Reports of the Academy of Sciences of the USSR

1. Vol. 114, No. 5

MATHEMATICS

V. K. ZAKHAROV

EMBEDDING THEOREMS FOR A SPACE WITH A METRIC DEGENERATING AT A FINITE NUMBER OF INTERIOR POINTS OF A BOUNDED DOMAIN

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

Let \(D'\) be a finite domain situated in the plane of the variables \((x,y)\). Denote by \(\Gamma'\) the closed curve bounding the domain \(D'\), and suppose that Sobolev’s embedding theorems \((^{1})\) hold for it. For simplicity of exposition we shall assume that the origin of coordinates lies inside the domain \(D'\) and that the complete boundary is \(\Gamma=\Gamma' + (0,0)\).

1. Let \(\bar{\Omega}^{0}\) be the manifold of all functions continuous in \(D'\), having bounded piecewise-continuous first derivatives and vanishing in some boundary strip of the domain and in some neighborhood of the point \((0,0)\). The strip and the neighborhood are taken separately for each function. Denote by \(G u^{0}\) the gradient of the function \(u^{0}\in \bar{\Omega}^{0}\):
   \[ G u^{0}=\left(\frac{\partial u^{0}}{\partial x},\frac{\partial u^{0}}{\partial y}\right). \]
   The manifold composed of the elements \(G u^{0}\) will be denoted by \(\bar{R}^{0}\). In \(\bar{R}^{0}\) we introduce a scalar product by the formula
   \[ \{G u^{0},G v^{0}\}=\iint_{D'} \left[ b_{11}\frac{\partial u^{0}}{\partial x}\frac{\partial v^{0}}{\partial x} +b_{12}\frac{\partial u^{0}}{\partial x}\frac{\partial v^{0}}{\partial y} +b_{12}\frac{\partial u^{0}}{\partial y}\frac{\partial v^{0}}{\partial x} + \right. \]
   \[ \left. +b_{22}\frac{\partial u^{0}}{\partial y}\frac{\partial v^{0}}{\partial y} \right]\,dx\,dy, \tag{1} \]
   where the following restrictions are imposed on the coefficients \(b_{11}\), \(b_{12}\), and \(b_{22}\):

1) \(b_{11}\), \(b_{12}\), and \(b_{22}\) are continuous in \(D^{\delta}=D'\cap (r\ge \delta)\), where \(\delta>0\) is arbitrary, \(r=\sqrt{x^{2}+y^{2}}\);

2) either \(b_{11}\) and \(b_{22}\) tend to infinity at the point \((0,0)\), or one of them tends to zero at the point \((0,0)\);

3) for any real numbers \(\xi_{1}\) and \(\xi_{2}\) such that \(\xi_{1}^{2}+\xi_{2}^{2}>0\), the quadratic form
\[ B_{1}(\xi_{1},\xi_{2};x,y)\equiv b_{11}\xi_{1}^{2}+2b_{12}\xi_{1}\xi_{2}+b_{22}\xi_{2}^{2}\ge 0 \tag{2} \]
everywhere in the domain \(\bar{D}'=D'\cup\Gamma\), and the equality sign is attained only at the point \((0,0)\) and in the case when either \(b_{11}\) or \(b_{22}\) tends to zero at this point. Denote by \(\dot{R}\) the closure of the space \(\bar{R}^{0}\) in the metric (1).

It is easy to see that any element \(g\in \dot{R}\) is equal to a system of generalized first partial derivatives of a function \(u\) in the domain \(D'\), i.e.
\[ g=G u=\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}\right). \]

For \(G u,G v\in \dot{R}\), the scalar product \(\{G u,G v\}\) can also be computed by formula (1), where the integral must be understood in the Lebesgue sense. Denote by \(\dot{\Omega}\) the manifold of functions \(u\) obtained as a result of the closure process indicated above. Obviously,
\[ G\dot{\Omega}=\dot{R}. \]

Theorem 1. 1) On the part of the boundary \(\Gamma'\), every function from \(\dot{\Omega}\) has mean value zero.

2) If, for \(\alpha_i \geq 0,\ i=1,2\), the conditions

\[ c^2 r^{\alpha_1} \leq b_{11} \leq C^2 r^{\alpha_1}, \qquad c_1^2 r^{\alpha_2} \leq b_{22} \leq C_1^2 r^{\alpha_2}, \tag{3} \]

are fulfilled, then any function \(f(x,y)\) having bounded piecewise-continuous first derivatives in \(D'\) and vanishing on \(\Gamma'\), with

a) \(\{Gf,Gf\}<+\infty\);

b) \(|f|\leq c^2 r^{-\bar{\alpha}/2}\), where \(\bar{\alpha}=\min(\alpha_1,\alpha_2)>0\); \(|f|\leq c^2|\ln r|^{1/2}\), where \(\bar{\alpha}=0\),

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

3) If, for arbitrary \(\xi_1\) and \(\xi_2\),

\[ (\xi_1^2+\xi_2^2)r^\alpha \leq c^2 B_1(\xi_1,\xi_2;x,y) \tag{4} \]

for \(\alpha<0\), then every function from \(\dot{\Omega}\) vanishes at the point \((0,0)\).

4) If, for \(\alpha\neq 0\), inequality (4) is satisfied, then for any function \(u\in\dot{\Omega}\) the estimate

\[ \iint_{D'} \sigma(x,y)u^2(x,y)\,dx\,dy \leq c^2\{Gu,Gu\}, \]

holds, where \(\sigma(x,y)>0\) for \(x^2+y^2>0\) and is a sufficiently smooth function, with

\[ \sigma(x,y)=O\bigl(r^{\alpha-2}|\ln r|^{-1-\varepsilon_0}\bigr), \qquad \varepsilon_0>0. \]

We note that \(\sigma\) may, for example, be taken equal to the expression under the \(O\)-sign.

It follows from this theorem that, when conditions (3) are satisfied for \(\alpha_i \geq 0\), the space \(\dot{\Omega}\) contains functions vanishing on \(\Gamma'\); at the point \((0,0)\), functions from \(\dot{\Omega}\) may take arbitrary values. If, for \(\alpha_i<0\), condition (4) is satisfied, then all functions from \(\dot{\Omega}\) vanish on \(\Gamma=\Gamma'+(0,0)\). If, for \(\alpha\neq 0\) and \(\alpha<2\), condition (4) is satisfied, then all functions from \(\dot{\Omega}\) are square-summable over the domain \(D'\).

1. Let \(\Omega_1^0\) be the manifold of all continuous functions in \(D'\) having bounded piecewise-continuous second derivatives and vanishing in some boundary strip of the domain \(D'\) and in some neighborhood of the point \((0,0)\). The strip and the neighborhood are chosen individually for each function. On the set of functions \(u^0\in\Omega_1^0\), define an operator of gradient type:

\[ Gu^0=\left(\frac{\partial^2u^0}{\partial x^2},\frac{\partial^2u^0}{\partial x\,\partial y},\frac{\partial^2u^0}{\partial y^2}\right). \]

As in remark (3), introduce the space \(\dot{\Omega}_1\) and the space \(\dot{R}_1\) with metric given by the scalar product

\[ \begin{aligned} \{Gu,Gv\} &=\iint_{D'}\Bigg[ a_{1111}\frac{\partial^2u}{\partial x^2}\frac{\partial^2v}{\partial x^2} +a_{1212}\frac{\partial^2u}{\partial x\,\partial y}\frac{\partial^2v}{\partial x\,\partial y} +a_{2222}\frac{\partial^2u}{\partial y^2}\frac{\partial^2v}{\partial y^2} \\ &\quad+\frac12 a_{1112}\frac{\partial^2u}{\partial x^2}\frac{\partial^2v}{\partial x\,\partial y} +\frac12 a_{1112}\frac{\partial^2u}{\partial x\,\partial y}\frac{\partial^2v}{\partial x^2} +\frac12 a_{1222}\frac{\partial^2u}{\partial x\,\partial y}\frac{\partial^2v}{\partial y^2} +\frac12 a_{1222}\frac{\partial^2u}{\partial y^2}\frac{\partial^2v}{\partial x\,\partial y} \\ &\quad+\frac12 a_{1122}\frac{\partial^2u}{\partial x^2}\frac{\partial^2v}{\partial y^2} +\frac12 a_{1122}\frac{\partial^2u}{\partial y^2}\frac{\partial^2v}{\partial x^2} \Bigg]\,dx\,dy, \end{aligned} \tag{5} \]

where the following restrictions are imposed on the coefficients of the integrand:

1) they are all continuous in the domain \(D'_\delta\), where \(\delta>0\) is arbitrary;

2) either \(a_{1111}, a_{1212}, a_{2222}\) tend to infinity at the point \((0,0)\), or one of them tends to zero at the point \((0,0)\);

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

\[ B(\xi_{11},\xi_{12},\xi_{22};x,y)\equiv \]

\[ \equiv a_{1111}\xi_{11}^{2}+a_{1212}\xi_{12}^{2}+a_{2222}\xi_{22}^{2} +a_{1112}\xi_{11}\xi_{12}+a_{1222}\xi_{12}\xi_{22}+a_{1122}\xi_{11}\xi_{22}\geq 0 \]

everywhere in the domain \(\overline{D'}=D'\cup\Gamma\), and the equality sign can occur only at the point \((0,0)\). It is easy to see that any element \(g\in \dot R_1\) is an operator of gradient type, i.e.,
\[ g=Gu=\left(\frac{\partial^{2}u}{\partial x^{2}},\frac{\partial^{2}u}{\partial y^{2}},\frac{\partial^{2}u}{\partial x\partial y}\right), \]
where the derivatives are understood in the generalized sense, and \(G\dot\Omega_1=\dot R_1\).

Sometimes we shall assume that

\[ c_1^2 r^{\alpha_1}\leq a_{1111}\leq C_1^2 r^{\alpha_1}, \tag{6} \]

\[ c_2^2 r^{\alpha_2}\leq a_{1212}\leq C_2^2 r^{\alpha_2}, \tag{7} \]

\[ c_3^2 r^{\alpha_3}\leq a_{2222}\leq C_3^2 r^{\alpha_3}, \tag{8} \]

\[ 0\leq r^{\alpha_1}\xi_{11}^{2}\leq c^2 B(\xi_{11},\xi_{12},\xi_{22};x,y), \tag{9} \]

\[ 0\leq r^{\alpha_2}\xi_{12}^{2}\leq c^2 B(\xi_{11},\xi_{12},\xi_{22};x,y), \tag{10} \]

\[ 0\leq r^{\alpha_3}\xi_{22}^{2}\leq c^2 B(\xi_{11},\xi_{12},\xi_{22};x,y). \tag{11} \]

Theorem 2. 1) On the part of the boundary \(\Gamma'\), every function from \(\dot\Omega_1\) has mean value zero together with its first derivatives.

2) If, for \(0\leq\alpha<2\), for any real \(\xi_{11},\xi_{12},\xi_{22}\), the condition

\[ (\xi_{11}^{2}+\xi_{12}^{2}+\xi_{22}^{2})\,r^\alpha\leq c^2 B(\xi_{11},\xi_{12},\xi_{22};x,y), \tag{12} \]

is satisfied, then every function \(u\in\dot\Omega_1\) vanishes at the point \((0,0)\).

3) If for \(\alpha<0\) condition (12) is satisfied, then every function \(u\in\dot\Omega_1\) vanishes at the point \((0,0)\) together with its first-order derivatives.

4) If conditions (6), (7), (8), (9), (10), and (11) are satisfied for \(0\leq \alpha_i\), then every function \(f(x,y)\) which has bounded piecewise-continuous second derivatives in \(D'_\delta\) and vanishes on \(\Gamma'\) together with its first derivatives, and such that

a) \(\{Gf,Gf\}<+\infty\);

b) \(|f|\leq c_1^2 r^{(2-\bar\alpha)/2}\) for \(\bar\alpha\neq 2\), \(|f|\leq c_1^2|\ln r|^{1/2}\) for \(\bar\alpha=2\);

c) \(\left|\dfrac{\partial f}{\partial x}\right|\leq c_2^2|\ln r|^{1/2}\), \(\left|\dfrac{\partial f}{\partial y}\right|\leq c_3^2|\ln r|^{1/2}\) for \(\bar\alpha=0\);

d) \(\left|\dfrac{\partial f}{\partial x}\right|\leq c_2^2 r^{-\bar\alpha/2}\), \(\left|\dfrac{\partial f}{\partial y}\right|\leq c_3^2 r^{-\bar\alpha/2}\) for \(\bar\alpha>0\), where \(\bar\alpha=\min(\alpha_1,\alpha_2,\alpha_3)\),

belongs to \(\dot\Omega_1\).

5) If, for \(\alpha\neq 0\), condition (12) is satisfied, then for functions \(u\in\dot\Omega_1\) the estimates

\[ \iint_{D'} \sigma_0(x,y)u^2(x,y)\,dx\,dy\leq c^2\{Gu,Gu\}, \]

\[ \iint_{D'} \sigma_1(x,y)\left(\frac{du}{dx}\right)^2 dx\,dy\leq c_1^2\{Gu,Gu\}, \]

\[ \iint_{D'} \sigma_1(x,y)\left(\frac{du}{dy}\right)^2 dx\,dy\leq c_2^2\{Gu,Gu\}, \]

where \(c^2\), \(c_1^2\), and \(c_2^2\) do not depend on the function \(u\); \(\sigma_0\) and \(\sigma_1\) are sufficiently smooth functions, \(\sigma_0>0\) and \(\sigma_1>0\) for \(x^2+y^2>0\), and

\[ \sigma_0(x,y)= \begin{cases} O\left(r^{\alpha-4}|\ln r|^{-1-\varepsilon_0}\right), & \text{for } \alpha \ne 2,\\ O\left(r^{-2}|\ln r|^{-2-\varepsilon_0}\right), & \text{for } \alpha=2; \end{cases} \]

\[ \sigma_1(x,y)=O\left(r^{\alpha-2}|\ln r|^{-1-\varepsilon_0}\right), \qquad \varepsilon_0>0 \text{ arbitrary}. \]

Assertion 1 follows from the equivalence of the metrics (5) and \(W_2^{(2)}\), \(D'_\delta\), and from the embedding theorems of S. L. Sobolev \((^1)\).

To prove assertion 2, let us extend the functions \(u\in \dot{\Omega}_1\) by zero to the domain \(D'\). We shall say that \(u\in W_2^{(2-\alpha/2)}(D')\) if \(u\) has generalized first-order derivatives \(u_x\) and \(u_y\) satisfying the conditions

\[ \left(\iint_{D'} |u_x(x+\Delta x,y+\Delta y)-u_x(x,y)|^2\,dx\,dy\right)^{1/2} \le c^2(\Delta x^2+\Delta y^2)^{(2-\alpha)/4}, \]

\[ \left(\iint_{D'} |u_y(x+\Delta x,y+\Delta y)-u_y(x,y)|^2\,dx\,dy\right)^{1/2} \le c^2(\Delta x^2+\Delta y^2)^{(2-\alpha)/4}, \]

where \(c^2\) does not depend on \(\Delta x\) and \(\Delta y\). It is shown that if \(u\in \dot{\Omega}_1\), then \(u\in W_2^{(2-\alpha/2)}(D')\). It follows that the function \(u\) will be continuous in the domain \(D'\) \((^2,^4)\), and consequently assertion 2 of the theorem is proved.

To prove assertion 4, introduce the function

\[ \chi_\delta(x,y)= \begin{cases} 0, & 0\le r<\delta,\\ \{1-[(\ln|\ln r|)^\varepsilon-(\ln|\ln\delta_1|)^\varepsilon]^2\}^2, & \delta\le r\le \delta_1,\\ 1, & r>\delta_1, \end{cases} \]

where \((\ln|\ln\delta|)^\varepsilon-(\ln|\ln\delta_1|)^\varepsilon=1\), \(0<\varepsilon<1/2\). Obviously, the function \(f_\delta=f\chi_\delta\in\dot{\Omega}_1\). It is shown that \(Gf_\delta\) converges in the metric (5) to \(Gf\). Hence it follows that \(f\in\dot{\Omega}_1\).

Assertions 3 and 5 are proved by means of ordinary estimates.

On the basis of this theorem one can draw the following conclusions.

If condition (12) is satisfied for \(\alpha<0\), then every function from \(\dot{\Omega}_1\) vanishes, together with its first derivatives, on the entire boundary \(\Gamma\).

If condition (12) is satisfied for \(0\le \alpha<2\), then every function from \(\dot{\Omega}_1\) vanishes, together with its first derivatives, on \(\Gamma'\); at the point \((0,0)\) only the function itself vanishes.

If conditions (6), (7), (8), (9), (10), and (11) are satisfied for \(\alpha_i\ge 0\), then \(\dot{\Omega}_1\) contains functions that vanish, together with their first derivatives, on \(\Gamma'\); at the point \((0,0)\) both the function and its first derivatives may take arbitrary values.

If condition (12) is satisfied, then for \(\alpha<4\) all functions from \(\dot{\Omega}_1\) are square-summable over the domain \(D'\); for \(\alpha<2\) their first derivatives are square-summable over the domain \(D'\).

All the results of the second part of the note are easily generalized to the case when derivatives of order \(m\) enter the scalar product.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
13 XI 1956

REFERENCES

1. S. L. Sobolev, Some applications of functional analysis in mathematical physics, Leningrad, 1950.
2. S. M. Nikol’skii, Mat. sbornik, 33 (75), 261 (1953).
3. V. K. Zakharov, DAN, 114, No. 3 (1957).
4. A. A. Dezin, DAN, 88, No. 5 (1953).