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UDC 517.544
MATHEMATICS
M. B. MALIUTOV
ON THE PROBLEM WITH AN OBLIQUE DERIVATIVE IN THREE-DIMENSIONAL SPACE
(Presented by Academician A. N. Kolmogorov, March 14, 1966)
1. Let \(D\) be a bounded three-dimensional domain of class \(C^3\). On its boundary \(\partial D\) there is given a vector field \(l(x)\) \((x\in \partial D)\) of class \(C^3\), not vanishing at 0. Denote by \(M\) the set of points at which \(l(x)\) lies in the tangent plane \(T_x\) to \(\partial D\).
We assume that: 1) \(M\) consists of a finite number of closed curves of class \(C^2\); 2) if a curve \(\gamma\subset M\), then \(l(x)\) is not tangent to \(\gamma\) at any point of \(\gamma\). Let
\[ L=\sum_{i,j=1}^{3} a_{ij}\,\partial^2/\partial x_i\partial x_j +\sum_{i=1}^{3} b_i\,\partial/\partial x_i \]
be a nondegenerate elliptic operator whose coefficients belong to \(C^3(\overline D)\), where \(\overline D=D\cup \partial D\). Consider the following boundary-value problem A: \(LF=f\) in \(D\), \(f\in C^3(\overline D)\), and \(dF/dl=0\) on \(\partial D\setminus M\). By a solution of it we shall mean a bounded function \(F(x)\), smooth in \(D\), such that \(LF=f\) in \(D\), \(F\in C^1(\overline D\setminus M)\) and \(dF/dl=0\) for \(x\in \partial D\setminus M\).
In the paper it is shown that the solution of problem A is uniquely determined by its limiting values on a certain part of the set \(M\). An existence theorem for a solution of problem A is also proved for a broad class of boundary values.
For some special cases, smooth solutions of the problem with an oblique derivative were studied by A. V. Bitsadze and A. Janashiauskas. When the present paper was being submitted for publication, I became aware of the general results of the work of Yu. V. Egorov and V. A. Kondrat’ev \((^2)\) on the problem with an oblique derivative. But they too are restricted to finding smooth solutions of problem A, which in the general case does not make it possible to obtain existence theorems. The question of selecting, among the solutions of problem A that are continuous in \(D\) and smooth in \(D\), remains open.
The three-dimensional case is considered only for simplicity; analogous arguments can also be carried out for a space of \(n\) dimensions.
2. Let \(n(x)\) be the vector of the inner normal to \(\partial D\) at the point \(x\). Introduce the vector field \(l_1(x)=l(x)\), if \((l(x),n(x))>0\); \(l_1(x)=-l(x)\) in the opposite case.
Let \(M_\delta\) be the \(\delta\)-neighborhood of \(M\) and \(S_\delta=M_\delta\cap \partial D\). If \(\delta\) is sufficiently small, then \(S_\delta\) consists of disjoint neighborhoods \(\gamma_\delta\) of the curves \(\gamma\) (relative to \(\partial D\)), and the domain \(\gamma_\delta\) is divided by the curve \(\gamma\) into parts \(\gamma_\delta^i\) \((i=1,2)\). We split each point \(a\) of the curve \(\gamma\) into points \(a^i\) \((i=1,2)\) and shall call the \(\varepsilon\)-neighborhood \((\varepsilon<\delta)\) of the point \(a^i\) the set of points from \(\gamma_\delta^i\) that are removed from the point \(a\) by a distance less than \(\varepsilon\). We shall denote by
\[ \lim_{x\to a}\Phi(x) \]
or simply by \(\Phi(a^i)\) the limiting value of the function \(\Phi(x)\) at the point \(a^i\). The collection of points \(a^i\) \((a\in\gamma)\) will be called the side \(\gamma^i\) of the curve \(\gamma\). Let \(n_i(a)\) be the vector of the normal to \(\gamma\) at the point \(a\), lying in the plane \(T_a\) and directed toward \(\gamma_\delta^i\). From condition 2) it follows that the scalar product \((l_1(a^i), n^i(a))\ne 0\), and the sign of the scalar product
\[ \text{* In the natural metric defined by the operator } L. \]
is constant for all points of the side \(\gamma^i\). We shall call the side \(\gamma^i\) of the curve \(\gamma\) positive (respectively negative) if the scalar product is negative (positive). If both sides of \(\gamma^i\) are positive, then \(\gamma\) is called positive (negative); if the sides of \(\gamma^i\) have different signs, then it is called zero. Let us renumber the special curves and denote by \(\gamma_k^i\) the \(i\)-th side of the \(k\)-th special curve. The set of points of the positive sides will be denoted by \(M_+\).
The homogeneous problem A \((f=0)\) will be called problem \(\mathrm{A}_0\).
Theorem. Suppose \(M_+\) is nonempty. Assign to each positive side \(\gamma_k^i\) a continuous function \(\varphi_k^i(a)\), \(a\in\gamma_k\). There exists a unique solution \(F(x)\) of problem A such that, for all \(\alpha^i\in\gamma_k^i\),
\[ \lim_{x\to \alpha^i} F(x)=\varphi_k^i(\alpha). \]
If \(M_+\) is empty, then the solutions of problem \(\mathrm{A}_0\) are only constants.*
Remark. It follows easily from the theorem that if for all points \(\alpha\) of a positive curve \(\gamma_k\) one has \(\varphi_k^1(\alpha)=\varphi_k^2(\alpha)\), then the solution \(F(x)\) is continuous in a three-dimensional neighborhood of \(\gamma_k\). The solution is discontinuous, generally speaking, at each point of the negative curves, even if the \(\varphi_k^i\) are arbitrarily smooth functions.
3. Let \(U\subset D\) be a smooth domain, and on a closed set \(G\) with smooth boundary \(\partial G\) (in the topology of \(\partial U\)) let there be given a smooth vector field \(\lambda(x)\), not lying in the tangent plane to \(\partial U\). We shall say that a function \(\Phi(x)\) is a solution of the mixed boundary-value problem C for the domain \(U\) if \(L\Phi=f\) in \(U\), and on \((\partial U\setminus G)\cup \partial G\) the Dirichlet condition \(\Phi(x)=\varphi(x)\) is prescribed, while on \(G\setminus \partial G\),
\[ d\Phi/d\lambda=0. \]
The theorem on existence and uniqueness of the solution of problem C for a sufficiently smooth function \(\varphi\) follows, for example, from (5). By passage to the limit one can prove the theorem of existence and uniqueness for more general boundary values \(\varphi\), in particular for discontinuous boundary functions \(\varphi\), used below. The proof of the theorem is obtained by passage to the limit from problem C for the domain \(\partial D\), in which \(\bar S_\delta\) is taken as \(G\). Let us first consider problem \(\mathrm{A}_0\).
Let
\[ U=D\cap\{x:\|x-a\|<\varepsilon\}, \]
where \(a_1\) is a point of the curve \(\gamma\) having the positive side \(\gamma^1\), and let
\[ \Gamma_\delta=\gamma_\delta^1\cap \partial U \qquad (\delta<\varepsilon). \]
Consider the solution \(U_\delta\) of problem \(\mathrm{C}_0\) (i.e., problem C with \(f=0\)) for the domain \(U\) with the following boundary conditions:
\[ U_\delta=1 \text{ on } \Gamma_\delta;\qquad U_\delta=0 \text{ on } \partial U\setminus \gamma_\varepsilon^1;\qquad du_\delta/dl=0 \text{ on } \gamma_\varepsilon^1\setminus \Gamma_\delta . \]
As \(\delta\to0\) the functions \(u_\delta\) decrease. This follows at once from the following lemma (see (3), p. 14).
Lemma 1. If \(L\Phi=0\), the function \(\Phi(x)\) \((x\in U)\) is not constant, attains a minimum at a point \(\xi\in\partial U\), and in some neighborhood of the point \(\xi\) is differentiable, then for any vector \(l\) forming an acute angle with the inner normal at \(\xi\),
\[ d\Phi/dl>0 \]
at the point \(\xi\).
Denote
\[ u_0=\lim_{\delta\to0} u_\delta . \]
Lemma 2. \(u_0(\alpha^1)=1\).
Let us note that it is enough to prove Lemma 2 for an arbitrarily small domain \(U\). In its proof a device close to the known method of “freezing” the coefficients is used, thanks to which the problem is essentially reduced to a two-dimensional one and is solved analogously to the two-dimensional case (see (1), proof of Theorem 1).
Let \(\Delta\) be a segment of the curve \(\gamma\) having the positive side \(\gamma^1\), and
\[ \Delta^1=\{\alpha^1;\ \alpha\in\Delta\}. \]
From Lemma 2 follows the existence of a solution \(F_\Delta\) of problem \(\mathrm{A}_0\) satisfying the conditions: the function \(F_\Delta(\alpha)\) is equal to \(0\) on \(M_+\setminus\Delta^1\), and \(F_\Delta(\alpha^1)=1\), if \(\alpha^1\) is an interior point of \(\Delta^1\). Indeed, let \(\mu_\Delta\) be the set of points of \(\gamma_\delta^1\) whose distance to the segment \(\Delta\) is less than to \(M_+\setminus\Delta\). Consider the solution \(\Phi_\delta\) of problem \(\mathrm{C}_0\) for the domain \(\partial D\) with conditions:
\[ \Phi_\delta=1 \text{ on } \mu_\Delta;\qquad \Phi_\delta=0 \text{ on } S_\delta-\mu_\Delta . \]
It is clear that inside a sufficiently small—
* If \(M_+\) is empty, then the solution of problem A exists under an additional condition on the right-hand side of the same type as in the Neumann problem.
neighborhood \(U\) of an interior point \(a \in \Delta\), the solution \(\Phi_\delta\) will be greater than the solution constructed in Lemma 2, and inside a sufficiently small neighborhood \(W\) of the point \(\beta \in M_+ \setminus \Delta^1\) it will be less than \(1-w_\delta\), where \(w_\delta\) is the solution constructed in Lemma 2 for \(W\).
Therefore, contracting first \(\mu_\Delta\) to the segment \(\Delta\), and then \(S_\delta \setminus \mu_\Delta\) to \(M \setminus \Delta^1\), we obtain a function \(F_\Delta\) with the required boundary values, satisfying, by virtue of the monotonicity of passage to the limit, the equation \(L F_\Delta=0\) in \(D\). It is not difficult to prove that \(F_\Delta\) also satisfies the condition \(dF_\Delta/dl=0\) on \(\partial D \setminus M\).
Using the functions \(F_\Delta\), it is easy to prove, by means of the same monotone limiting passage, that there exists a solution of problem \(A_0\) with arbitrary continuous boundary values*.
- Consider problem \(C_\delta\) with the same boundary conditions as in Lemma 2, but now let \(\gamma^1\) be the negative side. Denote the corresponding solution by \(v_\delta\).
Lemma 3. \(v_0=\lim_{\delta\to 0} v_\delta \equiv 0\).
If the field of conormal vectors to \(\partial D\) is taken as the vector field \(l(x)\), then this result is well known. We shall prove that the solution \(v_\delta\) in the case of the negative side \(\gamma^1\) is less than the corresponding solution \(v_\delta^1\) for the same domain, but for conormal vectors.
It is easy to verify that \(d(v_\delta^1-v_\delta)/dl \le 0\), if the domain \(U\) is chosen sufficiently small. Therefore, by Lemma 1, the minimum of \(v_\delta^1-v_\delta\) is attained on \(\Gamma_\delta \cup (\partial U-\gamma_\varepsilon^1)\), where the function \(v_\delta^1-v_\delta\) is equal to 0 by assumption.
An even stronger result is true. Let again \(\gamma^1\) be the negative side and let \(w_\delta\) be the solution of problem \(C_0\) with boundary conditions \(w_\delta=1\) on \(\Gamma_\delta \cup (\gamma_\varepsilon^2 \cap \partial U)\); \(w_\delta=0\) on \(\partial U \setminus (\gamma_\varepsilon^1 \cup \gamma_\varepsilon^2)\); \(dw_\delta/dl=0\) on \((\gamma_\varepsilon^1 \cap \partial U)\setminus \Gamma^\delta\).
Let \(w_0=\lim_{\delta\to 0} w_\delta\).
Lemma 4. \(\displaystyle \lim_{x\to a^1} w_0(x)<1.\)
Let the domain \(U\) and the field \(l_1\) be the same as in Lemma 3, but let arbitrary smooth functions be prescribed on \(\Gamma_\delta\) and \(\partial U \setminus \gamma_\varepsilon^1\). Then the solution of the \(\Gamma_\delta\)-problem \(C_0\) with such boundary conditions converges, as \(\delta\to 0\), to a function \(F(x)\), and, by Lemma 3, the extremum of the function \(F(x)\) is attained on \(\partial U \setminus \gamma_\varepsilon^1\). Hence, by Lemma 1, the maximum principle follows: if the solution \(F(x)\) of problem \(A_0\) is continuous in \(M_+\), then the extremum of \(F(x)\) is attained on \(M_+\). From this maximum principle the uniqueness of the solution of problem A follows at once.
Using the results of the next item, one can prove a stronger formulation of the maximum principle. Let \(F(x)\) be a function bounded from below, satisfying in \(D\) the equation \(LF=0\), \(dF/dl=0\) on \(\partial D \setminus M\), and \(\lim_{x\to M_+} F(x) \ge 0\). Then \(F(x)\ge 0\).
- In proving the existence of a particular solution of the nonhomogeneous problem A, considerations based on the probabilistic interpretation of the problem with oblique derivative are used. We start from the diffusion process \(X^\delta\), constructed by M. I. Freidlin in (5), with reflection along the field \(l(x)\), governed inside \(D\) by the operator \(L\), and terminated at the moment \(\xi^\delta\) of hitting \(S_\delta\). In the book (4) a method is given for constructing such a process \(X\), such that the trajectories of \(X^\delta\) coincide with the trajectories of \(X\) up to the moment \(\zeta^\delta\), with \(\zeta^\delta \uparrow \zeta\), where \(\zeta\) is the termination time of the process \(X\).
Lemma 1, in its probabilistic interpretation, means that the probability of the trajectory \(X_t\) of the process \(X\) hitting the curve \(\gamma\), starting from the point \(x \in \overline U\) before exit to \(\partial U \setminus \gamma_\varepsilon^1\), tends to 1 if \(x\to a^1\). Indeed, the function \(u_\delta\) is the probability of hitting \(\Gamma_\delta\), starting from \(x\), before exit to \(\partial U \setminus \gamma_\varepsilon^1\).
* In this way one can obtain a solution of problem \(A_0\) for a measurable bounded function as the “boundary values,” but, as in the Dirichlet problem, the question of the behavior of the solution near the boundary is more complicated.
It is not difficult to prove that the mathematical expectation \(m_\delta\) of the time \(\tau_\delta\) of reaching \(\Gamma_\delta \cup (\partial U \setminus \gamma^{\varepsilon_1})\) is bounded uniformly in \(\delta\).
Therefore the time \(\tau = \lim_{\delta\to0} \tau_\delta\) is finite with probability 1, and the trajectory up to the time \(\tau\) is a compact set. Since, with probability \(u_0(x)=\lim_{\delta\to0} u_\delta(x)\), the trajectory falls into any of the sets \(\Gamma_\delta\), it follows, by the compactness of the trajectory, that with probability \(u_0(x)\) the trajectory \(x_t\) falls on \(\gamma\). Let now \(\zeta_u=\tau\) if the trajectory has reached \(\gamma\), and otherwise \(\zeta_u=\infty\). From Lemma 1 it follows that, for some \(T>0\),
\[ \lim_{x\to \alpha^1} P_x\{\zeta_u<T\}\beta>0. \]
Hence it is easily derived that the mathematical expectation of \(\zeta\) is bounded over all initial points from \(\overline D\). Further, just as in (5), it is proved that if \(f(x)\in C^3(\overline D)\), then
\[ F(x)=M_x\int_0^\zeta f(x_t)\,dt \]
satisfies the equation
\[ LF=-f \]
and the boundary conditions \(dF/dl=0\), while \(F(x)\) tends to 0 when the point \(x\) approaches \(M_+\).
Thus, a particular solution of problem A has been constructed.
Moscow State University
named after M. V. Lomonosov
Received
10 III 1966
References
- M. B. Malyutov, DAN, 156, No. 6 (1964).
- Yu. V. Egorov, V. A. Kondrat’ev, DAN, 170, No. 4 (1966).
- C. Miranda, Partial Differential Equations of Elliptic Type, Moscow, 1957.
- E. B. Dynkin, Foundations of Markov Processes, Moscow, 1961.
- M. I. Freidlin, Dissertation, V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 1962.