Abstract
Full Text
UDC 517.514+517.945.9
MATHEMATICS
G. N. YAKOVLEV
ON THE DENSITY OF FINITE FUNCTIONS IN WEIGHTED SPACES
(Presented by Academician I. I. Vinogradov on January 7, 1966)
1. Let \(G\) be an open bounded domain of the \(n\)-dimensional Euclidean space \(E^n\); let \(S\) be its \((n-1)\)-dimensional boundary; let \(T\) be an \(m\)-dimensional bounded manifold of smoothness 2 (without boundary) \((^2)\). We shall assume that \(S\) is Lipschitz \((^3)\), and that for \(m=n-1\) the manifold \(T\) is such that \(T\cap G=0\). In the remaining cases, i.e. for \(0\le m<n-1\), \(T\) is situated in \(E^n\) in an arbitrary way. By \(x\) we shall denote an arbitrary point in \(E^n\).
Let \(a=a(t)\) be a nonnegative measurable function, defined for all \(t>0\) and possessing the following property: for every interval \([\delta,A]\), \(0<\delta<A<\infty\), there exist constants \(c_1\) and \(c_2\) such that
\(0<c_1<a(t)<c_2<+\infty\) for all \(t\in[\delta,A]\).
Let \(g\) be some closed subdomain of the domain \(G\), all points of which are at a positive distance from the set \(T\cup S\). We shall say that \(f(x)\in W^1_{p,\alpha}(G,T)\), \(1\le p<\infty\), if \(f(x)\) has first generalized derivatives in the sense of S. L. Sobolev \((^1)\) and if the norm is finite
\[ \|f,W^1_{p,\alpha}(G,T)\|=\|\alpha(r)|\operatorname{grad} f|\|_G+\|f\|_g, \tag{1} \]
where \(r=r(x)\) is the distance from the point \(x\) to \(T\), and \(\|\ \|_G\) is the norm in \(L_p(G)\). It is easy to see that if \(T\cap \overline{G}=0\), where \(\overline{G}\) is the closure of the domain \(G\), then \(W^1_{p,\alpha}(G,T)\) coincides with the usual space \(W^1_p(G)\) of S. L. Sobolev \((^1)\). If \(m=n-1\), then either all of \(T\) lies on \(S\), or part of \(T\) is a part of \(S\), and in this case the weight \(\alpha\) affects the behavior of the function \(f(x)\) only when approaching that part of the boundary \(S\) which lies on \(T\). If \(0\le m<n-1\), then all of \(T\), or some part of it, may lie both in \(G\) and on \(S\).
If \(f(x)\in W^1_{p,\alpha}(G,T)\), then the trace of the function \(f(x)\) on \(S\setminus T\) exists, while on \(T\cap\overline{G}\), depending on \(\alpha\), two cases are possible.
Let
\[ \varphi(t)=|\alpha(t)|^{-qt(1-q)(n-m-1)},\qquad q=p/(p-1). \]
If the integral
\[ \int_0^1 \varphi(t)\,dt \tag{2} \]
converges, then the trace of \(f(x)\) on \(T\cap\overline{G}\) exists; if the integral (2) diverges, then, generally speaking, the trace of \(f(x)\) on \(T\cap\overline{G}\) does not exist. By \(\overset{\circ}{W}{}^1_{p,\alpha}(G,T)\) we shall denote the space of functions \(f(x)\in W^1_{p,\alpha}(G,T)\) for which: 1) if the integral (2) converges, then the trace on \(S\cup(T\cap\overline{G})\) is equal to zero; 2) if the integral (2) diverges, then the trace on \(S\setminus T\) is equal to zero.
By \(\overset{\circ}{C}{}^\infty(G,T)\) we shall denote the set of infinitely differentiable functions, each of which is equal to zero outside some closed subdomain of the domain \(G\), all points of which are at a positive dis-
away from \(S \cup T\). (This subdomain is, generally speaking, different for each function.)
Theorem 1. The set \(\dot C^\infty(G,T)\) is dense in the space \(\dot W^1_{p,\alpha}(G,T)\), \(1<p<\infty\).
In the case \(\alpha \equiv 1\) this assertion was proved in the paper [3]. In the case when \(a(r)\) is a certain power of \(r\) and \(T\) coincides with all of \(S\), an analogous assertion is found in [4]. In proving Theorem 1 one uses a generalization of Hardy’s inequality [4] and the following lemma.
Lemma. If \(f(x) \in \dot W^1_{p,\alpha}(G,T)\), then \(f(x)\) is such that \(\beta(r)f \in L_p(G)\), where
\[ \beta(r)=a(r)\varphi(r)\left[\int_r^{2d}\varphi(t)\,dt\right]^{-1}, \]
\(d\) is the diameter of the domain \(G\).
2. Let
\[ L(u)=\sum_{i,j=1}^n \frac{\partial}{\partial x_i}\left(a_{ij}\frac{\partial u}{\partial x_j}\right), \]
where the functions \(a_{ij}=a_{ji}\) \((i,j=1,2,\ldots,n)\) are measurable. Suppose there exist constants \(c_1>0\) and \(c_2>0\) such that for almost all \(x\in G\) and all vectors \(\xi=(\xi_1,\ldots,\xi_n)\) the inequality
\[ c_1\alpha|\xi|^2\le \sum_{i,j=1}^n a_{ij}\xi_i\xi_j\le c_2\alpha|\xi|^2,\qquad |\xi|^2=\xi_1^2+\cdots+\xi_n^2. \]
holds.
Here we shall assume that the closure of the set \(T\cap S\) does not coincide with \(S\).
A function \(u(x)\) is said to be a generalized solution of the equation \(L(u)=0\) if \(u\in \dot W^1_{p,\alpha}(G,T)\) and
\[ \int_G \sum_{i,j=1}^n a_{ij}\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_j}\,dx=0 \]
for every function \(v\in \dot C^\infty(G,T)\).
Problem D. Suppose \(m<n-1\), or, if \(m=n-1\), the integral (2) converges. Find a solution of the equation \(L(u)=0\) (classical or generalized) taking prescribed values on \(S\).
Problem E. Suppose \(T\) is an \((n-1)\)-dimensional manifold and the integral (2) diverges. Find a solution of the equation \(L(u)=0\) taking prescribed values on \(S\setminus T\).
Theorem 1 makes it possible to prove the following uniqueness theorem for solutions of Problems D and E.
Theorem 2. The solution (classical or generalized) of Problems D and E in the class of functions \(\dot W^1_{2,\alpha}(G,T)\) is unique.
An analogous assertion in the case when \(\alpha(t)=t^\gamma\), \(\gamma\ne 1/2\), was proved in the paper of L. D. Kudryavtsev [2].
Moscow Institute of Physics and Technology
Received
30 XII 1965
REFERENCES
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