MATHEMATICS

V. K. ZAKHAROV

EMBEDDING THEOREMS FOR A SPACE WITH A METRIC DEGENERATING ON A RECTILINEAR PART OF THE BOUNDARY OF THE DOMAIN

(Presented by Academician S. L. Sobolev on 16 XI 1956)

Let \(D\) be a finite domain located in the upper half-plane and having a part of the boundary \(\Gamma_0\) on the \(Ox\) axis. We shall denote the remaining part of the boundary by \(\Gamma_1\). In this case the complete boundary is \(\Gamma=\Gamma_0\cup\Gamma_1\). We assume that \(\Gamma_1\) is such that the embedding theorems of S. L. Sobolev \((^1)\) hold for it.

Let \(\Omega^0\) be the manifold of all functions continuous in the domain \(D\), having bounded piecewise-continuous second derivatives and vanishing in some boundary strip of the domain \(D\) (their own for each function). On the set of functions \(u^0\in\Omega^0\) we define a gradient-type operator:
\[ Gu^0=\left(\frac{\partial^2 u^0}{\partial x^2},\frac{\partial^2 u^0}{\partial x\,\partial y},\frac{\partial^2 u^0}{\partial y^2}\right). \]
Thus, the operator \(G\) assigns to the function \(u^0\in\Omega^0\) the system of all its partial derivatives of second order. The manifold composed of the elements \(Gu^0\), where \(u^0\in\Omega^0\), will be denoted by \(R^0\).

Introduce in \(R^0\) the scalar product by the formula:
\[ \{Gu^0,Gv^0\}= \iint_D\left[ a_{1111}\frac{\partial^2u^0}{\partial x^2}\frac{\partial^2v^0}{\partial y^2} +a_{1212}\frac{\partial^2u^0}{\partial x\,\partial y}\frac{\partial^2v^0}{\partial x\,\partial y} +a_{2222}\frac{\partial^2u^0}{\partial y^2}\frac{\partial^2v^0}{\partial y^2} +\right. \]
\[ \left. +\frac12 a_{1112}\frac{\partial^2u^0}{\partial x^2}\frac{\partial^2v^0}{\partial x\,\partial y} +\frac12 a_{1112}\frac{\partial^2u^0}{\partial x\,\partial y}\frac{\partial^2v^0}{\partial x^2} +\frac12 a_{1222}\frac{\partial^2u^0}{\partial y^2}\frac{\partial^2v^0}{\partial x\,\partial y} +\right. \]
\[ \left. +\frac12 a_{1222}\frac{\partial^2u^0}{\partial x\,\partial y}\frac{\partial^2v^0}{\partial y^2} +\frac12 a_{1122}\frac{\partial^2u^0}{\partial x^2}\frac{\partial^2v^0}{\partial y^2} +\frac12 a_{1122}\frac{\partial^2u^0}{\partial y^2}\frac{\partial^2v^0}{\partial x^2} \right]dx\,dy, \tag{1} \]
where the following restrictions are imposed on the coefficients \(a_{1111}, a_{1112}, a_{1122}, a_{1212}, a_{1222}, a_{2222}\):

1) all of them are continuous in the closed domain \(\overline D=D\cup\Gamma\);

2) either \(a_{1111}\to0\) as \(y\to0\), or \(a_{2222}\to0\) as \(y\to0\);

3) for any real numbers \(\xi_{11}, \xi_{12}, \xi_{22}\) such that \(\xi_{11}^2+\xi_{12}^2+\xi_{22}^2>0\), the quadratic form
\[ B(\xi_{11},\xi_{12},\xi_{22};x,y)\equiv a_{1111}\xi_{11}^2+a_{1212}\xi_{12}^2+a_{2222}\xi_{22}^2 +a_{1112}\xi_{11}\xi_{12} +a_{1222}\xi_{12}\xi_{22} +a_{1122}\xi_{11}\xi_{12}>0 \tag{2} \]
everywhere in the domain \(\overline D\), with the equality sign attained only at points \((x,y)\in\Gamma_0\). We introduce the metric in \(R^0\) in the usual way, through the scalar product, and shall call it “metric (1).”

Denote by \(\dot R\) the closure of the space \(R^0\) in metric (1). By virtue of the equivalence of metric (1) and the metric \(W_2^{(2)}\) in \(D^\delta=D\cap(y\geq\delta)\), \(\delta>0\),

and also on the basis of the inequality

\[ \iint_{D^\delta} (u^0)^2\,dx\,dy \leq C^2 \iint_{D^\delta} \left[ \left(\frac{\partial^2 u^0}{\partial x^2}\right)^2 +2\left(\frac{\partial^2 u^0}{\partial x\,\partial y}\right)^2 +\left(\frac{\partial^2 u^0}{\partial y^2}\right)^2 \right]\,dx\,dy, \tag{3} \]

which follows from the embedding theorems of S. L. Sobolev \((^1)\), it follows that in every domain \(D^\delta\) the functions \(u_n^0\in\Omega^0\) converge in the mean to a certain function \(u\), which has generalized second derivatives in this domain. Hence it follows that the element \(g\in \dot R\) is equal to the system of generalized derivatives of second order (in the sense of S. L. Sobolev) of the function \(u\) in the domain \(D^\delta\).

Since the conclusion is valid for any domain \(D^\delta\), where \(\delta>0\) is arbitrary, it follows from this that the element \(g\in\dot R\) is equal to the system of generalized partial derivatives of second order of the function \(u\) in the domain \(D\), i.e.,

\[ g=Gu=\left(\frac{\partial^2 u}{\partial x^2},\, \frac{\partial^2 u}{\partial x\,\partial y},\, \frac{\partial^2 u}{\partial y^2}\right). \]

For \(Gu,\ Gv\in\dot R\), the scalar product \(\{Gu,Gv\}\) can also be computed by formula (1), where the integral must be understood in the sense of Lebesgue. Denote by \(\dot\Omega\) the manifold of all functions \(u\) obtained as a result of the above-described completion process. Obviously, \(G\dot\Omega=\dot R\), and consequently \(\dot\Omega\) is the domain of definition of the gradient-type operator \(G\).

Sometimes we shall assume that

\[ c^2 y^\alpha \leq a_{2222}\leq C^2 y^\alpha; \tag{4} \]

\[ \bar c^2 y^\beta \leq a_{1212}\leq \bar C^2 y^\beta; \tag{5} \]

and also that, for any real \(\xi_{11},\xi_{12},\xi_{22}\),

\[ 0\leq y^\alpha \xi_{22}^2\leq C_1^2 B(\xi_{11},\xi_{12},\xi_{22};\,x,y); \tag{6} \]

\[ 0\leq y^\beta \xi_{12}^2\leq C_2^2 B(\xi_{11},\xi_{12},\xi_{22};\,x,y); \tag{7} \]

\[ 0\leq a_{1111}\xi_{11}^2\leq C_3^2 B(\xi_{11},\xi_{12},\xi_{22};\,x,y). \tag{8} \]

Theorem 1. 1) On the part of the boundary \(\Gamma_1^\delta=\Gamma_1\cap (y\geq\delta)\), where \(\delta>0\) is arbitrary, every function from \(\dot\Omega\) takes, in the mean, the value zero together with its first derivatives.

2) If condition (6) is satisfied for \(0\leq\alpha<1\) and (7) for \(0\leq\beta<1\), then on \(\Gamma_0\) every function from \(\dot\Omega\) takes, in the mean, the value zero together with its first derivatives.

3) If condition (7) is satisfied for \(0\leq\beta<1\), then on \(\Gamma_0\) every function from \(\dot\Omega\) takes, in the mean, the value zero together with the first derivative with respect to \(x\).

4) If condition (6) is satisfied for \(0\leq\alpha<1\), then on \(\Gamma_0\) every function from \(\dot\Omega\) takes, in the mean, the value zero together with the first derivative with respect to \(y\).

5) If condition (6) is satisfied for \(1\leq\alpha<3\), then every function \(u\in\dot\Omega\) takes, in the mean, the value zero on \(\Gamma_0\).

6) If for \(\alpha\geq1\), \(\beta\) arbitrary, or for \(\beta\geq1\), \(\alpha\) arbitrary, conditions (4), (5), (6), (7), and (8) are satisfied, then any function \(\varphi(x,y)\) that has bounded piecewise-continuous second derivatives in \(D^\delta\), where \(\delta>0\) is arbitrary, and vanishes together with its first derivatives on \(\Gamma_1\), moreover:

a) \(\{G\varphi,G\varphi\}<+\infty\);

b) \(|\varphi|\leq C^2 y^{\frac{3-\alpha}{2}}\) for \(\alpha\neq3\); \(|\varphi|\leq c^2|\ln y|^{1/2}\) for \(\alpha=3\);

c) \(\left|\dfrac{\partial\varphi}{\partial x}\right|\leq C_1^2 y^{\frac{1-\beta}{2}}\) for \(\beta\neq1\); \(\left|\dfrac{\partial\varphi}{\partial x}\right|\leq C_1^2|\ln y|^{1/2}\) for \(\beta=1\);

c) \(\left|\dfrac{\partial\varphi}{\partial y}\right|\leq C_2^2 y^{\frac{1-\alpha}{2}}\) for \(\alpha\neq 1\); \(\left|\dfrac{\partial\varphi}{\partial y}\right|\leq C_2^2|\ln y|^{1/2}\) for \(\alpha=1\),

belongs to \(\dot{\Omega}\).

The proof of item 1 follows from the embedding theorems of S. L. Sobolev \(\left({}^{1}\right)\) and the equivalence of the metrics (1) and \(W_2^{(2)}\) in \(D^\delta\).

The proofs of items 2, 3, 4, and 5 do not essentially differ from one another. We shall therefore restrict ourselves to proving item 5 for \(1<\alpha<3\). Extend the function \(u\in\dot{\Omega}\) by zero into the region \((y>0)\setminus D\), and let the number \(A\) be so large that the point \((x,A)\) lies outside the domain \(D\).

For functions \(u\in\Omega^0\) the following estimate holds:
\[ [u(x,y)]^2\leq C^2 y^{3-\alpha}\int_0^A y^\alpha\left(\frac{\partial^2 u}{\partial y^2}\right)^2\,dy. \tag{9} \]

Integrating both sides of inequality (9) over \(\Gamma_h=D\cap (y=h)\) and using (6), we shall have:
\[ \int_{\Gamma_h} u^2\,d\Gamma_h \leq C_1^2 h^{3-\alpha}\{Gu,Gu\}. \tag{10} \]

By means of passage to the limit we verify the validity of estimate (10) for almost all \(h\) for any function \(u\in\dot{\Omega}\). This implies item 5 of the theorem for \(1<\alpha<3\).

For the proof of item 6 of the theorem, introduce the function
\[ \chi_\delta=\chi_\delta(y)= \begin{cases} 0, & 0\leq y<\delta,\\ \{1-[(\ln|\ln y|)^\varepsilon-(\ln|\ln\delta_1|)^\varepsilon]^2\}^2, & \delta\leq y\leq\delta_1,\\ 1, & y>\delta_1, \end{cases} \]
where
\[ (\ln|\ln\delta|)^\varepsilon-(\ln|\ln\delta_1|)^\varepsilon=1,\qquad 0<\varepsilon<1/2. \]
It is obvious that the function \(\varphi_\delta=\varphi\chi_\delta\in\dot{\Omega}\). It is shown that \(G\varphi_\delta\) converges in the metric (1) to \(G\varphi\). Hence it will follow that \(G\varphi\in\dot{R}\), \(\varphi\in\dot{\Omega}\).

It follows from the theorem that, when conditions (6) and (7) are satisfied, for \(0\leq\alpha<1,\ 0\leq\beta<1\), \(\dot{\Omega}\) contains functions that vanish together with their first derivatives on the entire boundary \(\Gamma=\Gamma_1\cup\Gamma_0\). When conditions (4), (5), (6), (7), and (8) are satisfied, \(\dot{\Omega}\) contains functions: in the case \(\alpha\geq 1,\ 0\leq\beta<1\), equal to zero on \(\Gamma\) together with the first derivative with respect to \(x\); the derivatives with respect to \(y\) of these functions vanish only on \(\Gamma_1\), while on \(\Gamma_0\) they may tend to infinity; in the case \(0\leq\alpha<1,\ \beta\geq 1\), equal to zero on \(\Gamma\) together with the derivative with respect to \(y\); the derivative with respect to \(x\) of these functions vanishes on \(\Gamma_1\), while on \(\Gamma_0\) it may or may not vanish; in the case \(1\leq\alpha<3,\ \beta\geq 1\), equal to zero on \(\Gamma\), whose first derivatives with respect to \(x\) and \(y\) vanish on \(\Gamma_1\); the first derivatives on \(\Gamma_0\) may tend to infinity; in the case \(\alpha\geq 3,\ \beta\geq 1\), equal to zero on \(\Gamma_1\) together with the first derivatives and tending to infinity together with the first derivatives as \(y\to 0\).

Vanishing is understood in the mean, in the sense of S. L. Sobolev.

Theorem 2. If inequalities (6) and (7) are satisfied, then for functions \(u\in\dot{\Omega}\) the following estimates hold:
\[ \iint_D \sigma_0(x,y)u^2(x,y)\,dx\,dy\leq C^2\{Gu,Gu\}; \]
\[ \iint_D \sigma_1(x,y)\left(\frac{\partial u}{\partial y}\right)^2\,dx\,dy\leq C_1^2\{Gu,Gu\}; \]

\[ \iint_D \sigma_2(x,y)\left(\frac{\partial u}{\partial x}\right)^2\,dx\,dy \leq C_2^2\{Gu,Gu\}, \]

where \(C^2, C_1^2, C_2^2\) do not depend on the function \(u\); \(\sigma_i(x,y)\), \(i=0,1,2\), are sufficiently smooth functions; \(\sigma_i(x,y)>0\) for \(y>0\), and

\[ \sigma_0(x,y)= \begin{cases} O\!\left(y^{\alpha-4}|\ln y|^{-1-\varepsilon_0}\right), & \text{for } \alpha\neq 1,\ \alpha<3,\ \beta \text{ arbitrary};\\ O\!\left(y^{-3}|\ln y|^{-2-\varepsilon_0}\right), & \text{for } \alpha=1,\ \beta \text{ arbitrary};\\ O\!\left(y^{\beta-2}|\ln y|^{-1-\varepsilon_0}\right), & \text{for } \alpha\geq 3,\ \beta<1;\\ O\!\left(y^{-1}|\ln y|^{-2-\varepsilon_0}\right), & \text{for } \alpha\geq 3;\ \beta=1;\ \alpha=3,\ \beta\geq 1;\\ O\!\left(y^{\alpha-4}|\ln y|^{-1-\varepsilon_0}\right)\ \text{or}\ O\!\left(y^{\beta-2}|\ln y|^{-1-\varepsilon_0}\right), & \text{for } \alpha>3,\ \beta>1; \end{cases} \]

\[ \sigma_1(x,y)= \begin{cases} O\!\left(y^{\alpha-2}|\ln y|^{-1-\varepsilon_0}\right), & \text{for } \alpha\neq 1;\\ O\!\left(y^{-1}|\ln y|^{-2-\varepsilon_0}\right), & \text{for } \alpha=1; \end{cases} \]

\[ \sigma_2(x,y)= \begin{cases} O\!\left(y^{\beta-2}|\ln y|^{-1-\varepsilon_0}\right), & \text{for } \beta\neq 1;\\ O\!\left(y^{-1}|\ln y|^{-2-\varepsilon_0}\right), & \text{for } \beta=1, \end{cases} \qquad \varepsilon_0>0 \text{ and arbitrary.} \]

It follows from Theorem 2 that, in the case \(\alpha<4\), for any \(\beta\geq 0\), all functions \(u\in\dot{\Omega}\), under condition (6), are square-summable over the domain \(D\); if \(\alpha<2\), then, under condition (6), the first derivatives with respect to \(y\) are also square-summable over the domain \(D\); if \(\beta<2\), for any \(\alpha\geq 0\), then all functions \(u\in\dot{\Omega}\), under condition (7), are square-summable over the domain \(D\) together with the first derivatives with respect to \(x\).

We note that there are examples showing that the order of the weights \(\sigma_i(x,y)\) is exact with the degree of accuracy indicated in [2].

All the results of the present note are easily generalized to the case when derivatives of order \(m\) participate in the scalar product, and the coefficients and derivatives depend on \(n\) variables.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
13 XI 1956

References

1. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
2. M. I. Vishik, Matem. sborn., 35 (77), 513 (1954).