Yu. G. Reshetnyak

On Stability in Liouville’s Theorem on Conformal Mappings of Space

(Presented by Academician S. L. Sobolev on 4 IV 1963)

1. Let \(E^n\) be \(n\)-dimensional Euclidean space, \(|x|\) the length of a vector \(x \in E^n\), and \(Q_r\) the ball \(\{|x|<r\}\), \(0 \le r < 1\). In the space \(E^n\) we assume a fixed Cartesian orthonormal coordinate system. Let \(f(x)\) be a mapping of a domain \(M \subset E^n\) into \(E^n\). We shall say that the mapping \(f(x)\) belongs to the class \(W_n^1(M)\) if the coordinates \(f_1(x), f_2(x), \ldots, f_n(x)\) of the vector function \(f(x)\) have first derivatives, generalized in the sense of S. L. Sobolev \((^1)\), summable to the power \(n\) in the domain \(M\). If, in addition, \(f(x)\) is a topological mapping of the domain \(M\) into itself, then we shall say that \(f(x) \in W_{n,T}^1(M)\).

For an arbitrary mapping \(f(x) \in W_n^1(M)\) we set:

\[ \lambda(x,f)=\sum_{i=1}^{n}\sum_{j=1}^{n}\left(\frac{\partial f_i}{\partial x_j}\right)^2,\qquad D(f,M)=\int_M[\lambda(x,f)]^{n/2}\,dx, \]

\[ J(x,f)=\frac{\partial(f_1,f_2,\ldots,f_n)}{\partial(x_1,x_2,\ldots,x_n)} =\det\left\|\frac{\partial f_i}{\partial x_j}\right\|,\qquad V(f,M)=\int_M J(x,f)\,dx. \]

If the mapping \(f(x) \in W_{n,T}^1(M)\), then \(V(f,M)\ne 0\). We set

\[ \theta(f)=\frac{D(f,M)}{n^{n/2}|V(f,M)|}. \]

Theorem 1. There exists a universal function \(\alpha(\varepsilon,r)\ge 0\), defined for \(\varepsilon>0\), \(0\le r<1\), with \(\alpha(\varepsilon,r)\to 0\) as \(\varepsilon\to 0\), such that for every mapping \(f(x)\in W_{n,T}^1\) of the ball \(Q_1\) into itself for which \(\theta(f)\le 1+\varepsilon\), there exists a Möbius mapping \(g(x)\) such that, for \(|x|\le r<1\), the inequality

\[ |f(x)-g(x)|\le \alpha(\varepsilon,r) \]

holds.

Remark. The function \(\alpha(\varepsilon,r)\) in the theorem just formulated cannot be replaced by a function \(\alpha(\varepsilon)\) independent of \(r\).

Theorem 2. If a mapping \(f(x)\in W_{n,T}^1\) and \(\theta(f)=1\), then the mapping \(f(x)\) is Möbius.

Theorem 1 contains, as a special case, the theorem from the author’s work \((^2)\). The proofs of Theorems 1 and 2 are based on considerations analogous to those given in \((^2)\).

The equicontinuity of a sequence of mappings of the class \(W_{n,T}^1\) of the ball \(Q_1\) into itself is ensured by the following lemma.

Lemma. Let \(f(x)\) be a mapping of the class \(W_n^1(Q_1)\) of the ball \(Q_1\) such that \(D(f,Q_1)\le D_0<\infty\). Then for every point \(x_0\in D_1\) and for every

\(\delta \in [0,1]\) there exists an \(r\) such that \(\delta < r < \sqrt{\delta}\), and the vector-function \(f(x)\) is continuous on the sphere \(S_r = \{|x - x_0| = r\}\). Moreover, for any \(x_1, x_2 \in S_r\) the inequality

\[ |f(x_1) - f(x_2)| \leq C_n \left( \frac{D_0}{\ln 1/\delta} \right)^{1/n} \left| \frac{x_1 - x_2}{r} \right|^{1/n}, \]

holds, where \(C_n\) is a constant.

In the case \(n = 2\), this lemma coincides with the well-known lemma of R. Courant (³). The proof of the lemma in the general case is based on considerations analogous to those used when \(n = 2\). In doing so it is necessary to apply the embedding theorems of S. L. Sobolev (¹).

Received
23 II 1963

CITED LITERATURE

¹ S. L. Sobolev, Some Applications of Functional Analysis to Mathematical Physics, 1951.
² Yu. G. Reshetnyak, in: Some Problems of Mathematics and Mechanics, Novosibirsk, 1961, p. 219.
³ R. Courant, Dirichlet’s Principle, Conformal Mappings, and Minimal Surfaces, IL, 1953.