Abstract
Full Text
MATHEMATICS
A. L. FUKSMAN
ON THE APPROXIMATION OF FUNCTIONS OF MANY VARIABLES WITH PRESERVATION OF BOUNDARY CONDITIONS
(Presented by Academician V. I. Smirnov, 17 VII 1961)
1. Let \(D\) be an \(m\)-dimensional domain with boundary \(\Gamma\), and let \(\varphi(x)=0\) be the equation of \(\Gamma\). We consider the approximation of functions \(u(x)\), equal to zero on \(\Gamma\) together with derivatives up to order \(s-1\), by expressions of the form \(\varphi^s(x_1,x_2,\ldots,x_m)P_n(x_1,x_2,\ldots,x_m)\), where \(P_n(x_1,x_2,\ldots,x_m)\) is a polynomial of degree not exceeding \(n\) in each of the arguments \(x_1,x_2,\ldots,x_m\). This problem, connected with variational methods for solving boundary-value problems for elliptic equations, was considered in papers \((^{1-3})\).
The results set forth below are based essentially on a certain new method of extending a function with preservation of differential properties from a domain whose boundary contains singular points of a definite kind.
Denote by \(C_\omega^k(D)\) the space in which the unit sphere is the set of functions \(f\) having in the domain \(D\) continuous partial derivatives up to order \(k\), with
\[ \max_{0\le l\le k}\max_{x\in \overline D}|D^l f(x)|\le 1,\qquad \omega_{C(D)}(D^l f;\varepsilon)\le \omega(\varepsilon) \]
\[ (0\le l\le k). \]
If \(\omega(\varepsilon)=\varepsilon\), then we shall use the notation \(C^{k,1}(D)\). By \(\widetilde C_\omega^k\) and \(\widetilde C_\omega^{k,1}\) we denote the corresponding spaces of functions that are \(2\pi\)-periodic in all arguments.
Under certain restrictions on the function \(f(x)\) the following holds.
Proposition 1. For every function \(u(x)\) from \(C_\omega^k(D)\), equal to zero on the boundary together with derivatives up to order \(s-1\) inclusive \((0\le s-1\le k)\), there exists a sequence of polynomials \(P_n(x)\) of degree not exceeding \(n\) in each variable, for which
\[ \|u-\varphi^sP_n\|_{C^l(D)} \le N_1\|u\|_{C_\omega^k(D)}\, n^{-(k-l)}\omega(1/n), \]
where \(l=0,1,\ldots,k\), and \(N_1\) does not depend on \(n\) or on \(u\).
In paper \((^3)\) it was proved:
Theorem 1. Proposition 1 is valid if \(\varphi(x)\) is such that: a) \(\varphi(x)\in C^{k,1}(D)\); b) \(\varphi(x)>0\) in \(D\); c) for each point \(y\) of the boundary \(\Gamma\) there exists a number \(p=p(y)\) and a neighborhood \(\Omega_y\), in which the function \(\varphi(x)\) is representable in the form \(\varphi(x)=\varphi_1(x)\varphi_2(x)\cdots\varphi_p(x)\), the functions \(\varphi_i(x)\) satisfying the conditions: 1) \(\varphi_i(x)>0\) in \(\Omega_y\cap D\); 2) \(\varphi_i(x)\in C^{k,1}(\Omega_y)\); 3) \(\varphi_i(y)=0\); 4) \(\operatorname{grad}\varphi_i(y)\ne0\); d) the vectors \(\operatorname{grad}\varphi_i(y)\) \((i=1,2,\ldots,p)\) are linearly independent.
In papers \((^{1,2})\) an analogous result was established under the assumption that \(p(y)=1\) at every point \(y\) of the boundary \(\Gamma\), for the cases \(s=1\) and \(s=2\).
It follows from the conditions of Theorem 1 that the boundary \(\Gamma\), in a sufficiently small neighborhood of an arbitrary point \(y\) of it, is formed by the intersection of \(p\) surfaces whose normals are linearly independent. In the present note the class of domains and functions \(\varphi(x)\) for which Proposition 1 is valid is enlarged. For some domains, for which the theorem is certainly false, results of an analogous kind are established.
- Everywhere below we assume that the functions \(\varphi(x)\) satisfy conditions a) and b) of Theorem 1.
We shall call a boundary point \(x_0\) of the domain \(D\) admissible for the function \(\varphi(x)\) if there exists a neighborhood \(\Omega_{x_0}\) such that Proposition 1 is valid for the given \(D\) and \(\varphi(x)\) for every function \(u(x)\) satisfying the additional requirement: \(u(x)\equiv 0\) outside \(\Omega_{x_0}\).
If all boundary points are admissible for the function \(\varphi(x)\), then Proposition 1 is valid for it. In particular, any point \(y\) at which conditions c) and d) of Theorem 1 are fulfilled for the function \(\varphi(x)\) is admissible. From condition d) it follows that \(p \leq m\), so that, for example, Theorem 1 does not include among the domains under consideration a polyhedron in three-dimensional space at some vertex of which more than three faces meet. This restriction is removed by the following
Theorem 2. If at a boundary point \(y\) condition c) of Theorem 1 is fulfilled, and \(p>m\), but any \(m\) of the vectors \(\operatorname{grad}\varphi_i(y)\) \((i=1,2,\ldots,p)\) are linearly independent and \(k \geq ps\), then the point \(y\) is admissible.
The proof is carried out as in \((^3)\). The main difficulty lies in the proper extension of the function \(u(x)\) from the domain \(D\cap \Omega_y\) to the whole neighborhood \(\Omega_y\), namely in such a way that the extended function is equal to zero together with its derivatives up to order \(s-1\) on the surfaces \(\varphi_i(x)=0\) \((i=1,2,\ldots,p)\). If for \(p\leq m\) it was sufficient for this purpose to apply Hestenes’ method (see \((^4)\)), here one has to use more delicate arguments. We shall need a number of lemmas.
Lemma 1. If an \(m\)-dimensional domain \(D\) satisfies the condition: any two of its points \(x,y\) can be joined by a curve lying in the domain of length \(l\leq \lambda(x,y)\), where \(|xy|\) is the distance between the points and \(\lambda\) does not depend on \(x\) and \(y\), then any function \(f(x)\) from \(C_\omega^k(D)\) can be extended to the whole space \(E_m\) with preservation of smoothness, and moreover
\[ \|f\|_{C_\omega^k(E_m)}\leq N_2\|f\|_{C_\omega^k(D)}, \]
where \(N_2\) depends only on \(m,k,\lambda\).
The proof of this lemma is obtained by a certain detailing of Whitney’s method \((^5)\), simplified by Hestenes \((^6)\).
Lemma 2. Let \(S\) be the unit sphere of \(m\)-dimensional Euclidean space and \(f\in C_\omega^k(S)\), and suppose that at the origin \(D^i f=0\), \(i=0,1,\ldots,k\); let the behavior of the function \(\chi(x)\) at the origin be determined by the estimate
\[ |D^l\chi(x)|\leq N_3(l)|x|^{-(l+\alpha)}, \]
where \(|x|\) is the distance of the point \(x\) from the origin, \(l=0,1,2,\ldots\), and \(\alpha\) is some integer, \(0\leq \alpha\leq k\). Then the function \(f_1(x)=f(x)\chi(x)\) belongs to the class \(C_\omega^{k-\alpha}(S)\), and
\[ \|f\chi\|_{C_\omega^{k-\alpha}(S)}\leq N_4\|f\|_{C_\omega^k(S)}, \]
where \(N_4\) does not depend on \(f\).
Lemma 3. Let \(\Delta_1,\Delta_2\) be domains on the surface \(\Sigma\) of the sphere \(S\) from Lemma 2 whose closures do not intersect; let \(\lambda(x)\) be a function given on the surface \(\Sigma\), infinitely differentiable, equal to \(0\) in \(\Delta_1\) and equal to \(1\) in \(\Delta_2\). Then the function
\[ \chi(x)=\lambda\!\left(\frac{x}{|x|}\right) \]
satisfies the condition
\[ |D^l(\chi(x))|\leq N_5|x|^{-l}, \]
where \(N_5\) depends only on \(l\) and on the function \(\lambda(x)\).
Lemma 4. Under the conditions of Theorem 2, the function \(u(x)\) from Proposition 1 is representable in the form \(u(x)=\varphi^s q(x)+u_1(x)\), where \(q(x)\) is a polynomial of degree \(k-ps\), and the function \(u_1(x)\) at the point \(y\) is equal to zero together with all derivatives up to order \(k\) inclusive.
The principal part of the function \(u(x)\) at the point \(y\) singled out in Lemma 4, i.e. \(\varphi^s q\), has already been defined in the whole neighborhood \(\Omega_y\); it remains only to extend the function \(u_1(x)\). For this it suffices to extend it from the domain \(D\), while preserving smoothness and the zero conditions on the surfaces \(\varphi_i(x)=0\) \((1\leq i\leq p)\), to a function \(\bar u_1(x)\) defined in some conical neighborhood \(\Lambda\) of the domain \(D\) (this can be done by Hestenes’ method), choose an intermediate conical neighborhood \(\Lambda_1\) \((\bar D\subset \Lambda_1\subset \bar\Lambda_1\subset \Lambda)\), construct by Lemma 3
the function \(\chi(x)\), equal to 1 inside \(\Lambda_1\) and equal to 0 outside \(\Lambda\), and then set \(u_2=\overline{u}_1\chi\) inside \(\Lambda\) and \(u_2\equiv 0\) outside \(\Lambda\). Applying Lemma 2, we see that the smoothness of the function \(u_2\) is the same as that of \(u_1\), while the satisfaction of the zero conditions on the surfaces \(\varphi_i(x)\) is obvious. Consequently, the function \(\overline{u}(x)=\varphi(x)q(x)+u_2(x)\) will be the required extension.
Theorem 3. Suppose that in a neighborhood of a point \(y\) the function \(\varphi(x)\) is such that, under a smooth change of coordinates \(x=\sigma(\xi)\) (the \(x_i(\xi)\) are functions of class \(C^{k+1,1}\), \(\xi_i(y)=0\), and the Jacobian \(|\sigma(\xi)|\) is different from zero), it becomes the function
\[
\hat{\varphi}(\xi)=\varphi(\sigma(\xi))=\xi_m^2-(\xi_1^2+\xi_2^2+\cdots+\xi_{m-1}^2),
\]
and, in the domain \(\hat D\cap\hat\Omega_y\), we have: \(\hat\varphi(\xi)>0,\ \xi_m>0\). If \(k\ge 2s\), then for the function \(\varphi(x)\) the point \(y\) is admissible.
For the proof, first, as in Theorem 1, we extend the function \(\hat u(\xi)=u(\sigma(\xi))\), defined only in that half of the cone where \(\hat\varphi(\xi)>0,\ \xi_m>0\), to the whole neighborhood \(\hat\Omega_y\) by the method of separating the principal part. However, unlike Theorem 2, here one cannot directly use the method of the paper \((^3)\). True, the polynomials \(P_n(x)\) are constructed from the function \(u(x)\) as in \((^3)\): extending \(u\) and \(\varphi\) in the proper way to the cube \(|x_i|\le a\) containing \(\bar D\), we form the function \(v=\varphi^{-s}u\); we pass to periodic functions \(\tilde u,\tilde\varphi,\tilde v\), setting \(x_i=a\cos\xi_i,\ i=1,2,\ldots,m\); finally, we set
\[
\widetilde P_n(\xi_i)=\tilde v-(I-T_n)^k\tilde v,
\]
where \(T_n\) is an integral operator with a Jackson-type kernel. But in order to prove that the polynomials \(P_n(x_i)=\widetilde P_n(\arccos x_i)\) are the desired ones, the reasoning has to be further complicated. We prove, using the method of separating the principal part and Lemma 2, that for the functions
\[
v_{l,p}=\varphi^{-l}(D_{i_1}\varphi D_{i_2}\varphi\cdots D_{i_p}\varphi)\,u,
\]
where \(0\le p\le l\le s,\ 1\le i_r\le m\), and \(D_i\) denotes differentiation with respect to \(x_i\), the estimate
\[
\|v_{l,p}\|_{\widetilde C_\omega^{\,k-2l+p}(\Omega_y)}
\le
N_6\|u\|_{C_\omega^k(\Omega_y)}
\]
is valid, and then the following lemma is applied:
Lemma 5. Let \(\varphi\) be from \(\widetilde C^{k+1,1}\), and let \(u\) be from \(\widetilde C_\omega^k\), and suppose these are periodic functions such that the function of the form
\[
v_{l,p}=\varphi^{-l}(D_{i_1}\varphi D_{i_2}\varphi\cdots D_{i_p}\varphi)u,
\]
where \(0\le p\le l\le s\), belongs to the class \(\widetilde C_\omega^{k-2l+p}\). Then the estimate is valid:
\[
\|\varphi^s(I-T_n)^k v_{s,0}\|_{\widetilde C^r}
\le
N_7 A(u)\,\omega(1/n)\,n^{-(k-r)},
\]
where
\[
A(u)=\max_{0\le p\le l\le s}\|v_{l,p}\|_{\widetilde C_\omega^{\,k-2l+p}},
\qquad 0\le r\le k.
\]
Theorem 4. Let \(m=2\), and suppose that at a point \(y\) the boundary \(\Gamma\) has the form of a “beak,” while the function \(\varphi(x)\) satisfies condition c) of Theorem 1, but the vectors \(\operatorname{grad}\varphi_1(y)\), \(\operatorname{grad}\varphi_2(y)\) are collinear. Then for the function \(\varphi(x)\) the point \(y\) is admissible if \(k\ge 2s\).
The proof is carried out exactly as in Theorems 2, 3. Under the conditions of Theorem 1, the neighborhood \(\Omega_y\) is divided by the surfaces \(\varphi_i(x)=0\) \((1\le i\le p)\) into \(2^p\) parts, only one of which belongs to the domain \(D\). If, however, the remaining \(2^p-1\) parts belong to the domain \(D\), then Proposition 1 is false, no matter how the function \(\varphi(x)\) is chosen. However, if, except for \(y\), all points of the boundary \(\Gamma\) are admissible, then the following is valid:
Proposition 2. For every function \(u(x)\) from \(C_\omega^k(D)\), equal to zero on the boundary together with derivatives up to order \(s-1\), there exist sequences \(P_n\) and \(Q_{n,i}\) of polynomials for which
\[
\left\|
u-\varphi^sP_n-\sum_{i=1}^p \psi_i Q_{n,i}
\right\|_{C^l(D)}
\le
N_8\|u\|_{C_\omega^k(D)}\,\omega(1/n)\,n^{-(k-l)},
\]
where \(l=0,1,\ldots,k\); \(N_8\) does not depend on \(u,n\); the function \(\psi_i(x)\) in the neighborhood \(\Omega_y\) has the form
\[
\psi_i(x)=[\varphi_i(x)]^{k+1}
\quad\text{if }\varphi_i(x)\le 0,\qquad
\psi_i(x)\equiv 0
\quad\text{if }\varphi_i(x)>0,
\]
and outside \(\Omega_y\) it is defined arbitrarily, provided only that it belongs to \(C^{k,1}(D)\) and vanishes on the boundary together with derivatives up to order \(s-1\).
An analogous proposition is valid if, in the neighborhood \(\Omega_y\), the domain \(D\) contains not \(2^p-1\), but another number of the mentioned \(2^p\) parts.
Let us also note that the boundary of the domain may contain any set of points of the indicated kind. It is only necessary that the \(\psi_i(x)\) have the corresponding representation in a neighborhood of each such point.
- Let us point out some unsolved questions. In the case \(m=2\) we studied the approximation of functions with preservation of boundary conditions in domains with piecewise smooth boundaries for various values of the angle \(\alpha\) between neighboring curves, directed toward the domain \(D\). The case \(0<\alpha<\pi\) corresponds to paper \({}^{(3)}\), and the cases \(\alpha=0\) and \(\pi<\alpha<2\pi\) correspond, respectively, to Theorem 4 and Proposition 2 of the present note. Thus the cases \(\alpha=\pi\) and \(\alpha=2\pi\) remain. In the second of these, the main difficulty lies in the inapplicability of Lemma 1 on the extension of functions. If one restricts oneself to the class of functions \(u\) admitting an extension from the domain \(D\) with preservation of smoothness, then one can obtain a result of the type of Proposition 2 for a function \(\varphi(x)\) satisfying the conditions of Theorem 1, and for appropriately chosen functions \(\psi_i(x)\). In the case \(\alpha=\pi\), however, not every function \(u(x)\) can be extended from the domain \(D\) with preservation of smoothness and simultaneous preservation of the zero conditions on the lines \(\varphi_1(x)=0\), \(\varphi_2(x)=0\). Nevertheless, here too one can obtain a result of the type of Proposition 2, but this requires essentially new methods, and the scope of this note does not allow us to dwell on them in greater detail.
Rostov-on-Don
State University
Received
12 VII 1961
REFERENCES
- I. Yu. Kharrik, Matem. sborn., 37 (79), 353 (1955).
- I. Yu. Kharrik, Matem. sborn., 47 (89), 177 (1959).
- A. L. Fuksman, DAN, 134, 289 (1960).
- G. M. Fichtenholz, Course of Differential and Integral Calculus, 1, Moscow–Leningrad, 1958.
- H. Whitney, Trans. Am. Math. Soc., 36, 63 (1934).
- M. R. Hestenes, Duke Math. J., 8, 183 (1941).