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This note studies an instability property of Newtonian harmonic capacity for open subsets of three-dimensional Euclidean space and its consequences for uniform approximation by harmonic functions. It shows that if the local capacity of an open set is sufficiently large in a limsup sense at almost every point, then every ball has full capacity after intersection with that set, and extends this conclusion to bounded open sets. The proof uses the Keldysh, Brelot criterion for regular boundary points in the Dirichlet problem to relate regularity to equality of capacities. The resulting criterion states that continuous functions on a compact set are uniformly approximable by harmonic functions precisely when the complement has full capacity in every ball, equivalently in every bounded open set, with analogous formulations in higher dimensions.
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UDC 517.947.42
MATHEMATICS
A. A. GONCHAR
ON A PROPERTY OF “INSTABILITY” OF HARMONIC CAPACITY
(Presented by Academician M. A. Lavrent'ev, 9 IV 1965)
- Let \(e\) be a Borel subset of three-dimensional Euclidean space \(C^3\). We shall use the following notation: \(\gamma(e)\) is the capacity (relative to the Newtonian potential; see, for example, (\(^1\))) of the set \(e\); \(\bar e\) is the closure, \(\partial e\) the boundary of the set \(e\). The open ball of radius \(\delta\) with center at the point \(P\) will be denoted by \(K(P,\delta)\). We note that \(\gamma[K(P,\delta)] = \delta\).
The present note is adjacent to the author’s works (\(^3,\ ^4\)); see also there for a discussion of the results.
Let \(D\) be an open subset of the space \(C^3\). In (\(^4\)) the following property of “instability” of capacity was proved (cf. (\(^4\)), Theorem 2):
Let
\(1^\circ.\) For almost all points \(P \in C^3\) (or, what is the same, for almost all \(P\) belonging to the complement of \(D\)),
\[ \varlimsup_{\delta\to 0}\frac{\gamma[D\cap K(P,\delta)]}{\delta^3}=\infty. \]
Then
\(2^\circ.\) For all \(P \in C^3\) and \(\delta>0\),
\[ \gamma[D\cap K(P,\delta)] \geq A_0\delta, \]
where \(A_0>0\) is an absolute constant.
With the aid of a theorem of Keldysh–Brelot it is not difficult to show (this will be done in § 2) that from property \(2^\circ\) there follows the following property:
\(3^\circ.\) For all \(P \in C^3\) and \(\delta>0\),
\[ \gamma[D\cap K(P,\delta)] = \gamma[K(P,\delta)] = \delta. \]
Thus, the following is true.
Theorem 1. If for almost all (with respect to Lebesgue measure in \(C^3\)) points \(P \in C^3\)
\[ \varlimsup_{\delta\to 0}\frac{\gamma[D\cap K(P,\delta)]}{\delta^3}=\infty, \]
then for all \(P \in C^3\) and \(\delta>0\)
\[ \gamma[D\cap K(P,\delta)] = \delta. \]
Consequently, conditions \(1^\circ\) and \(3^\circ\) are equivalent.
Theorem 1 makes it possible to formulate a necessary and sufficient condition for the possibility of uniform approximation of continuous functions by harmonic functions in the following form (cf. (\(^4\)), Theorems A and B). Let \(E \subset C^3\) be a compact set; \(C(E)\) the space of all functions continuous on \(E\), \(f(P)\), \(P \in E\), with norm \(\|f\|=\max_{P\in E}|f(P)|\); \(H(E)\) the subspace of \(C(E)\) consisting of all functions admitting uniform approximation on \(E\) by functions harmonic on \(E\). Applying Theorem 1 to the complement \(\mathcal D\) of the compact set \(E\) and Theorem B of (\(^4\)), we obtain:
Theorem 2. In order that \(C(E)=H(E)\), it is necessary and sufficient that for all \(P \in C^3\) and \(\delta>0\)
\[ \gamma[K(P,\delta)\setminus E] = \delta. \]
Theorem 1 can be strengthened somewhat.
Theorem 3. If property \(1^\circ\) is satisfied, then for any bounded open set \(G \subset C^3\)
\[ \gamma(D \cap G)=\gamma(G). \]
It follows from this that Theorem 2 also remains valid if, instead of \(K(P,\delta)\), one takes an arbitrary open set \(G\). More precisely, the following holds.
Theorem 4. In order that \(C(E)=H(E)\), it is necessary and sufficient that, for every bounded open set \(G\),
\[ \gamma(G\setminus E)=\gamma(G). \]
- Let \(F\) be a closed set; let \(g_n,\ n=1,2,\ldots,\ F \subset \overline{g}_{n+1} \subset g_n\), be a sequence of open sets converging to \(F\) in the sense that each point of the complement of \(F\) belongs to only a finite number of the sets \(g_n\). Suppose that each of the sets \(g_n\) is bounded by a finite number of smooth closed Jordan surfaces. Let \(\varphi(P)\) be an arbitrary function, defined and continuous on the boundary \(\partial F\) of the set \(F\); let \(\widetilde{\varphi}(P)\) be its continuous extension to the whole space \(C^3\); let \(h_n(P)\) be the solution of the Dirichlet problem in \(g_n\) with boundary data \(\varphi(P)\), \(P\in\partial g_n\) (if \(g_n\) contains the exterior of some ball, then \(h_n(P)\to 0\) as \(\overline{OP}\to\infty\)). The following theorem was proved by M. V. Keldysh \((^7)\) for closed domains and by M. Brelot \((^2)\) for arbitrary closed sets:
If each point \(P\in\partial F\) is a regular point for the complement of \(F\), i.e.
\[ \int_0 \frac{\gamma[K(P,\delta)\setminus F]}{\delta^2}\,d\delta=\infty, \]
then the sequence \(h_n(P),\ n=1,2,\ldots,\) converges uniformly on \(\partial F\) to the function \(\varphi(P)\).
From the Keldysh–Brelot theorem there follows
Lemma. Let \(g\) be a bounded open set. If each point \(P\in\partial g\) is a regular point of the set \(g\), i.e.
\[ \int_0 \frac{\gamma[g\cap K(P,\delta)]}{\delta^2}\,d\delta=\infty, \tag{1} \]
then \(\gamma(g)=\gamma(\overline{g})\).
Indeed, let \(F\) be the complement of \(g\); let \(W(P)\) be the potential of the set \(\overline{g}\). From the regularity condition for the points \(P\in\partial g\) it follows that \(W(P)=1\) for \(P\in\overline{g}\), in particular, \(W(P)=1\) on the boundary of the set \(F\). Put \(\widetilde{\varphi}(P)=W(P)\). Let \(g_n\) be a sequence of “good” open sets converging to \(F\) (see above); let \(e_n\) be the complement of \(g_n\). The solution of the Dirichlet problem in \(g_n\) with boundary data \(\widetilde{\varphi}(P)=1,\ P\in\partial g_n\), is the potential \(W_n(P)\) of the set \(e_n\). From the Keldysh–Brelot theorem it follows that \(W_n(P)\) converges uniformly to \(W(P)\) on \(\partial F\), and consequently also on \(F\). Hence it follows that \(\gamma(e_n)\to\gamma(\overline{g})\); since \(e_n\subset g\), we obtain the assertion of the lemma.
We now show that \(2^\circ \Rightarrow 3^\circ\). Fix a point \(P_0\in C^3\) and \(\delta_0>0\). Put \(g=D\cap K(P_0,\delta_0)\). Obviously, we have \(\overline{g}=\overline{K(P_0,\delta_0)}\) (from \(2^\circ\) it follows that \(D\) is an everywhere dense set); hence \(\gamma(\overline{g})=\delta_0\). We show that at every point \(P\in K(P_0,\delta_0)\) condition (1) is satisfied. At every point \(P\in K(P_0,\delta_0)\) condition (1) is obviously satisfied, since for sufficiently small \(\delta>0\)
\[ \gamma[g\cap K(P,\delta)]=\gamma[D\cap K(P,\delta)]\ge A_0\delta. \]
Let \(P\) be a boundary point of the ball \(K(P_0,\delta_0)\). Consider the ball \(K(P,\delta)\); for sufficiently small \(\delta>0\), \(K(P,\delta)\) contains a ball \(K(P',\delta/4)\) which belongs entirely to \(K(P_0,\delta_0)\). We have
\[ \gamma[g\cap K(P,\delta)]\geq \gamma\left[g\cap K\left(P',\frac{\delta}{4}\right)\right] = \gamma\left[D\cap K\left(P',\frac{\delta}{4}\right)\right]\geq \frac{A_0}{4}\delta, \]
whence (1) follows. Thus (1) holds for all \(P\in \partial g \in \overline{K}(P_0,\delta_0)\). By the lemma,
\[ \gamma(g)=\gamma(\overline g)=\delta_0. \]
Property \(3^0\) is proved.
In a completely analogous way one can show that if property \(2^0\) is satisfied, then for open sets \(\Gamma\) with a sufficiently good boundary
\[ \gamma(D\cap \Gamma)=\gamma(\Gamma)=\gamma(\overline{\Gamma}). \]
This implies Theorem 3.
- Analogues of the results obtained above are also valid for \(m\)-dimensional Euclidean space \(C^m\) (we retain the notation adopted above for the case \(m=3\)). The capacity \(\gamma\) (with respect to the Newtonian potential in \(C^m\)) of the ball \(K(P,\delta)\) of radius \(\delta\) in the space \(C^m\) is equal to \(\delta^{m-2}\). Accordingly, analogues of Theorems 1 and 2 in \(m\)-dimensional space are formulated as follows*.
Theorem \(1'\). If for almost all (with respect to Lebesgue measure in \(C^m\)) points \(P\in C^m\)
\[ \overline{\lim_{\delta\to 0}}\, \frac{\gamma[D\cap K(P,\delta)]}{\delta^m} =\infty, \]
then for all \(P\in C^m\) and \(\delta>0\)
\[ \gamma[D\cap K(P,\delta)]=\delta^{m-2}. \]
Theorem \(2'\). In order that \(C(E)=H(E)\), \(E\subset C^m\), it is necessary and sufficient that for all \(P\in C^m\) and \(\delta>0\)
\[ \gamma[K(P,\delta)\setminus E]=\delta^{m-2}. \]
Theorem 4 for \(E\subset C^m\) is valid in the same formulation.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
11 III 1965
References
- M. V. Keldysh, UMN, vol. 8 (1941), 73, 55 (1945).
- M. Brelot, Bull. Soc. Math. de France.
- A. A. Gonchar, DAN, 154, No. 3 (1964).
- A. A. Gonchar, Izv. AN SSSR, ser. matem., 27, No. 4 (1963).
* We note that in works (3, 4), in the formulation of the approximation theorem for the case \(C^m\) there is an inaccuracy; the inequality for the capacity in Theorem B′, b) should be as follows: \(\gamma(P,\delta)\geq A_0\delta^{m-2}\).