Mathematics

V. P. MIKHAILOV

ON THE EXTENSION OF FUNCTIONS

(Presented by Academician I. G. Petrovskii, 28 X 1961)

In the theory of boundary-value problems for partial differential equations, an essential role is played by the question of the possibility of extending a function \(u(x)\), \(x=(x_1,\ldots,x_n)\), given in some bounded domain \(Q\), to the whole space while preserving its smoothness. A function \(u^*(x)\), defined on the whole space \(R_n\), will be called an extension from the domain \(Q\) of the function \(u(x)\), if \(u^*(x)=u(x)\) for \(x\in Q\). When \(u(x)\) belongs to the Sobolev class \(W_p^{(l)}(Q)\) \((^1)\) for some integer \(l\), and the boundary \(\Gamma\) of the domain \(Q\) is sufficiently smooth, its extension \(u^*(x)\) to \(R_n\), belonging to the class \(W_p^{(l)}(R_n)\), was constructed by V. M. Babich \((^2)\) by the methods of Whitney and Hestenes \((^3)\). It was also proved that there exists a constant \(C\), independent of the function \(u(x)\), such that

\[ \left\|u^*\right\|_{W_p^{(l)}(R_n)} \leq C \left\|u\right\|_{W_p^{(l)}(Q)}. \tag{1} \]

L. N. Slobodetskii \((^4)\) investigated an analogous question on the extension of functions from the class \(W_{(x_1,\ldots,x_n),2}^{(l_1,\ldots,l_n)}(Q)\), where the \(l_i\) are arbitrary positive numbers, to the whole space in the case when \(Q\) is a domain of special form. If all \(l_i\), \(i=1,\ldots,n\), are distinct, then the domain \(Q\) in \((^4)\) is assumed to be a parallelepiped with faces parallel to the coordinate planes. If \(\{l_i,\ i=1,\ldots,n\}\) are divided into \(k\) groups of numbers equal among themselves
\(l_1=l_2=\cdots=l_{s_1}\), \(l_{s_1+1}=\cdots=l_{s_2},\ldots,l_{s_{k-1}+1}=\cdots=l_n\), then the domain \(Q\) in \((^4)\) is the intersection of cylinders \(\Omega_1,\ldots,\Omega_k\),

\[ Q=\prod_{r=1}^{k} Q_r, \]

where \(\Omega_r\) is a cylinder with generators parallel to the coordinate planes
\(x_{s_{r-1}+1}=x_{s_{r-1}+2}=\cdots=x_{s_r}=0\), and with a sufficiently smooth directrix surface lying in the plane \(\{x_{s_{r-1}+1},\ldots,x_{s_r}\}\). In \((^5)\), in solving a certain boundary-value problem for a parabolic equation of order \(2p\), an extension of a function \(u(t,x_1,\ldots,x_n)\) from \(W_{(t,x_1,\ldots,x_n),2}^{(1,2p,\ldots,2p)}(Q)\) to the whole space was carried out with preservation of the norm. In this case the domain \(Q\) in \((^5)\) (if, for simplicity, it is assumed convex) is bounded by a smooth surface \(\Gamma\), which with the tangent planes \(t=t_B\) and \(t=t_H\), \(t_H=\inf\{t,\ (x_1,\ldots,x_n,t)\in Q\}\), \(t_B=\sup\{t,\ (x_1,\ldots,x_n,t)\in Q\}\), has contact of at least order \(2p\).

In the present note the question is considered of extending, from a two-dimensional domain \(Q\), a function \(u(x,y)\) belonging to the space \(W_{(x,y),2}^{(m,n)}(Q)\) \((^4)\) and satisfying certain boundary conditions, to the whole plane with preservation of the norm. For simplicity in the formulation of the result we assume the domain \(Q\) convex, and \(m<n\) (the case \(m=n\) was considered in \((^2,^4)\)).

Theorem. Let \(u(x,y)\in W_{(x,y),2}^{(m,n)}(Q)\), \(u|_{\Gamma}=D_xu|_{\Gamma}=\cdots=D_x^{[m]-1}u|_{\Gamma}=0\), where \(\Gamma\) is a sufficiently smooth boundary of the domain \(Q\). If, in neighborhoods of the points \(\Lambda=(x_\Lambda,y_\Lambda)\) and \(\Pi=(x_\Pi,y_\Pi)\), \(x_\Lambda=\inf\{x,\ (x,y)\in\Gamma\}\), \(x_\Pi=\sup\{x,\ (x,y)\in\Gamma\}\), the equation \(\Gamma\) can be represented in the form \(x-x_\Lambda=O(|y-y_\Lambda|^{n/m})\), \(x_\Pi-x=\)

\[ = O\left(|y-y_{\Pi}|^{n/m}\right), \]
respectively, then there exists a function \(u^*(x,y)\), which is an extension of the function \(u(x,y)\) to the whole plane \(R\) and is such that
\[ \|u^*\|_{W^{(m,n)}_{(x,y),2}(R)} \leq C\|u\|_{W^{(m,n)}_{(x,y),2}(Q)}, \tag{2} \]
where \(C\) is a constant depending only on the domain \(Q\).

Before proceeding to the proof of the theorem, which for brevity will be carried out for integer \(m\) and \(n\), we formulate several auxiliary assertions concerning fractional derivatives. Let \(f(x)\) be a function of one variable \(x\) of class \(L_2(0,1)\).

We shall say that \(f(x)\) has a derivative \(f^{(\alpha)}(x)\) of order \(\alpha\), \(0<\alpha<1\), from \(L_2(0,1)\), if \(f^{(\alpha)}(x)\in L_2(0,1)\) and
\[ \int_0^x f^{(\alpha)}(x)\,dx = \frac{1}{\Gamma(1-\alpha)} \int_0^x \frac{f(\xi)}{(x-\xi)^\alpha}\,d\xi . \tag{3} \]

Lemma 1. If the function \(f(x)\in L_2(0,1)\) has a derivative \(f^{(\alpha)}(x)\) of order \(\alpha\), \(0<\alpha<1\), from \(L_2(0,1)\), then
\[ \int_0^x f(\xi)\,d\xi = \frac{1}{\Gamma(1+\alpha)} \int_0^x (x-\xi)^{(\alpha)} f^{(\alpha)}(\xi)\,d\xi . \tag{4} \]

Lemma 2. If the function \(f(x)\in L_2(0,1)\) has a derivative \(f^{(\alpha)}(x)\), \(0<\alpha<1\), from \(L_2(0,1)\), then
\[ \left\|\frac{f}{x^\alpha}\right\|_{L_2(0,1)} \leq C_1\|f^{(\alpha)}\|_{L_2(0,1)}, \tag{5} \]
where \(C_1\) is an absolute constant.

Lemma 1 is proved analogously to the corresponding assertions on fractional derivatives in the book \((^6)\), and Lemma 2 can be obtained from Theorem 319 of the book \((^7)\).

Lemma 3. The set of functions \(f(x)\in L_2(0,1)\) for which there exists a derivative of order \(\alpha\), \(0<\alpha<1\), from \(L_2(0,1)\), coincides with Slobodetskii’s space \(W_2^{(\alpha)}(0,1)\).

Lemma 3 is proved by applying the Fourier transform to (3) and using the corresponding theorem of Slobodetskii \((^4)\).

Lemma 4. Let \(u(x,y)\) be a finite function, concentrated in the domain \(Q\) and belonging to the space \(W^{(m,n)}_{(x,y),2}(R)\), and let \(\alpha\) and \(\beta\) be integers, \(\alpha\geq 0\), \(\beta\geq 0\), such that
\[ 1-\frac{1}{n}\leq \frac{\alpha}{m}+\frac{\beta}{n}\leq 1. \]
Then
\[ \left\| \frac{D_x^\alpha D_y^\beta u} {x^{m\left(1-\frac{\alpha}{m}-\frac{\beta}{n}\right)}} \right\|_{L_2(Q)} \leq C_2\|u\|_{W^{(m,n)}_{(x,y),2}(Q)}, \tag{6} \]
where \(C_2\) is a constant depending only on the domain \(Q\).

For the proof of the theorem, cut the domain \(Q\) by the straight lines \(x=x_\Lambda+\delta\) and \(x=x_\Pi-\delta\) into the parts
\[ Q_1^{(\delta)}=Q\cap(x_\Lambda\leq x\leq x_\Lambda+\delta),\quad Q_2^{(\delta)}=Q\cap(x_\Lambda+\delta<x<x_\Pi-\delta), \]
\[ Q_3^{(\delta)}=Q\cap(x_\Pi-\delta\leq x\leq x_\Pi), \]
where \(\delta\) is some positive sufficiently small number, \(\delta<(x_\Pi-x_\Lambda)/2\). From the condition of the theorem it follows that the equations of the curves \(\Gamma\cap(x_\Lambda\leq x\leq x_\Lambda+\delta)\) and \(\Gamma\cap(x_\Pi-\delta\leq x\leq x_\Pi)\), parts of the boundary of \(Q_1^{(\delta)}\) and \(Q_3^{(\delta)}\) in neighborhoods of the points \(\Lambda\) and \(\Pi\), respectively, can be represented in the form
\[ x-x_\Lambda=\psi_\Lambda(|y-y_\Lambda|) \quad\text{and}\quad x_\Pi-x=\psi_\Pi(|y-y_\Pi|), \]
where
\[ \psi_\Lambda(t)\leq At^{n/m},\quad \psi_\Pi(t)\leq At^{n/m},\quad 0\leq t\leq \min\left(\psi_\Lambda^{-1}(\delta),\psi_\Pi^{-1}(\delta)\right), \]
and \(A\) is a constant independent of \(\delta\), provided \(\delta\) is less than some sufficiently small \(\delta_0\).

Lemma 5. If the function \(u(x,y)\) satisfies the conditions of the theorem, then for it there exists an extension from the domain \(Q_2^{(\delta)}\) into the strip
\[ R_2^{(\delta)}=(x_\Lambda+\delta<x<x_\Pi-\delta). \]
Moreover,
\[ \|u^*\|_{W^{(m,n)}_{(x,y),2}(R_2^{(\delta)})} \leq C(\delta)\|u\|_{W^{(m,n)}_{(x,y),2}(Q_2^{(\delta)})}, \tag{7} \]
where the constant \(C(\delta)\) does not depend on \(u(x,y)\).

Lemma 5 is proved analogously to the corresponding theorems in \((^{2,4,8})\) by means of a local straightening of the boundary \(\Gamma \cap (x_\Lambda+\delta<x<x_\Pi-\delta)\) of the domain \(Q_2^{(\delta)}\). In extending \(u(x,y)\) from the domain \(Q_1^{(\delta)}\) (as well as from the domain \(Q_3^{(\delta)}\)) we shall assume that \(x_\Lambda=y_\Lambda=0\) and that the function \(u(x,y)\), together with all its existing derivatives, vanishes on the line \(x=0\).

Define in \(\widetilde Q_1^{(\delta)}=(0\le x\le \delta,\ y\ge 0)\setminus Q_1^{(\delta)}\cap (y\ge 0)\) the function

\[ v_1(x,y)=\sum_{i=1}^{N}\lambda_i e^{M_i(1-y/\varphi(x))}\,u\bigl(x,\varphi(x)e^{m_i(1-y/\varphi(x))}\bigr), \tag{8} \]

where \(y=\varphi(x)\) is the equation \(x=\psi_\Lambda(y)\), solved for \(y\) for \(y>0\); \(m_i\ge 1,\ M_i\ge m_i,\ i=1,\ldots,N\), are certain positive numbers. It has been proved that, with a suitable choice of \(m_i\) and \(M_i\), one can find numbers \(N,\lambda_i,\ i=1,\ldots,N\), such that the function \(v_1(x,y)\), defined in \(\widetilde Q_1^{(\delta)}\), will coincide with the function \(u(x,y)\), given in \(Q_1^{(\delta)}\), along their common boundary \(y=\varphi(x)\) with any number of derivatives with respect to \(x\) and to \(y\). In the same way one constructs the function \(v_2(x,y)\), which carries out the extension of \(u(x,y)\) from \(Q_1^{(\delta)}\) to the domain \(\widetilde Q_1^{(\delta)}=(0\le x\le \delta,\ y\le 0)\setminus Q_1^{(\delta)}\cap (y\le 0)\). Denoting by \(u^*(x,y)\) the function defined in the strip \(R_1^{(\delta)}=(0\le x\le \delta)\), coinciding with \(u(x,y)\) in \(Q_1^{(\delta)}\), with \(v_1(x,y)\) in \(\widetilde Q_1^{(\delta)}\), and with \(v_2(x,y)\) in \(\widetilde Q_1^{(\delta)}\), and taking into account the regularity of the boundary \(\Gamma\), from (8) we obtain

\[ \left\|D_y^n u^*\right\|_{L_2(R_1^{(\delta)})}^{2} \le C_3^2 \sum_{k=0}^{n} \left\| \frac{D_y^k u}{x^{(n-k)m/n}} \right\|_{L_2(Q_1^{(\delta)})}^{2}, \tag{9} \]

\[ \left\|D_x^m u^*\right\|_{L_2(R_1^{(\delta)})}^{2} \le C_3^2 \sum_{\substack{0\le s/n+k/m\le 1\\ k+s=m}} \sum \left\| \frac{D_x^k D_y^s u}{x^{m(1-k/m-s/n)}} \right\|_{L_2(Q_1^{(\delta)})}^{2}, \tag{10} \]

where the constant \(C_3\), for sufficiently small \(\delta\), does not depend on \(\delta\). Analogous estimates are obtained:

\[ \left\|D_y^s u^*\cdot\theta(x)\right\|_{L_2(R_1^{(\delta)})}^{2} \le C_3^2\sum_{k=0}^{s} \left\| \frac{D_y^k u}{x^{(s-k)m/n}}\,\theta(x) \right\|_{L_2(Q_1^{(\delta)})}^{2}, \qquad 0\le s\le n; \tag{9_s} \]

\[ \left\|D_x^s u^*\cdot\theta(x)\right\|_{L_2(R_1^{(\delta)})}^{2} \le C_3^2 \sum_{\substack{k_1+k_2=s\\ 0\le k_1+k_2m/n\le s}} \sum \left\| \frac{D_x^{k_1}D_y^{k_2}u\cdot\theta(x)} {x^{s-k_1-k_2m/n}} \right\|_{L_2(Q_1^{(\delta)})}^{2}, \qquad 0\le s\le m, \tag{10_s} \]

for an arbitrary \(\theta(x)\ge 0\) for which \((9_s)\) and \((10_s)\) make sense.

Let us note that the function \(u^*(x,y)\) in \(R_1^{(\delta)}\), for sufficiently large \(|y|\), may be taken to be zero, without violating the estimates (9), (10), \((9_s)\), and \((10_s)\). Moreover, without violating these estimates, one may also regard \(u^*(x,y)\) as being defined not only in \(R_1^{(\delta)}\), but in the whole half-strip \(x\le x_\Lambda+\delta\), setting it equal to zero for \(x\le x_\Lambda\). This follows directly from (8), since \(e^{M_i(1-y/\varphi(x))}\) for \(x=0,\ y\ne 0\) vanishes together with all its derivatives. In connection with this, the norms standing on the right-hand side of (9), for \(k<n\), can be estimated as follows:

\[ \left\| \frac{D_y^k u}{x^{(n-k)m/n}} \right\|_{L_2(Q_1^{(\delta)})}^{2} \le \left\| \frac{D_y^k u^*}{x^{(n-k)m/n}} \right\|_{L_2(R_1^{(\delta)})}^{2} \le \]

\[ \le \varepsilon \left\| \frac{D_y^{k+1}u^*}{x^{(n-k-1)m/n}} \right\|_{L_2(R_1^{(\delta)})}^{2} + \frac{1}{\varepsilon} \left\| \frac{D_y^{k-1}u^*}{x^{(n-k+1)m/n}} \right\|_{L_2(R_1^{(\delta)})}^{2}. \]

for any \(\varepsilon>0\). Hence, with the aid of \((9_s)\), for a suitable choice of \(\theta(x)\),

\[ \left\|\frac{D_y^n u}{x^{(n-k)m/n}}\right\|_{L_2(Q_1^{(\delta)})}^{2} \leq \varepsilon \|D_y^n u^*\|_{L_2(R_1^{(\delta)})}^{2} + C_\varepsilon \left\|\frac{u}{x^m}\right\|_{L_2(Q_1^{(\delta)})}^{2}, \tag{11} \]

where \(C_\varepsilon\) depends only on \(\varepsilon>0\). From (11) and (9) it follows that

\[ \|D_y^n u^*\|_{L_2(R_1^{(\delta)})}^{2} \leq B\left( \|D_y^n u\|_{L_2(Q_1^{(\delta)})}^{2} + \left\|\frac{u}{x^m}\right\|_{L_2(Q_1^{(\delta)})}^{2} \right), \tag{12} \]

where the constant \(B>0\).

Similarly, with the aid of \((10_s)\), estimates are obtained for the terms on the right-hand side of (10) when

\[ \frac{k}{m}+\frac{s}{n}>1-\frac{1}{n}: \]

\[ \left\| \frac{D_x^k D_y^s u}{x^{m(1-k/m-s/n)}} \right\|_{L_2(Q_1^{(\delta)})}^{2} \leq \varepsilon \left\| \frac{D_x^k D_y^{s+\lambda} u}{x^{m(1-k/m-(s+\lambda)/n)}} \right\|_{L_2(R_1^{(\delta)})}^{2} + C_\varepsilon \left\| \frac{D_x^k u}{x^{m-k}} \right\|_{L_2(Q_1^{(\delta)})}^{2}, \tag{13} \]

where \(\lambda\) is such an integer that

\[ 1-\frac{1}{n}\leq \frac{k}{m}+\frac{s+\lambda}{n}\leq 1. \]

Applying Lemmas 3 and 4 to the remaining terms of the right-hand side of (10), as well as to the first term of (13), we finally obtain

\[ \|D_x^m u\|_{L_2(R_1^{(\delta)})}^{2} \leq B\left( \|D_y^n u\|_{L_2(Q_1^{(\delta)})}^{2} + \|D_x^m u\|_{L_2(Q_1^{(\delta)})}^{2} + \sum_{r=0}^{m} \left\| \frac{D_x^r u}{x^{m-r}} \right\|_{L_2(Q_1^{(\delta)})}^{2} \right). \tag{14} \]

By means of integration by parts with respect to \(x\) over the domain \(Q_1^{(\delta)}\), one can prove the existence of a constant \(C_4\), independent of \(u(x,y)\), such that

\[ \left\| \frac{D_x^r u}{x^{m-r}} \right\|_{L_2(Q_1^{(\delta)})}^{2} \leq C_4^2 \|D^m u\|_{L_2(Q_1^{(\delta)})}^{2}. \tag{15} \]

Taking into account (15), (14), (12), and (10), we obtain

\[ \|u\|_{W_{(x,y),2}^{(m,n)}(R_1^{(\delta)})}^{2} \leq C_5^2 \|u\|_{W_{(x,y),2}^{(m,n)}(Q_1^{(\delta)})}^{2}, \tag{16} \]

with a constant \(C_5\) independent of \(u(x,y)\) and of \(\delta\) for \(\delta<\delta_0\).

Since estimate (16) remains valid if in it the domains \(Q_1^{(\delta)}\) and \(R_1^{(\delta)}\) are replaced by \(Q_3^{(\delta)}\) and \(R_3^{(\delta)}\), respectively, it follows immediately from (16) and Lemma 5, in which \(\delta\) should be replaced by \(\delta_0\), that the assertion of the theorem is obtained.

Remark. From this theorem, with the aid, for example, of \((4)\) or \((8)\), there immediately follows the existence in \(Q\) of the generalized derivatives \(D_x^\mu D_y^\nu u\in L_2(Q)\) for

\[ \frac{\mu}{m}+\frac{\nu}{n}\leq 1 \]

and

\[ \|D_x^\mu D_y^\nu u\|_{L_2(Q)} \leq C_1 \|u\|_{W_{(x,y),2}^{(m,n)}(Q)}, \qquad \frac{\mu}{m}+\frac{\nu}{n}\leq 1, \]

where \(C_1\) does not depend on \(u(x,y)\). Moreover, in \(Q\) the corresponding results proved for \(R_2\) on the behavior of \(u(x,y)\) on manifolds of a smaller number of dimensions (on curves in the domain \(Q\)) turn out to be valid.

Moscow State University
named after M. V. Lomonosov

Received
26 X 1961

CITED LITERATURE

1. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.
2. V. M. Babich, UMN, 8, no. 2 (54), 111 (1953).
3. G. M. Fichtenholz, Course of Differential and Integral Calculus, 1, 1948.
4. L. N. Slobodetskii, Uch. zap. Leningrad State Pedagogical Institute named after A. I. Herzen, 197, 54 (1958).
5. V. P. Mikhailov, DAN, 140, No. 2 (1961).
6. A. Zygmund, Trigonometric Series, 1939.
7. G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, IL, 1948.
8. S. M. Nikol’skii, Matem. sbornik, 40, no. 2 (1956).