UDC 517.946.2

MATHEMATICS

Yu. V. EGOROV, V. A. KONDRAT’EV

ON A PROBLEM WITH AN OBLIQUE DERIVATIVE

(Presented by Academician I. G. Petrovskii on 6 I 1966)

In a bounded closed domain \(G\) with smooth boundary \(\Gamma\) in \(n\)-dimensional space, we consider the elliptic equation

\[ Lu \equiv \sum_{i,j=1}^{n} a_{ij}(x)\frac{\partial^2 u}{\partial x_i \partial x_j} +\sum_{i=1}^{n} b_i(x)\frac{\partial u}{\partial x_i} +c(x)u=f(x) \tag{1} \]

with the boundary condition

\[ \left.\frac{\partial u}{\partial \mu}\right|_{\Gamma} =\left.\sum_{j=1}^{n}\mu_j(x)\frac{\partial u}{\partial x_j}\right|_{\Gamma} =g(x). \tag{2} \]

The coefficients of the equation and the functions \(\mu_j(x)\) are assumed, for simplicity, to be infinitely differentiable.

In the case where the direction of the field \(\mu\) is nowhere tangent to the boundary, the problem under consideration has been well studied. We shall allow the field \(\mu\) to be tangent to the boundary \(\Gamma\) at points of certain smooth manifolds \(\Gamma_i\) \((i=0,1,\ldots,k)\), having dimension \(n-2\). Let \(P\) be an arbitrary point of the manifold \(\Gamma_i\). By means of a transformation of the independent variables one can ensure that this point is the origin of coordinates and that, in some neighborhood of the point \(P\), the domain \(G\) is defined by the inequality \(x_n \geq 0\), while the manifold \(\Gamma_i\) is given by the equation \(x_n=x_{n-1}=0\). In the new coordinates condition (2) has the form

\[ \left.\sum_{j=1}^{n}\nu_j(x)\frac{\partial u}{\partial x_j}\right|_{\Gamma} =g(x). \tag{2'} \]

In the work \((^1)\) L. Hörmander showed that in the case where the principal part of the operator \(L\) is the Laplace operator and

\[ \sum_{1}^{n}\nu_j\frac{\partial \nu_n}{\partial x_j} \leq r\nu_n \]

with some smooth function \(r\), there exists a solution \(u\in H_s(G)\) of problem (1)—(2), if \(f\in H_{s-1}(G)\) and \(g\in H_{s-1/2}(\Gamma)\). If, however,

\[ \sum_{j=1}^{n}\nu_j(x)\frac{\partial \nu_n}{\partial x_j} \geq r\nu_n, \]

then every solution \(u\) of such a problem which is equal to zero outside a sufficiently small neighborhood of a boundary point belongs to the space \(H_s(G)\). Moreover, he showed that if at points where \(\nu_n=0\),

\[ \sum_{1}^{n}\nu_j\frac{\partial \nu_n}{\partial x_j}<0, \]

then problem (1)—(2) has an infinite-dimensional kernel, while if

\[ \sum_{1}^{n}\nu_j\frac{\partial \nu_n}{\partial x_j}>0, \]

then it has an infinite-dimensional cokernel.

Some special cases of problem (1)—(2) were considered in the works of A. V. Bitsadze \((^{2-4})\) and Janusauskas \((^5)\).

1. We shall assume that the field \(\mu\) at the points of the manifolds \(\Gamma_i\) is not tangent to these manifolds. In addition, for simplicity we shall assume that there is only one manifold \(\Gamma_0\) on which the field is tangent to the boundary \(\Gamma\). In view of our conditions, \(v_{n-1}(x) \ne 0\) on \(\Gamma_0\). We may therefore assume that \(v_{n-1}(x)>0\). The properties of the solutions of problem (1)—(2) depend essentially on the sign of the function \(v(Q)=x_{n-1}v_n\) in a neighborhood of the point \(P\).

2. Theorem 1. If everywhere in a neighborhood of \(P\) the function \(v(Q)\le 0\), then:

a) for \(u(x)\in H_s(G)\) the estimate
\[ \|u\|_s^G \le C\left(\|f\|_{s-1}^G+\|g\|_{s-1/2}^{\Gamma}+\|u\|_0^G\right) \qquad (s>1/2); \tag{3} \]

b) the space of solutions of the homogeneous problem (1)—(2) belonging to \(H_s(G)\) has finite dimension;

c) the intersection of the range of the operator
\[ (L,\partial/\partial v):\quad H_s(G)\to H_{s-2}(G)\times H_{s-3/2}(\Gamma) \]
with the space \(H_{s-1}(G)\times H_{s-1/2}(\Gamma)\) is closed;

d) if \(f\in H_s(G)\), \(g\in H_{s+1/2}(\Gamma)\), then every solution \(u\) of problem (1)—(2) from \(H_s(G)\) belongs to the space \(H_{s+1}(G)\).

The following example shows the impossibility, in the case under consideration, of obtaining estimates for the solution of problem (1)—(2) with the left-hand side of inequality (3) replaced by \(\|u\|_{s+\delta}^G\) for \(\delta>0\).

Let the boundary of a three-dimensional domain \(G\) in a neighborhood of the origin coincide with the plane \(x_3=0\). Consider the sequence of functions
\[ u_m(x_1,x_2,x_3)=e^{(ix_2-x_3)m}\varphi\left(rm^{1/(p+1)}\right), \]
where the function
\[ \varphi(r)=\varphi\left(\sqrt{x_1^2+x_2^2+x_3^2}\right)\in C^\infty(G) \]
vanishes outside this neighborhood. It is easy to verify that, as \(m\to+\infty\),
\[ \|u_m\|_s^G \ge C m^{s-1/2-1/(p+1)}, \]
\[ \|\Delta u_m\|_{s-1}^G=O\left(m^{s-1/2}\right), \]
\[ \left\|x_1^p\,\partial u_m/\partial x_3+\partial u_m/\partial x_1\right\|_{s-1/2}^{\Gamma} =O\left(m^{s-1/2}\right). \]

1. Let us now consider the case when \(v(Q)\ge 0\) in a neighborhood of \(\Gamma_0\). In this case the homogeneous problem (1)—(2) has an infinite-dimensional space of solutions. Therefore we shall require that the solution of problem (1)—(2) satisfy the additional condition
   \[ u|_{\Gamma_0}=u_0(x). \tag{4} \]

Theorem 2. If \(v(Q)\ge 0\) in a neighborhood of \(\Gamma_0\), then:

a) for \(u(x)\in H_s(G)\) the estimate
\[ \|u\|_s^G\le C\left(\|f\|_{s-1}^G+\|g\|_{s-1/2}^{\Gamma}+\|u_0\|_{s-1/2}^{\Gamma}+\|u\|_0^G\right) \tag{5} \]
holds for all \(s>1\);

b) the space of solutions of the homogeneous problem (1), (2), (4) belonging to \(H_s(G)\), \(s>1\), is finite-dimensional;

c) the intersection of the range of the operator
\[ u\mapsto (Lu,\partial u/\partial v|_{\Gamma},u|_{\Gamma_0}), \]
acting from \(H_s(G)\) to
\[ H_{s-2}(G)\times H_{s-3/2}(\Gamma)\times H_{s-1}(\Gamma_0), \]
with the space
\[ H_{s-1}(G)\times H_{s-1/2}(\Gamma)\times H_{s-1/2}(\Gamma_0) \]
is closed;

d) if \(f\in H_s(G)\), \(g\in H_{s+1/2}(\Gamma)\), \(u_0\in H_{s+1/2}(\Gamma_0)\), then every solution of problem (1), (2), (4) from \(H_s(G)\) belongs to the space \(H_{s+1}(G)\);

e) if \(f\in H_{s-1}(G)\), \(g\in H_{s-1/2}(\Gamma)\), \(u_0\in H_{s-1/2}(\Gamma_0)\), and the vector \((f,g,u_0)\) is orthogonal to some finite-dimensional subspace in
\[ H_{s-1}(G)\times H_{s-1/2}(\Gamma)\times H_{s-1/2}(\Gamma_0), \]
then there exists a solution \(u\) of problem (1), (2), (4) belonging to the space \(H_s(G)\).

As follows from the next example, in the left-hand side of inequality (5) one cannot replace \(\|u\|_s^G\) by \(\|u\|_{s+\delta}^G\) for \(\delta>0\).

Example. In the domain \(G=\{(x_1,x_2,x_3):\ x_1^2+x_2^2+x_3^{2p}\leqslant 1\}\) \((p\geqslant 1)\) the sequence of harmonic functions

\[ u_m=(x_1+ix_2)^m x_3 \]

satisfies, as \(m\to\infty\), the inequalities

\[ \|u\|_s^G \geqslant c m^{s-1/2-3/4p}, \]

\[ \|\partial u/\partial x_3\|_{s-1/2}^{\Gamma} =O\bigl(m^{s-1/2-1/4p}\bigr), \]

\[ \|u_0\|_{s-1/2}^{\Gamma}=0. \]

1. Let now \(\nu(Q)\) change sign as \(Q\) passes through \(\Gamma_0\) (i.e., the function \(\nu_n(x)\) has a constant sign in a neighborhood of \(\Gamma_0\)).

Theorem 3. If \(\nu(Q)\) changes sign when passing through \(\Gamma_0\), then:

a) for functions \(u(x)\in H_s(G)\) the estimate

\[ \|u\|_s^G \leqslant C\left(\|f\|_{s-1}^G+\|g\|_{s-1/2}^{\Gamma}+\|u\|_0^G\right)\qquad (s>1/2); \tag{6} \]

holds;

b) the space of solutions of the homogeneous problem (1)—(2) belonging to \(H_s(G)\) is finite-dimensional;

c) the intersection of the range of the operator \((L,\partial/\partial \nu): H_s(G)\to H_{s-2}(G)\times H_{s-3/2}(\Gamma)\) with the space \(H_{s-1}(G)\times H_{s-1/2}(\Gamma)\) is closed;

d) if \(f\in H_{s-1}(G)\), \(g\in H_{s-1/2}(\Gamma)\), then every solution \(u\) of problem (1)—(2) from \(\dot H_s(G)\) belongs to the space \(H_{s+1}(G)\);

e) if \(f\in H_{s-1}(G)\), \(g\in H_{s-1/2}(\Gamma)\), and the vector \((f,g)\) is orthogonal to a certain finite-dimensional subspace in \(H_{s-1}(G)\times H_{s-1/2}(\Gamma)\), then there exists a solution \(u\) of problem (1)—(2) belonging to the space \(H_s(G)\).

Estimate (6) cannot be improved. This follows from the example in item 3, if the parameter \(p\) takes even values.

1. The main point of the proof consists in studying the solution in a sufficiently small neighborhood of a point \(P\) belonging to \(\Gamma_0\). In this neighborhood we consider the function \(w=\partial u/\partial \mu^*\), where the field \(\mu^*\) is a local extension of the field \(\mu\) given on the boundary. The function \(w\) satisfies a certain linear equation, “close” to equation (1), and boundary conditions on \(\Gamma\) of Dirichlet type. Using the results of the theory of elliptic boundary-value problems, we obtain the required estimates for the function \(w\), and further, depending on the structure of the field \(\mu\) in a neighborhood of the point \(P\), obtain the corresponding theorems for the solution of problem (1)—(2).

Moscow State University
named after M. V. Lomonosov

Received
3 I 1966

CITED LITERATURE

1. L. Hörmander, Pseudodifferential operators and nonelliptic boundary problems, preprint, 1965.
2. A. V. Bitsadze, DAN, 148, No. 4, 749 (1963).
3. A. V. Bitsadze, DAN, 155, No. 4, 730 (1964).
4. A. V. Bitsadze, DAN, 157, No. 6, 1273 (1964).
5. A. Janushauskas, DAN, 164, No. 4, 753 (1965).