MATHEMATICS

Yu. G. RESHETNYAK

SPATIAL MAPPINGS WITH BOUNDED DISTORTION

(Presented by Academician A. D. Aleksandrov, August 1966)

1. In what follows \(R^n\) denotes \(n\)-dimensional arithmetic Euclidean space. Let \(L\) be a linear mapping of the space \(R^n\) into itself such that \(\det L \ne 0\). The mapping \(L\) transforms every sphere into an ellipsoid. The ratio of the largest semiaxis of this ellipsoid to its smallest semiaxis is called the coefficient of distortion of the mapping \(L\) and will be denoted below by \(q(L)\).

A real-valued function \(F(L)\), defined on the set of all square matrices of order \(n\), is called a conformal norm if \(F\) is a norm in the vector space of matrices of order \(n\) and there exists a constant \(\varkappa_F > 0\) such that for every matrix
\[ F(L) \ge \varkappa_F |\det L|^{1/n}, \]
with equality occurring if and only if \(L=\alpha P\), where \(\alpha\) is a number and \(P\) is an orthogonal matrix.

Set
\[ q_F(L) = [F(L)]^n / \varkappa_F^n |\det L|. \]
The conformal norm of a linear mapping \(L: R^n \to R^n\) is the conformal norm of its matrix.

For every conformal norm \(F\) the inequalities
\[ q(L) \le \varphi_1\bigl(F(L)/\varkappa_F |\det L|^{1/n}\bigr), \qquad F(L)/\varkappa_F |\det L|^{1/n} \le \varphi_2(q(L)), \]
hold, where the functions \(\varphi_1\) and \(\varphi_2\) are such that as \(x \to 1\), \(\varphi_1(x) \to 1\), \(\varphi_2(x) \to 1\).

2. Let \(U\) be a domain, i.e. a connected open set in \(R^n\). We shall say that a mapping \(f: U \to R^n\) belongs to the class \(W_n^1\) if each of the coordinates \(f_1, f_2, \ldots, f_n\) of the vector function \(f\) has in \(U\) generalized first derivatives that are locally summable in \(U\) to the power \(n\).

If \(f: U \to R^n\) is a mapping of class \(W_n^1\), then for almost all \(x \in U\) the linear mapping
\[ df_x(X) = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(x) X_i,\quad X=(X_1, X_2,\ldots,X_n), \]
is defined; we shall call it the formal differential of the mapping \(f\) at the point \(x\). Set
\[ \lambda(x,f) = \sum_{i=1}^n \left|\frac{\partial f}{\partial x_i}(x)\right|^2, \]
\[ J(x,f) = \det(df_x). \]

A mapping \(f: U \to R^n\) is called a mapping with bounded distortion (abbreviated, a b.d. mapping) if \(f \in W_n^1\) and there exists a constant \(K\), \(1 \le K < \infty\), such that for almost all \(x \in U\) the inequality
\[ [\lambda(x,f)]^{n/2} \le n^{n/2} KJ(x,f). \tag{1} \]
holds.

If \(f: U \to R^n\) is an o.i. mapping, then the function \(q(df_x)\) is bounded in \(U\). Its exact upper bound in \(U\) is called the coefficient of distortion of the mapping \(f\) and is denoted by \(q(f,U)\). If \(F\) is a conformal norm, then the function \(q_F(df_x)\) is also bounded in \(U\). Its exact upper bound in \(U\) is called the coefficient of distortion of the mapping \(f\) in the norm \(F\) and is denoted by \(q_F(f,U)\).

A special case of mappings with bounded distortion is formed by the so-called quasiconformal mappings.

A set \(G \subset R^n\) is called a compact domain if \(G\) is compact, its open kernel is connected, and \(G\) is the closure of its open kernel. Let \(f: U \to R^n\) be a continuous mapping, and let \(G \subset U\) be a compact domain. Then \(\mu(y,f|G)\), where \(y \notin \operatorname{Fr} G\) (\(\operatorname{Fr} G\) is the boundary of \(G\)), denotes the degree of the mapping \(f: G \to R^n\) with respect to the point \(y\).

Theorem 1. Every mapping \(f: U \to R^n\) with bounded distortion is continuous and, for almost all \(x \in U\),

\[ f(x+X)=f(x)+df_x(X)+o(|X|). \]

Theorem 2. Let \(f: U \to R^n\) be an o.i. mapping. Then, for every compact domain \(G \subset R^n\) such that \(\operatorname{mes}\operatorname{Fr} G=0\), and for every bounded measurable function \(u(y)\), the equality holds

\[ \int_G u(f(x))\,J(x,f)\,dx = \int_{R^n} u(y)\,\mu(y,f|G)\,dy. \]

Let \(A\) be a compact set in \(R^n\). One says that \(A\) is a set of zero capacity if, for every \(\varepsilon>0\) and every open set \(B \supset A\), one can specify such an infinitely differentiable function \(\varphi\) that \(\varphi(x)=0\) for \(x \notin B\), \(\varphi(x)\ge 1\) for \(x \in A\), and

\[ \int_{R^n}\sum_{i=1}^n \left|\frac{\partial \varphi}{\partial x_i}(x)\right|^n dx <\varepsilon. \]

Let \(A\) be an arbitrary set contained in the open domain \(U\) and closed relative to \(U\). We shall say that \(A\) is a set of zero capacity if every compact subset of it has zero capacity.

Theorem 3. Let \(f: U \to R^n\) be an o.i. mapping, not constant in \(U\), such that, for every compact domain \(G \subset U\), the function \(\mu(y,f|G)\) is bounded. Then, for any set \(A\) of zero capacity, the set \(f^{-1}(A)\) is also a set of zero capacity.

Theorem 4. Let \(f: U \to R^n\) be an o.i. mapping, not constant in \(U\), and such that, for every compact domain \(G \subset U\), the function \(\mu(y,f|G)\) is bounded. Then \(f\) is an open mapping.

For every point \(y\) and every compact domain \(G \subset U\), \(y \notin f(\operatorname{Fr}G)\), the set \(f^{-1}(y)\cap G\) consists of no more than \(\mu(y,f|G)\) elements.

Theorem 5. Let \(\{f_m: U \to R^n\}\), \(m=1,2,\ldots\), be a sequence of o.i. mappings such that the sequence \(\{q(f_m,U)\}\) is bounded and, as \(m \to \infty\), the mappings \(f_m\) converge to some mapping \(f: U \to R^n\), the convergence being uniform on every compact set \(A \subset U\). Then the limiting mapping \(f\) is a mapping with bounded distortion. Moreover, for every conformal norm \(F\) the inequality holds:

\[ q_F(f,U)\le \varliminf_{m\to\infty} q_F(f_m,U). \]

The following two theorems concern the so-called quasiconformal mappings. The class of quasiconformal mappings coincides with the class of topological mappings with bounded distortion.

Let \(f: U \to R^n\) be a quasiconformal mapping. Then, for almost all \(x \in U\), the linear mapping \(df_x\) is nondegenerate. Denote by

\(E_f(x)\) is an ellipsoid which is transformed by the mapping \(df_x\) into the unit sphere of the space \(R^n\).

Theorem 6. Let \(f: U \to R^n\) and \(g: U \to R^n\) be quasiconformal mappings of the domain \(U\). Then, if for almost all \(x \in U\) the ellipsoids \(E_f(x)\) and \(E_g(x)\) are similar, there exists a Möbius mapping \(\varphi(y)\) such that
\[ g(x)=\varphi[f(x)] \]
for all \(x \in U\).

This theorem can be supplemented by a certain stability theorem.

Fix a bounded domain \(U \subset R^n\) and a certain bounded quasiconformal mapping \(f: U \to R^n\). Denote by \(W_n^1(U,M)\) the set of all mappings \(g\) of class \(W_n^1\) of the domain \(U\) into \(R^n\) such that for almost all \(x \in U\), \(|g(x)| \le M\). For \(g \in W_n^1\) put
\[ \|g(x)\|_{W_n^1(G)} = \int_G \sum_{i=1}^n \left|\frac{\partial g}{\partial x_i}\right|^n dx, \]
where \(G \subset U\),
\[ V(g,U)=\int_U J(x,g)\,dx. \]

We also prescribe a certain conformal norm \(F\). The ellipsoid \(E_f(x)\) can be given by an equation of the form
\[ |T(x,f)X|=\operatorname{const}, \]
where \(T(x,f)\) is a positive definite symmetric matrix such that
\[ \det T(x,f)=1. \]
The coefficients of this matrix are measurable functions of the variable \(x\). All eigenvalues of the matrix \(T(x,f)\) lie in some interval \((\alpha,\beta)\), where \(0<\alpha<\beta\), and \(\alpha\) and \(\beta\) depend only on the quantity \(q(f,U)\). Put, for \(g \in W_n^1\),
\[ D(g;f,F)=\int_U \{F[dg_x \circ (T(x,f))^{-1}]\}^n dx. \]

Let us note that the mapping \(f(x)\) gives the least value to the functional \(D(g;f,F)\) in the class of mappings \(g \in W_n^1\) coinciding with \(f\) in a neighborhood of the boundary of the domain \(U\).

Theorem 7. For every \(\varepsilon>0\), every compact domain \(G \subset U\), and every \(M>0\), there exists \(\delta>0\) such that, for every mapping \(g \in W_n^1(U,M)\) such that
\[ D[g;f,F]\le \varkappa_F^n V(g,U)(1+\delta), \]
one can indicate a Möbius mapping \(\varphi(y)\) for which the inequality
\[ \|g(x)-\varphi[f(x)]\|_{W_n^1(G)}<\varepsilon \]
holds.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
23 VI 1966