its continuous values are prescribed in advance on the set of points of the sphere \(S\) at which the direction of the vector
\(\mathbf P(x-a,y,z)\) goes out into the plane tangent to \(S\).

Problem (1) is investigated analogously also in the case when

\[ p=p_0+ax+by+cz,\qquad q=q_0-bx+ay+dz,\qquad r=r_0-cx-dy+az. \]

It should be noted that, in the case considered above, the Kronecker index characterizing the rotation of the vector field \(\mathbf P(x-a,y,z)\) prescribed on \(S\) is equal to \(+1\) or \(0\) according as \(|a|<1\) or \(|a|>1\). In the case \(|a|=1\), although the concept of the Kronecker index loses its meaning, problem (1) is investigated in exactly the same way as in the case \(|a|<1\).

The degree of overdetermination or underdetermination of the oblique derivative problem (1) for \(\mathbf P\ne 0\) is most closely connected with the Kronecker index of the vector field \(\mathbf P(p,q,r)\) on the surface \(S\). We shall return to this question in another note.

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

Received
3 I 1964

References

¹ A. V. Bitsadze, DAN, 148, No. 4 (1963). ² A. V. Bitsadze, Outlines of the Joint Soviet-American Symposium on Partial Differential Equations, 1963, Novosibirsk, p. 46.