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UDC 517.53:517.947.42
MATHEMATICS
I. S. OVCHINNIKOV
METRIC PROPERTIES OF MAPPINGS THAT LEAVE CERTAIN INTEGRAL FUNCTIONALS BOUNDED
(Presented by Academician M. A. Lavrent′ev, January 20, 1969)
1. Let a continuous vector-function \(y=f(x)\) with values in a set \(\Delta \subset E^m\) be given in a domain \(D\) of \(n\)-dimensional Euclidean space \(E^n\) \((n \geq 2)\), where \(x=(x_1,\ldots,x_n)\), \(y=(y_1,\ldots,y_m)\), \(y_i=f_i(x)\) \((i=1,2,\ldots,m)\). Suppose that this function leaves bounded the functional
\[ I(f,D,F)=\int_D F\left(x,f,\frac{df}{dx}\right)\,dx, \]
where
\[ \frac{df}{dx}=\left(\frac{\partial f_i}{\partial x_j}\right) \]
is a rectangular \(n \times m\) matrix whose elements are the partial derivatives, understood in the sense of S. L. Sobolev, and \(dx=dx_1\ldots dx_n\) is the volume element. With respect to \(F(x,y,Z)\) we assume that it is a measurable function of its arguments, and moreover such that the inequality
\[ F(x,y,Z)\geq h^n(x,y)\|Z\|^n \]
is satisfied, where \(h(x,y)\) is a certain continuous nonnegative function defined in \(D\times\Delta\); \(Z=(z_{ij})\) is a rectangular \(n\times m\) matrix;
\[ \|Z\|=\left(\sum_{i=1}^{n}\sum_{j=1}^{m}z_{ij}^{2}\right)^{1/2}. \]
Consider on the set \(\Delta\) a non-Euclidean metric with length element \(ds=h\,dl\), where \(dl\) is the length element in \(E^m\); \(h=h(f^{-1}(y),y)\) in the case when \(y=f(x)\) is a homeomorphism; if, however, \(y=f(x)\) is not a homeomorphism, then we assume that \(h(x,y)\) does not depend on \(x\), and put \(h=h(x,y)\). We shall call the metric thus introduced the metric \(h\).
Let \(\{S_r\}\) be a family of concentric spheres of radii \(r\) with center at some point \(b\in E^n\): \(S_r=\{x\in E^n: |x-b|=r\}\), \(r_1\leq r\leq r_2\), \(r_1\ne r_2\), and such that the sets \(S'_r=S_r\cap D\) are nonempty for all \(r\in [r_1,r_2]\). Suppose that on each set \(S'_r\) an open spherical disk \(K_r\) of spherical radius \(R(r)\leq \pi r/2\) has been chosen. Let measurable nonnegative functions \(\Omega(r)\) and \(\beta(r)\) be defined on the segment \([r_1,r_2]\), such that \(\Omega(r)\leq d_h(f(K_r))\), \(\beta(r)\geq 2R(r)/\pi r\), where \(d_h(G)\) is the diameter of the set \(G\subset \Delta\) in the metric \(h\).
Theorem. Under the assumptions made, the inequality
\[ \int_{r_1}^{r_2}\frac{\Omega^n(r)}{r\beta(r)}\,dr \leq M_n I(f,D_{r_1r_2},F), \tag{1} \]
holds, where \(D_{r_1r_2}=\bigcup_{r_1\leq r<r_2} S'_r\); \(M_n\) is an absolute constant depending only on \(n\).
Proof is based on an inequality from paper \((^1)\) and is a generalization of the inequality of paper \((^2)\).
Inequality (1) makes it possible to obtain a number of important metric properties of mappings.
We shall use the following notation: \(\rho(M_1,M_2)\) is the distance between sets in \(E^n\); \(|x'-x''|\) is the distance between points in \(E^n\); \(\overline M\) is the closure of the set \(M\) in \(E^n\); \(\partial D\) is the boundary of the domain \(D\) in \(E^n\); \(d(M)\) is the diameter of the set \(M\) in \(E^n\); \(\rho_h(M_1,M_2)\) is the distance between sets in the metric \(h\); \(\widetilde E^n\) is the completion of \(n\)-dimensional space with respect to the spherical metric \(\widetilde\rho(x',x'')\); \(\partial\widetilde D\) is the boundary of the domain \(D\) in \(\widetilde E^n\).
Below we consider a family \(\{f\}\) of homeomorphic mappings \(y=f(x)\), defined in a domain \(D\subset E^n\) and with values in a domain \(\Delta\subset E^n\), \(f(D)=\Delta_f\subset\Delta\) (in particular, it may be that \(\Delta=E^n\)). We assume that the boundary of the domain \(D\) is connected in \(\widetilde E^n\).
To consider various normalizations of the family of functions \(\{f\}\), we introduce the following classes of mappings. Suppose that \(a\) is some point of \(D\), and that \(M,\delta,\delta_1,\delta_2\) are arbitrary positive numbers.
Let \(\infty\notin D\) (the domain \(D\) contains no exterior of any ball in \(E^n\)). We shall say that \(\{f\}\subset A_1(a,M,\delta)\) if \(|f(a)|\le M\), \(\rho(f(a),\partial\Delta_f)\le\delta\) for all \(f\in\{f\}\); \(\{f\}\subset A_2(a,\delta)\) if \(\rho(f(a),\partial\Delta_f)\ge\delta\) for all \(f\in\{f\}\); \(\{f\}\subset A_3(a,M,\delta_1,\delta_2)\) if \(|f(a)|\le M\), \(\delta_1\le\rho(f(a),\partial\Delta_f)\le\delta_2\) for all \(f\in\{f\}\).
Let \(\infty\in D\). In this case we shall assume that \(f(\infty)=\infty\) and that the functions \(f\) are continuous in \(\widetilde E^n\) at the point \(x=\infty\). We shall say that \(\{f\}\subset B_1(M,\delta)\) if \(\rho(\partial\Delta_f,0)\le M\) and \(d(\partial\Delta_f)\le\delta\) for all \(f\in\{f\}\); \(\{f\}\subset B_2(\delta)\) if \(d(\partial\Delta_f)\ge\delta\) for all \(f\in\{f\}\); \(\{f\}\subset B_3(M,\delta_1,\delta_2)\) if \(\rho(\partial\Delta_f,0)\le M\) and \(\delta_1\le d(\partial\Delta_f)\le\delta_2\) for all \(f\in\{f\}\).
2. Order of growth. Let
\[ I(f^{-1},\Delta_f,\Phi)=\int_{\Delta_f}\Phi\left(y,f^{-1},\frac{df^{-1}}{dy}\right)\,dy\le K, \tag{2} \]
where \(K\) does not depend on \(f\in\{f\}\), and \(\Phi(y,x,Z)\ge h^n(x)\|Z\|^n\), with \(h(x)\) a nonnegative continuous function defined in \(D\).
Let \(\{f\}\subset A_1(a,M,\delta)\). Then for every connected compact set \(G\subset D\) containing the point \(a\), we shall have
\[ |f|_G\le M+\delta\exp\left[M_nK\rho_h^{-n}(G,\partial D)\right], \tag{3} \]
where
\[ |f|_G=\sup_{x\in G}|f(x)|. \]
Let now \(\{f\}\subset B_1(M,\delta)\). Then for every bounded connected set \(G\subset D\) such that the set \(\partial D\cap \overline G\) is nonempty, the inequality
\[ |f|_G\le M+\delta\exp\left[M_nKq_h^{-n}(G,D)\right], \tag{4} \]
holds, where \(q_h(G,D)=\inf d_h(S)\), and the infimum is taken over all Jordan surfaces \(S\subset D\) such that \(\partial D\) is contained inside \(S\) and the set \(S\cap G\) is nonempty.
Remark 1. Suppose that the set of zeros of the function \(h(x)\) is compact in \(D\). Then \(\rho_h(G,\partial D)>0\) for any \(G\), and inequality (3) shows that the family \(\{f\}\) is uniformly bounded on compact subsets of the domain \(D\). Here \(h(x)\) may tend to zero if \(\rho(x,\partial D)\to0\).
Suppose that the function \(h(x)\) is positive in \(D\), and let there exist a point \(b\in\partial D\) and a positive number \(\beta\) such that, in some neighborhood of the point \(b\), the inequality \(h(x)\ge\beta\) holds. Then \(q_h(G,D)>0\) for any \(G\), and inequality (4) gives uniform boundedness of the family \(\{f\}\) on every set \(G\).
Consider an example. Let \(\Phi(y,x,Z)=h^n(x)\|Z\|^n\), where \(h(x)=\dfrac{1}{1+|x|^2}\) defines the spherical metric in \(E^n\). Let \(\{f\}\) be a fami-
of \(Q\)-quasiconformal mappings. Then
\(I(f^{-1}, \Delta_f, \Phi) \leq n^{n/2}Q^{n-1}\tilde mD\),
where by \(\tilde mD\) is denoted the spherical volume of the domain \(D\). In this case we may put
\(K=n^{n/2}Q^{n-1}\tilde mD\), and inequalities (3) and (4) give estimates for the order of growth for the family of mappings \(\{f\}\).
3. Estimate from below of the distortion of the distance to the boundary. Let condition (2) be satisfied for the functions \(f\in\{f\}\). If \(\{f\}\subset A_2(a,\delta)\), then for every connected compact set \(G\subset D\) containing the point \(a\), the inequality
\[ \rho(f(G),\partial\Delta_f)\geq \delta\exp[-M_nK\rho_h^{-n}(G,\partial D)] . \tag{5} \]
holds.
If \(\{f\}\in B_2(\delta)\), then for every connected set \(G\subset D\) such that \(\rho(G,\partial D)>0\), inequality (5) also holds.
4. Covering theorem. Suppose that \(\{f\}\subset A_2(a,\delta)\). Then for every closed domain \(G\subset D\) such that \(a\in G\), \(\rho(a,\partial G)>0\), we have
\[ \rho(f(a),\partial f(G))\geq \delta\exp[-M_nK\varphi_h^{-n}(\rho(a,\partial G),D)], \]
where \(\varphi_h(a,D)=\inf d_h(S)\), and the infimum is taken over all Jordan surfaces \(S\subset D\) for which the point \(a\) lies inside \(S\) and \(d(S)\geq \alpha\).
Remark 2. Let the function \(h(x)\) be positive in \(D\), and let there exist a point \(b\in\partial D\) and positive numbers \(\varepsilon\) and \(\beta\) such that on the set
\(\{x\in D:\rho(x,b)\leq \varepsilon\}\) one has \(h(x)\geq \beta\). Then on the interval \((0,d(D))\) the function \(\varphi_h(a,D)\) is positive.
5. Equicontinuity. Let, for the family \(\{f\}\), the condition
\[ I(f,D,F)\leq K_1,\qquad I(f^{-1},\Delta_f,\Phi)\leq K_2, \]
be satisfied, where
\(F(x,y,Z)\geq u^n(x,y)\|Z\|^n\),
\(\Phi(y,x,Z)\geq h^n(x)\|Z\|^n\), and the functions
\(u(x,y)\) and \(h(x)\) are defined, continuous, and positive in \(D\times\Delta\) and in \(D\), respectively.
If \(\{f\}\subset A_3(a,M,\delta_1,\delta_2)\), then for any two points \(x',x''\in G\) satisfying the condition
\[ |x'-x''|<\rho_1/2,\quad \rho_1=\rho(G,\partial D), \tag{6} \]
the inequality
\[ |f(x')-f(x'')|\leq \frac{1}{m_1}[M_nK_1]^{1/n}\ln^{-1/n}\frac{2\rho_1}{3|x'-x''|} \tag{7} \]
holds, where \(G\) is an arbitrary closed bounded domain from \(D\), with
\(a\in G\), \(\rho(a,\partial G)>0\), and the set \(\partial G\) connected;
\(m_1=\min_{x\in G_1,y\in H}u(x,y)\),
\(G_1=\{x\in D:\rho(x,G)<\rho_1/2\}\),
\(H=\{y\in\Delta:\rho(y,\partial\Delta)>\alpha,\ |y|<\gamma\}\),
\(\alpha=\delta_1\exp[-M_nK_2\rho_h^{-n}(G_1,\partial D)]\),
\(\gamma=M+\delta_2\exp[M_nK_2\rho_h^{-n}(G_1,\partial D)]\).
If \(\{f\}\subset B_3(M,\delta_1,\delta_2)\), then assume additionally that \(h(x)\) satisfies the condition of the remark. Then inequality (7) holds under condition (6), if as the set \(G\) there is taken a bounded closed domain from \(D\) such that \(D\setminus G\) consists of two components, one of which contains \(\infty\) and the boundary of the second contains the set \(\partial D\). In this case \(m_1\) in (7) is defined as before, only another \(\gamma\) is taken, namely
\(\gamma=M+\delta_2\exp[M_nK_2q_h^{-n}(G_2,D)]\), where
\(G_2=D\setminus G_3\), and \(G_3\) is the component of the set \(D\setminus G\) containing \(\infty\).
Corollary 1. Under the conditions of Sec. 5, from the family of mappings \(\{f\}\) one can choose a sequence which converges uniformly on every compact set of \(D\) to a continuous vector function defined in the domain \(D\).
6. Equiopenness. Suppose that the conditions of Sec. 5 are satisfied and, moreover, the function \(h(x)\) satisfies the conditions of Remark 2. Then, in the case where \(\{f\}\subset A_3(a,M,\delta_1,\delta_2)\), for any
two points \(x'\), \(x'' \in G\), satisfying the condition
\[ |x'-x''|<{}^{2}/_{3}\rho_{1}\exp\left\{-M_n K_1\left[\frac{2}{m_1\delta_1}\exp\left(\rho_h^{-n}(G,\partial G_1)+\varphi_h^{-n}(\rho(a,\partial G_1),D)\right)\right]^n\right\}; \tag{8} \]
the inequality
\[ |f(x')-f(x'')|\geq {}^{2}/_{3}\delta_1 \exp\left\{-M_nK_2\left[m_2^{-n}|x'-x''|^{-n} +\rho_h^{-n}(G,\partial G_1)+\varphi_h^{-n}(\rho(a,\partial G_1),D)\right]\right\}, \tag{9} \]
holds, where \(m_2=\min_{x\in G'} h(x)\), and the remaining notation is the same as in item 5.
In the case \(\{f\}\subset B_3(M,\delta_1,\delta_2)\), inequality (9) will hold under condition (8), if in (8) and (9) the term \(\varphi_h^{-n}(\rho(a,\partial G_1),D)\) is omitted.
Corollary 2. Under the conditions of the present item, the family \(\{f\}\) is uniformly open inside the domain \(D\), and relations (8) and (9) give the order of this uniform openness.
Remark 3. Under condition (8), together with inequality (9), inequality (7) is also satisfied, i.e., for the family \(\{f\}\) there is a two-sided estimate of the distortion of distances inside the domain \(D\).
Corollary 3. Let the conditions of item 6 be satisfied for the family \(\{f\}\). Consider some sequence of the family \(\{f\}\) converging at the point \(a\) to the value \(a^*\), if \(\infty \notin D\), and an arbitrary sequence from \(\{f\}\), if \(\infty\in D\). Then from this sequence one can choose a subsequence \(\{f^{(p)}\}\) (\(p=1,2,\ldots\)) such that the sequence of domains \(\{\Delta_{f^{(p)}}\}\) will converge to its kernel \(\Delta_0\) with respect to the point \(a^*\) (or with respect to the point \(y=\infty\), if \(\infty\in D\)), and the sequence \(\{f^{(p)}\}\) will converge uniformly inside \(D\) to a homeomorphic mapping \(y=f(x)\) of the domain \(D\) onto the domain \(\Delta_1\), contained in \(\Delta_0\). Moreover, \(\rho(f(a),\partial\Delta_1)\leq\delta_2\) (in the case \(\infty\notin D\)), and for the mapping \(y=f(x)\) all inequalities that held for the family of mappings \(\{f\}\) are satisfied, with the exception of inequality (5) in the case when \(\infty\in D\), but \(\infty\notin\Delta_1\).
Remark 4. Simple examples show that the domain \(\Delta_1\) need not coincide with \(\Delta_0\). ●
We give a sufficient condition for \(\Delta_1=\Delta_0\). Let \(u(x,y)=u_1(x)\cdot u_2(y)\). Consider the set
\[ E=\left\{c\in\partial\bar D:\ \lim_{\rho(x,c)\to 0}u_1(x)=0\right\}. \]
Note that the set \(E\) is compact in \(\widetilde E^n\). For the case \(\infty\in D\) we shall assume that
\[ \lim_{x\to\infty}u_1(x)>0. \]
Let \(b\) be an arbitrary point of the set \(\partial\bar D\setminus E\). If \(b\ne\infty\), we shall assume that there exists \(\varepsilon>0\) such that, for any sphere \(S_r\) of radius \(r<\varepsilon\) with center at the point \(b\), every component of the set \(S_r\cap D\) divides the domain \(D\) into two parts. If \(b=\infty\), then let spheres of sufficiently large radius with center at the origin of coordinates have the analogous property. Suppose also that \(\tilde\rho(f(x),\partial\Delta_f)\to0\) uniformly with respect to \(f\in\{f\}\) if \(\tilde\rho(x,E)\to0\). Then the domain \(\Delta_1\) coincides with the kernel \(\Delta_0\).
Donetsk Computing Center
Academy of Sciences of the Ukrainian SSR
Received
6 I 1969
References
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- I. S. Ovchinnikov, G. D. Suvorov, Sibirsk. matem. zhurn., 6, 6, 1292 (1965).