V. SOLONNIKOV

ON LINEAR DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER AT THE HIGHEST DERIVATIVES

(Presented by Academician V. I. Smirnov on 1 XI 1957)

In a number of works the behavior of solutions of various partial differential equations having, at the highest derivatives, a “small parameter” \(\varepsilon\), as \(\varepsilon \to 0\), is investigated. In Levinson’s paper \(\left({}^{1}\right)\) it is proved that the solutions of the Dirichlet problem

\[ \varepsilon\left(\frac{\partial^{2}u^{\varepsilon}}{\partial x^{2}}+\frac{\partial^{2}u^{\varepsilon}}{\partial y^{2}}\right) +A(x,y)\frac{\partial u^{\varepsilon}}{\partial x} +B(x,y)\frac{\partial u^{\varepsilon}}{\partial y} +C(x,y)u^{\varepsilon}=D(x,y), \tag{1} \]

\[ u^{\varepsilon}\big|_{S}=\varphi \tag{2} \]

tend, as \(\varepsilon \to 0\), to the solution of the “degenerate” equation

\[ A\frac{\partial u}{\partial x}+B\frac{\partial u}{\partial y}+Cu=D, \tag{3} \]

which preserves the value \(\varphi\) on a certain part of the boundary.

In papers \(\left({}^{3-5}\right)\), in the same direction as in \(\left({}^{1}\right)\), the second and third boundary-value problems for equation (1) and the first boundary-value problem for an equation of parabolic type are considered. In \(\left({}^{6}\right)\) Levinson’s restrictions on the regularity of the arrangement of the characteristics of equation (3) are removed. In papers \(\left({}^{7,8}\right)\) a “boundary layer” is constructed for equations of elliptic type under the assumption that the solution of the degenerate problem exists and is unique.

We proceed from the work of O. A. Ladyzhenskaya \(\left({}^{9}\right)\), in which another approach to such problems is given, based on integral estimates and the introduction of generalized solutions, which made it possible for her in some problems to avoid consideration of the boundary layer. This turned out to be possible, for example, in the problem

\[ \sum_{k=1}^{r}\varepsilon_{k}L_{2m_{k}}u^{\varepsilon}+L_{2m_{0}}u^{\varepsilon}=f, \qquad m_{k}>m_{k-1}, \]

\[ u^{\varepsilon}\big|_{S} =\frac{\partial u^{\varepsilon}}{\partial n}\bigg|_{S} =\cdots= \frac{\partial^{m_{r}-1}u^{\varepsilon}}{\partial n^{m_{r}-1}}\bigg|_{S} =0, \]

where \(L_{2m_{k}}\) are strongly elliptic operators of order \(2m_{k}\), and \(u\) is a vector-function. However, in the problem

\[ \sum_{i,j=1}^{n}\varepsilon\frac{\partial}{\partial x_{i}}a_{ij}(x)\frac{\partial u^{\varepsilon}}{\partial x_{j}} +\sum_{i=1}^{n}a_{i}\frac{\partial u^{\varepsilon}}{\partial x_{i}} +au^{\varepsilon}=f, \qquad u^{\varepsilon}\big|_{S}=0 \]

in paper \(\left({}^{9}\right)\), in order to prove the convergence of \(u^{\varepsilon}\) to \(u\), certain pointwise considerations of boundary-layer type are carried out.

In the present paper we prove the convergence of \(u^{\varepsilon}\) to a certain solution \(u\) of the degenerate problem for a number of linear problems, relying only on some new integral estimates. In contrast to preceding works, we do not use for this the ability to solve the degenerate problem and, on the contrary, show its solvability by means of passage to the limit as \(\varepsilon \to 0\). In this way it has been possible to prove the unique solvability of some-

second boundary-value problem for certain classes of differential equations of odd order (see Sec. 1).

The consideration is divided into two stages: 1) obtaining a priori estimates which make it possible to conclude that from \(\{u^\varepsilon\}\) one can extract a subsequence converging to some function \(u\), which may be regarded as a generalized solution of the degenerate problem, and 2) proving the uniqueness of such a generalized solution, which makes it possible to conclude that the entire sequence \(\{u^\varepsilon\}\) converges to \(u\) as \(\varepsilon \to 0\). For the case in which the elliptic equation degenerates into an equation of odd order, the second stage proved to be the most difficult.

A priori estimates for the problems of Sec. 1 with \(m_r=1\) and \(2\), Secs. 2 and 3, were also obtained independently of us by M. Sh. Birman.

1. Consider the problem

\[ L^\varepsilon u^\varepsilon \equiv \sum_{k=1}^{r}\varepsilon_k L_{2m_k}u^\varepsilon +\sum_{i=1}^{n} a_i \frac{\partial L_{2m_0}u^\varepsilon}{\partial x_i} +M_{2m_0}u^\varepsilon=f, \tag{4} \]

\[ u^\varepsilon\big|_S= \frac{\partial u^\varepsilon}{\partial n}\bigg|_S =\cdots= \frac{\partial^{m_r-1}u^\varepsilon}{\partial n^{m_r-1}}\bigg|_S=0; \tag{5} \]

\(L_{2m_k}, L_{2m_0}, M_{2m_0}\) are elliptic operators of orders \(2m_k\) and \(2m_0\):

\[ L_{2m_0}u = (-1)^{m_0} \frac{\partial^{m_0}}{\partial x_{\alpha_1}\cdots \partial x_{\alpha_{m_0}}} \, b_{\alpha_1\ldots \alpha_{m_0};\,\alpha_{m_0+1}\ldots \alpha_{2m_0}}\, \frac{\partial^{m_0}u}{\partial x_{\alpha_{m_0+1}}\cdots \partial x_{\alpha_{2m_0}}} + \sum_{p=0}^{2m_0-1} b_{\alpha_1\ldots \alpha_p} \frac{\partial^p u}{\partial x_{\alpha_1}\cdots \partial x_{\alpha_p}}; \]

\[ M_{2m_0}u = (-1)^{m_0} \frac{\partial^{m_0}}{\partial x_{\alpha_1}\cdots \partial x_{\alpha_{m_0}}} \, c_{\alpha_1\ldots \alpha_{m_0};\,\alpha_{m_0+1}\ldots \alpha_{2m_0}}\, \frac{\partial^{m_0}u}{\partial x_{\alpha_{m_0+1}}\cdots \partial x_{\alpha_{2m_0}}} + \sum_{p=0}^{2m_0-1} c_{\alpha_1\ldots \alpha_p} \frac{\partial^p u}{\partial x_{\alpha_1}\cdots \partial x_{\alpha_p}}; \]

summation over repeated indices is everywhere carried out from \(1\) to \(n\).

Suppose that problem (4)—(5) has a solution in \(W_2^{2m_r}(\Omega)\) for every \(f\) from some set dense in \(L_2(\Omega)\), and that the estimate

\[ \sum_{k=1}^{r}\varepsilon_k \|u^\varepsilon\|_{W_2^{m_k}(\Omega)}^2 + \|u^\varepsilon\|_{W_2^{m_0}(\Omega)}^2 \leq C_1\|f\|_{L_2(\Omega)}^2, \tag{6} \]

holds, where \(C_1\) does not depend on \(\varepsilon_k\). We shall assume that \(a_i\) and the coefficients entering into \(L_{2m_k}, L_{2m_0}\), and \(M_{2m_0}\) are sufficiently smooth functions. The following lemma is known:

Lemma 1. If
\[ (M_{2m_0}u,u)_{L_2(\Omega)} \geq \beta_0\|u\|_{W_2^{m_0}(\Omega)}^2 \quad \text{for } u\in W_2^{2m_0}(\Omega)\cap \overset{\circ}{W}{}_{2}^{m_0}(\Omega), \]
\(\beta_0>0\), and \(\beta_0\) is a sufficiently large number, then (6) holds.

Denote by \(S_1\) that part of \(S\) where \(a_i\cos nx_i<0\); by \(S_2\), that part where \(a_i\cos nx_i=0\), and by \(S_3\), that part where \(a_i\cos nx_i>0\).

Lemma 2. If (6) is valid, then also

\[ \sum_{i,\alpha_1\ldots \alpha_{m_0}=1}^{n} \int_{\Omega} \left( a_i \frac{\partial^{m_0+1}u^\varepsilon} {\partial x_{\alpha_1}\cdots \partial x_{\alpha_{m_0}}\partial x_i} \right) \zeta^2(x)\,dx \leq C_2\|f\|_{L_2(\Omega)}^2, \tag{7} \]

where \(\zeta(x)\) is a function continuously differentiable \(m_r\) times, equal to zero on \(S_3\); \(\zeta(x)\geq 0\).

(7) is obtained from consideration of the identity

\[ \int_{\Omega} L^\varepsilon u^\varepsilon \zeta^2 a_k \frac{\partial u^\varepsilon}{\partial x_k}\,dx = \int_{\Omega} f\zeta^2 a_k \frac{\partial u^\varepsilon}{\partial x_k}\,dx. \]

Using (6) and (7) and replacing (4) and (5) by an integral identity, we prove Theorem 1.

Theorem 1. There exists a subsequence \(\{u^{\varepsilon_k}\}\), converging weakly in the metric

\[ [u,v]_{\overset{\circ}{H}{}^{m_0}(\zeta)} = (u,v)_{\overset{\circ}{W}{}^{m_0}_2(\Omega)} + \int_\Omega a_i \frac{\partial^{m_0+1}u}{\partial x_i\partial x_{\alpha_1}\ldots \partial x_{\alpha_{m_0}}} a_k \frac{\partial^{m_0+1}v}{\partial x_k\partial x_{\alpha_1}\ldots \partial x_{\alpha_{m_0}}} \zeta^2\,dx \]

to a function \(u\) satisfying the integral identity

\[ \begin{aligned} \int_\Omega \Bigg[& b_{\alpha_1\ldots \alpha_{2m_0}}a_i \frac{\partial^{m_0+1}u}{\partial x_i\partial x_{\alpha_{m_0+1}}\ldots \partial x_{\alpha_{2m_0}}} \zeta \frac{\partial^{m_0}\Phi}{\partial x_{\alpha_1}\ldots \partial x_{\alpha_{m_0}}} +\\ &+a_i \frac{\partial b_{\alpha_1\ldots \alpha_{2m_0}}}{\partial x_i} \frac{\partial^{m_0}u}{\partial x_{\alpha_{m_0+1}}\ldots \partial x_{\alpha_{2m_0}}} \frac{\partial^{m_0}\Phi}{\partial x_{\alpha_1}\ldots \partial x_{\alpha_{m_0}}} \zeta -\\ &-b_{\alpha_1\ldots \alpha_{2m_0}} \frac{\partial^{m_0}u}{\partial x_{\alpha_{m_0+1}}\ldots \partial x_{\alpha_{2m_0}}} \frac{\partial}{\partial x_i} \left( \frac{\partial^{m_0}a_i\zeta\Phi}{\partial x_{\alpha_1}\ldots \partial x_{\alpha_{m_0}}} - a_i\zeta \frac{\partial^{m_0}\Phi}{\partial x_{\alpha_1}\ldots \partial x_{\alpha_{m_0}}} \right) +\\ &+\Phi\zeta a_i\frac{\partial}{\partial x_i} \sum_{p=0}^{m_0-1} b_{\alpha_1\ldots \alpha_p} \frac{\partial^p u}{\partial x_{\alpha_1}\ldots \partial x_{\alpha_p}} +\\ &+\sum_{p=m_0}^{2m_0-1} (-1)^{p-m_0+1} \frac{\partial^{m_0}u}{\partial x_i\partial x_{p-m_0+2}\ldots \partial x_{\alpha_p}} \frac{\partial^{p-m_0+1}b_{\alpha_1\ldots \alpha_p}\Phi\zeta} {\partial x_{\alpha_1}\ldots \partial x_{\alpha_{p-m_0+1}}} +\\ &+c_{\alpha_1\ldots \alpha_{2m_0}} \frac{\partial^{m_0}u}{\partial x_{\alpha_{m_0+1}}\ldots \partial x_{\alpha_{2m_0}}} \frac{\partial^{m_0}\Phi\zeta}{\partial x_{\alpha_1}\ldots \partial x_{\alpha_{m_0}}} + \zeta\Phi\sum_{p=0}^{m_0-1} c_{\alpha_1\ldots \alpha_p} \frac{\partial^p u}{\partial x_{\alpha_1}\ldots \partial x_{\alpha_p}} +\\ &+\sum_{p=m_0}^{2m_0-1} (-1)^{p-m_0} \frac{\partial^{m_0}u}{\partial x_{\alpha_{p-m_0+1}}\ldots \partial x_{\alpha_p}} \frac{\partial^{p-m_0}\Phi\zeta c_{\alpha_1\ldots \alpha_p}} {\partial x_{\alpha_1}\ldots \partial x_{\alpha_{p-m_0}}} \Bigg]\,dx = \int_\Omega f\zeta\Phi\,dx . \tag{8} \end{aligned} \]

for any \(\Phi\in \overset{\circ}{W}{}^{m_0}_2(\Omega)\).

Here by

\[ a_i \frac{\partial^{m_0+1}u}{\partial x_i\partial x_{\alpha_1}\ldots \partial x_{\alpha_{m_0}}} \]

is meant the weak limit in \(L_2(\Omega)\) with weight \(\zeta^2(x)\) of the same expression for \(u^{\varepsilon_k}\). The space \(\overset{\circ}{H}{}^{m_0}(\zeta)\) is obtained by closure, in the metric \(\overset{\circ}{H}{}^{m_0}(\zeta)\), of functions from \(\overset{\circ}{W}{}^{m_0+1}_2(\Omega)\). Every function \(w\in \overset{\circ}{H}{}^{m_0}(\zeta)\) satisfies (in the generalized sense) the boundary conditions

\[ w\big|_S = \frac{\partial w}{\partial n}\bigg|_S = \cdots = \frac{\partial^{m_0-1}w}{\partial n^{m_0-1}}\bigg|_S = \frac{\partial^{m_0}w}{\partial n^{m_0}}\bigg|_S = 0 . \tag{9} \]

The limiting function \(u\) belongs to \(\overset{\circ}{H}{}^{m_0}(\zeta)\).

If \(u\) had all derivatives entering the degenerate operator \(L\), then almost everywhere in the domain under consideration one would have \(Lu=f\), since identity (8) is formally obtained by integration by parts from \(\int Lu\cdot\Phi\,dx=\int f\Phi\,dx\). Therefore it is natural to call every function \(u\) from \(\overset{\circ}{H}{}^{m_0}(\zeta)\) satisfying identity (8) a generalized solution of the degenerate problem

\[ Lu\equiv a_i\frac{\partial L_{2m_0}u}{\partial x_i}+M_{2m_0}u=f, \tag{10} \]

\[ u\big|_S = \frac{\partial u}{\partial n}\bigg|_S = \cdots = \frac{\partial^{m_0-1}u}{\partial n^{m_0-1}}\bigg|_S = \frac{\partial^{m_0}u}{\partial n^{m_0}}\bigg|_{S_1} = 0 . \tag{11} \]

Theorem 2. If the function \(v \in \overset{0}{H}{}^{m_0}(\zeta)\) satisfies (8) for \(f=0\), then \(v=0\).

In proving this theorem we impose restrictions on the behavior of the characteristics near the boundary.

Let us list results pertaining to some other equations with a small parameter. They are also obtained on the basis of simple a priori estimates.

1. Solutions of the mixed problem for an equation of parabolic type

\[ \frac{\partial u^\varepsilon}{\partial t} -\varepsilon L_2 u +a_i\frac{\partial u^\varepsilon}{\partial x_i} +a u^\varepsilon=f,\qquad u^\varepsilon\big|_{t=0}=\varphi(x),\qquad u^\varepsilon\big|_S=0, \]

defined in the cylinder \(Q_l=\Omega\times[0,l]\), with lateral surface \(S\), converge weakly in \(\overset{0}{H}{}'(\zeta)\) to the generalized solution of the reduced equation, preserving the prescribed value on that part of the boundary where \(\operatorname{const}+a\cos nx_i<0\), i.e. on the lower base and on part of \(S\). The function \(\zeta\), entering into the definition of \(H'(\zeta)\), is equal to zero on that part of \(S\) where \(\cos nt+a_i\cos nx_i>0\).

1. Solutions of the Dirichlet problem for the equation

\[ -\Delta u^\varepsilon-\varepsilon u^\varepsilon_{tt} +u^\varepsilon_t +a_i\frac{\partial u^\varepsilon}{\partial x_i} +a u^\varepsilon=f,\qquad u^\varepsilon\big|_{\Gamma}=0, \]

defined in the cylinder \(Q_l=\Omega\times[0,l]\) with surface \(\Gamma\), converge weakly in the metric

\[ \{u,v\}= \int_{Q_l} \left( uv+ \frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_i} +\zeta\frac{\partial u}{\partial t}\frac{\partial v}{\partial t} \right)\,dx\,dt \]

to the generalized solution of the mixed problem for the reduced parabolic equation

\[ \frac{\partial u}{\partial t} -\Delta u +a_i\frac{\partial u}{\partial x_i} +a u=f,\qquad u\big|_{t=0}=0,\qquad u\big|_S=0. \]

Instead of \(-\Delta u^\varepsilon-\varepsilon u^\varepsilon_{tt}\) one may put

\[ -\sum_{i,j=0}^{n} \frac{\partial}{\partial x_i} a^\varepsilon_{ij} \frac{\partial u^\varepsilon}{\partial x_j}, \qquad \text{where } a^\varepsilon_{0j}=\varepsilon a_{0j}(x), \]

\(j=0,\ldots,n,\quad a^\varepsilon_{ij}=a_{ij}(x)\) for \(i,j=1,\ldots,n\).

1. Solutions of the mixed problem for an equation of hyperbolic type

\[ \varepsilon L_{2r}u^\varepsilon+u^\varepsilon_{tt}-L_2u^\varepsilon=f, \]

\[ u^\varepsilon\big|_S =\frac{\partial u^\varepsilon}{\partial n}\bigg|_S =\cdots =\frac{\partial^{r-1}u^\varepsilon}{\partial n^{r-1}}\bigg|_S =0,\qquad u^\varepsilon\big|_{t=0}=\varphi_0(x),\qquad u^\varepsilon_t\big|_{t=0}=\varphi_1(x) \]

(\(L_{2r}\) is an elliptic operator of order \(2r\)), defined in the cylinder \(Q=\Omega\times[0,l]\) with lateral surface \(S\), converge weakly in \(\overset{0}{W}{}^r_2(Q)\) to the generalized solution of the mixed problem for the reduced hyperbolic equation

\[ \frac{\partial^2 u}{\partial t^2} -L_2u=f,\qquad u\big|_S=0,\qquad u\big|_{t=0}=\varphi_0(x),\qquad u_t\big|_{t=0}=\varphi_1(x). \]

I take this opportunity to express my deep gratitude to Prof. O. A. Ladyzhenskaya, under whose supervision the present work was carried out.

Received
24 X 1957

REFERENCES

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