Abstract
Full Text
E. B. Dynkin
NONNEGATIVE SOLUTIONS OF A BOUNDARY-VALUE PROBLEM WITH OBLIQUE DERIVATIVE
(Presented by Academician A. N. Kolmogorov on 27 III 1964)
1. Let \(D\) be a plane domain bounded by a smooth closed contour \(C^*\), and let \(v(z)\) be a Hölder-continuous vector field on \(C\). Our aim is to study all nonnegative harmonic functions \(h\) in the domain \(D\) satisfying the boundary condition
\[ \frac{\partial h}{\partial v}=0. \tag{1} \]
More precisely, the problem is posed as follows. It is assumed that the vector \(v(z)\) is tangent to the contour \(C\) only at a finite number of points. If the projection of \(v(z)\) onto the outward normal to \(C\) changes sign at a point \(\alpha\), then the point \(\alpha\) is called exceptional. The set of all exceptional points is denoted by \(\Gamma\). We shall call solutions of problem A the harmonic functions \(h\) in the domain \(D\) that satisfy condition (1) at all points of the set \(C \setminus \Gamma\). (No restrictions are imposed on the behavior of \(h\) as \(z \to \alpha \in \Gamma\).) We wish to describe all nonnegative solutions of problem A.
2. Let \(s\) be the canonical parameter (arc length) on the contour \(C\), measured from the point \(\alpha \in \Gamma\) in the direction of the vector \(v(\alpha)\). The point of the contour \(C\) corresponding to the value of the parameter \(s\) will be denoted by \(c(s)\). Let \(\theta(s)\) be the angle between \(v[c(s)]\) and \(c'(s)\). The function \(\theta(s)\) changes sign at \(s=0\). Put \(\alpha \in \Gamma_+\) if the sign changes from plus to minus, and \(\alpha \in \Gamma_-\) if it changes from minus to plus. The number of points in the sets \(\Gamma_+\) and \(\Gamma_-\) will be denoted respectively by \(n_+\) and \(n_-\). We shall assume that the function \(\theta(s)\) has, in a neighborhood of zero, a Hölder-continuous derivative \(\theta'(s)\). Put \(\alpha \in \Gamma_+^0\) if \(\alpha \in \Gamma_+\) and \(\chi=\theta'(0)=0\).
Theorem 1. If \(n_+=0\), then problem A has no nonnegative solutions other than constants. If \(n_+>0\), then every nonnegative solution \(h\) of problem A is uniquely represented in the form
\[ h=\sum_{\alpha\in\Gamma_-\cup\Gamma_+^0} a_\alpha u_\alpha +\sum_{\alpha\in\Gamma_+^1}\left(c_\alpha^+p_\alpha^+ + c_\alpha^-p_\alpha^-\right), \]
where \(a_\alpha, c_\alpha^+, c_\alpha^-\) are nonnegative constants, \(u_\alpha\) \((\alpha\in\Gamma_-\cup\Gamma_+^0)\), \(p_\alpha^+\), \(p_\alpha^-\) \((\alpha\in\Gamma_+^1)\) are certain special solutions. The solution \(h\) is bounded if and only if \(a_\alpha=0\) \((\alpha\subset \Gamma_-\cup\Gamma_+^0)\).
We describe the behavior of the special solutions near exceptional points. By means of a conformal mapping one can reduce the general case to the case in which the domain \(D\) is the unit disk. In this disk consider the harmonic functions
\[ \varphi_\alpha(z)=\operatorname{Im}\ln(1-z/\alpha)=\arg(1-z/\alpha)=\operatorname{arc\,tg}(1-x)/y, \]
\[ \psi_\alpha(z)=\operatorname{Re}\ln(1-z/\alpha)=\ln|1-z/\alpha| =\tfrac12\ln\bigl[(1-x)^2+y^2\bigr], \]
\[ \omega_\alpha(z)=\operatorname{Re}(1-z/\alpha)^{-1} =\frac{1-x}{(1-x)^2+y^2}. \]
* It is sufficient to require that the periodic function \(c(t)\) \((-\infty<t<+\infty)\), which determines the contour \(C\), have a Hölder-continuous derivative \(c'(t)\).
\((x+iy=z/a)\). These functions are positive in \(D\) and continuous in \(D\cup C\) everywhere except the point \(a\). We shall agree to write \(f\equiv g\) if the difference \(f-g\) can be represented as the sum of a harmonic function \(h\), continuous in \(D\cup C\), and some linear combination of the functions \(\varphi_\alpha(z)\) \((\alpha\in\Gamma_-)\).
Theorem 2. We have
\[
u_\alpha(z)\equiv a_\alpha\,[\omega_\alpha(z)-\chi\psi_\alpha(z)]
\qquad (\alpha\in\Gamma_-\cup\Gamma_+^0),
\]
\[
p_\alpha^+(z)\equiv -c_\alpha\varphi_\alpha(z),\qquad
p_\alpha^-(z)\equiv c_\alpha\varphi_\alpha(z)
\qquad (\alpha\in\Gamma_+),
\]
where \(a_\alpha\) and \(c_\alpha\) are certain positive constants.
- Martin \((^1)\) indicated a method that makes it possible to describe all nonnegative harmonic functions in an arbitrary domain \(D\) of Euclidean space. In order to obtain the results formulated in § 2, it was necessary to extend Martin’s method to a certain class of boundary-value problems (including problem A). As in Martin’s case, a certain compact extension \(E\) of the domain \(D\) is constructed. The domain \(D\) is everywhere dense in \(E\). The set \(B=E\setminus D\) is called the Martin boundary. To each point \(b\in B\) there corresponds a nonnegative solution \(k_b(z)\) of problem A. If this solution cannot be represented as the sum of two linearly independent nonnegative solutions, the point \(b\) is called minimal. Every nonnegative solution can be represented as an integral over the functions \(k_b(z)\) corresponding to minimal points.
Theorem 3. Suppose \(n_+>0\). The Martin boundary \(B\) for the boundary-value problem A decomposes into connected components \(B_\alpha\) \((\alpha\in\Gamma)\). For \(\alpha\in\Gamma_-\), the component \(B_\alpha\) consists of one minimal point \(b_\alpha\). The corresponding nonnegative solution is proportional to \(u_\alpha\). For \(\alpha\in\Gamma_+\), the component \(B_\alpha\) is a segment. The endpoints \(b_\alpha^+\) and \(b_\alpha^-\) of this segment are minimal points; the corresponding solutions are proportional to \(p_\alpha^+\) and \(p_\alpha^-\). For \(\alpha\in\Gamma_+^0\), some interior point \(b_\alpha\) of the segment \(B_\alpha\) is also minimal; the solution corresponding to it is proportional to \(u_\alpha\).
For \(\alpha\in\Gamma_+\setminus\Gamma_+^0\), all solutions corresponding to points of the segment \(B_\alpha\) are expressed linearly through \(p_\alpha^+\) and \(p_\alpha^-\). For \(\alpha\in\Gamma_+^0\), the solutions corresponding to points of the segment \([b_\alpha^-, b_\alpha]\) are expressed linearly through \(p_\alpha^-\) and \(u_\alpha\), while the solutions corresponding to points of the segment \([b_\alpha, b_\alpha^+]\) are expressed through \(u_\alpha\) and \(p_\alpha^+\).
Introduce notation for the points of the segment \(B_\alpha\) \((\alpha\in\Gamma_+)\). In the case \(\alpha\in\Gamma_+\setminus\Gamma_+^0\), denote by \(b_\alpha^\lambda\) the point corresponding to the solution
\[
c\bigl[(2+\lambda+|\lambda|)p_\alpha^+ + (2+|\lambda|-\lambda)p_\alpha^-\bigr]
\]
(\(c\) is a constant depending on \(\lambda\)). In the case \(\alpha\in\Gamma_+^0\), denote by \(b_\alpha^\lambda\) \((\lambda\ge 0)\) the point of the segment \([b_\alpha,b_\alpha^+]\) corresponding to the solution \(c[u_\alpha+\lambda p_\alpha^+]\), and by \(b_\alpha^{-\lambda}\) the point of the segment \([b_\alpha^-,b_\alpha]\) corresponding to the solution \(c[u_\alpha+\lambda p_\alpha^-]\). (In addition, we agree to regard \(b_\alpha^{+\infty}=b_\alpha^+\), \(b_\alpha^{-\infty}=b_\alpha^-\).)
Let \(\alpha\in\Gamma_+\), and let \(s\) be the canonical parameter introduced in § 2. Let \(n(s)\) be the unit vector directed along the inward normal to the contour \(C\) at the point \(c(s)\), and let \(w(s,t)=c(s)+tn(s)\). Restricting the values of \(s\) and \(t\) to a sufficiently small interval \((-\varepsilon,+\varepsilon)\), we obtain a local coordinate system in some neighborhood of the point \(\alpha\). Put \(\theta(s,t)=\theta(s)\), \(\xi=2\pi st^{-1}|\ln(s^2+t^2)|^{-1}\) for \(\alpha\in\Gamma_+\setminus\Gamma_+^0\), and \(\xi=-2\pi t^{-1}\theta(s,t)\) for \(\alpha\in\Gamma_+^0\).
Theorem 4. If \(\alpha\in\Gamma_-\), then for \(w\) to converge to \(\alpha\) in the Martin topology it is necessary and sufficient that \(w\to\alpha\) in the ordinary topology of the plane. If \(\alpha\in\Gamma_+\), then for \(w\) to converge to \(\alpha\) in the Martin topology it is necessary and sufficient that \(w\to\alpha\) and \(\xi\to\lambda\).
- The results of § 3 are obtained by computing the Green function \(g(z,w)\) of problem A and studying its behavior as \(w\to\alpha\in\Gamma\).
Without loss of generality, one may assume that the domain \(D\) is the unit disk. The computation of the Green function breaks down into the following stages:
1) A pair of analytic functions \(S(z)\) and \(T(z)\) \((z\in D)\) is constructed, regular everywhere in \(D\), except at the point 0, where they may have poles; Hölder-continuous near the boundary \(C\) of the disk \(D\); connected by the relation \(ST=1\) and satisfying the condition: for \(z\in C\), \(S(z)\) differs by a positive factor from \(e^{i\theta}\), where \(\theta\) is the angle between \(\upsilon(z)\) and the positive direction of the tangent to \(C\) at the point \(z\).
2) For each \(\alpha\in\Gamma_+\), a bounded solution \(p_\alpha(z)\) of problem A is constructed, satisfying the conditions: \(p_\alpha(\alpha)=1,\ p_\alpha(\gamma)=0\) for \(\gamma\in\Gamma_+,\ \gamma\ne\alpha\).
3) The Green function \(g(z,w)\) is defined by the formula
\[ g(z,w)=q(z,w)-\sum_{\alpha\in\Gamma_+} q(\alpha,w)p_\alpha(z), \]
where
\[ q(z,w)=\operatorname{Re}\int_0^z T(z)z^{-1}\,[\overline{S(w)}L(z,\overline{w}^{-1})-S(w)L(z,w)]\,dz. \]
Here, if \(n_+\ge n_-\), then
\[ L(z,w)=\frac12\,\frac{z+w}{z-w}. \]
If \(n_->n_+\), then we set \(m=n_- - n_+\), choose some subset \(\widetilde{\Gamma}_-\) of the set \(\Gamma_-\), consisting of \(2m-1\) points, construct for each \(\gamma\in\widetilde{\Gamma}_-\) the function
\[ P_\gamma(w)=\gamma^{m-1}w^{1-m}\prod (w-\beta)(\gamma-\beta)^{-1} \]
(the product is taken over all \(\beta\in\widetilde{\Gamma}_-\) distinct from \(\gamma\)), and define the function \(L(z,w)\) by the formula
\[ L(z,w)=\frac12\,\frac{z+w}{z-w} -\frac12\sum_{\gamma\in\widetilde{\Gamma}_-}P_\gamma(w)\frac{z+\gamma}{z-\gamma}. \]
- In conclusion, let us indicate a probabilistic interpretation of some of the results described above. Consider the Wiener process in the domain \(D\) with reflection in the direction \(\upsilon(z)\) at a boundary point \(z\in C\setminus\Gamma\). We shall assume that the process is terminated as soon as the trajectory reaches one of the points of the set \(\Gamma\). It turns out that motion starting from the point \(z\) is terminated at the point \(\alpha\in\Gamma_+\) with probability \(p_\alpha(z)=p_\alpha^-(z)+p_\alpha^+(z)\). The probability of reaching a point \(\alpha\in\Gamma_-\) is zero. The trajectory can enter \(\alpha\in\Gamma_+\) only by touching the contour \(C\) either from the positive side (the probability of this is \(p_\alpha^+\)) or from the negative side (the probability of this is \(p_\alpha^-\)). These probabilistic conclusions were first obtained by another method in the work of M. B. Malyutov \((^2)\). From Theorem 4 it is not difficult to extract also more precise information concerning the order of contact of the trajectory with the contour \(C\).
Moscow State University
named after M. V. Lomonosov
Received
25 III 1964
REFERENCES
- R. S. Martin, Trans. Am. Math. Soc., 49, 137 (1941).
- M. B. Malyutov, DAN, 156, No. 6 (1964).