UDC 517.54

MATHEMATICS

I. S. OVCHINNIKOV

SOME PROPERTIES OF PLANE AND SPATIAL MAPPINGS

(Presented by Academician M. A. Lavrent’ev on 29 V 1969)

1. In this note we formulate results established on the basis of inequality (1) from the work (¹).

We shall consider a family \(\{f\}\) of homeomorphic mappings \(y=f(x)\), defined in a domain \(D\) of \(n\)-dimensional Euclidean space \(E^n\) (unless the contrary is stipulated). Denote

\[ I(f,D,F)=\int_D F\left(x,f,\frac{df}{dx}\right)\,dx \tag{1} \]

the functional defined on the mappings of the family \(\{f\}\), where in (1) \(x=(x_1,\ldots,x_n)\), \(f=(f_1(x),\ldots,f_n(x))\), \(df/dx=(\partial f_i/\partial x_j)\) is an \(n\times n\)-matrix whose elements are partial derivatives understood in the sense of S. L. Sobolev, \(F(x,y,Z)\) is a measurable function of its \(2n+n^2\) arguments, and \(Z=(z_{ij})\) is an \(n\times n\)-matrix.

Below the following notation will be used: \(\rho(M_1,M_2)\) 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 \(E^n\); \(\widetilde E^n\) is the completion of \(n\)-dimensional space with respect to the spherical metric \(\widetilde\rho(x',x'')\); \(d_{\sim}(M)\) is the diameter of the set \(M\) in the metric \(\widetilde\rho\). Suppose that in the domain \(D\) a positive continuous function \(h(x)\) is given, which generates in \(D\) a non-Euclidean metric with line element \(ds=h(x)\,dl\), where \(dl\) is the line element in \(E^n\). By \(\rho_h(M_1,M_2)\) we shall denote the distance between the sets \(M_1\) and \(M_2\) in this non-Euclidean metric, and by \(d_h(M)\) the diameter of the set \(M\) in this metric.

1. Theorem 1. Let, for a family of mappings \(\{f\}\) of a domain \(D\) onto a domain \(\Delta\), the functional

\[ I(f,\Delta^{-1},\Phi)=\int_{\Delta}\Phi\left(y,f^{-1},\frac{df^{-1}}{dy}\right)\,dy\leq K, \]

be bounded, where \(\Phi(y,x,Z)\geq h^n(x)\|Z\|^n\), the function \(h(x)\) is defined, continuous, and positive in \(D\),

\[ \|Z\|=\left(\sum_{i=1}^n\sum_{j=1}^n z_{ij}^{\,2}\right)^{1/2}. \]

Assume that there exist a continuum \(H\subset D\) and a number \(a>0\) such that

\[ d(f(H))>2\rho(f(H),\partial\Delta),\qquad d(f(H))\geq a, \tag{2} \]

for all \(f\in\{f\}\). Then

\[ \rho(f(H),\partial\Delta)\geq \frac{\alpha}{2}\exp[-M_nK\rho_h^{-n}(H,\partial D)], \tag{3} \]

where \(M_n\) is an absolute constant \((M_n>0)\).

Remark. Inequality (3) shows that, for any point \(a\in H\), under each mapping \(f\in\{f\}\) the ball of radius \(\frac12\alpha\exp[-M_nK\rho_h^{-n}(H,\partial D)]\) with center at the point \(f(a)\) lies entirely in the domain \(\Delta\).

We shall give, for the planar case, some sufficient conditions for the fulfillment of relations (2).

a) Let the domain \(D\) be simply connected, and suppose that on the boundaries of the domains \(D\) and \(\Delta\) three distinct prime ends \(e_i\) and \(e_i^*\) are fixed, with \(f(e_i)=e_i^*\) for each \(f\in\{f\}\). Let \(a\) be a certain principal point of the prime end \(e_1\). Suppose that there exists a domain \(U_1\subset D\), into which the prime end \(e_1\) enters, and a neighborhood \(U_2\) of the point \(a\) in \(E^2\), such that for all \(f\in\{f\}\) the inequality

\[ I(f,U_3,F)=\int_{U_3} F\left(x,f,\frac{df}{dx}\right)\,dx\leqslant K<\infty, \tag{4} \]

holds, where \(U_3=U_1\cap U_2\), \(F(x,y,Z)\geqslant h^2(y)\|Z\|^2\), and \(h(y)\) is a continuous function in \(E^2\) generating a metric \(\rho_h\) topologically equivalent to the spherical metric. Suppose that an inequality analogous to (4) is also satisfied for the prime end \(e_2\). Then there exists a continuum \(H\) for which the inequalities (2) are satisfied.

b) Suppose that on the boundaries of the domains \(D\) and \(\Delta\) three distinct prime ends \(e_i\) and \(e_i^*\) \((i=1,2,3)\) are fixed, and that an open arc of prime ends \(e_1e_2\), not containing \(e_3\), is carried under each mapping \(f\in\{f\}\) into the arc of prime ends \(e_1^*e_2^*\), not containing the prime end \(e_3^*\). Suppose that \(f(e_3)=e_3^*\) for all \(f\in\{f\}\). Let there be in the domain \(D\) a curve \(L:x=\pi(t)\) \((0<t<1)\). Put \(E_\varepsilon=[\varepsilon,1-\varepsilon]\) \((0<\varepsilon<1/2)\). Suppose that \(f(\pi(t))\) tends uniformly with respect to \(f\in\{f\}\) to \(e_3^*\) as \(t\to1\), and that \(f(\pi(t))\) tends uniformly with respect to \(f\in\{f\}\) to the arc of prime ends \(e_1^*e_2^*\) as \(t\to0\). Then, for sufficiently small \(\varepsilon\), the continuum \(\pi(E_\varepsilon)\) will satisfy the inequalities (2).

1. Theorem 2. Let there be given a family \(\{f\}\) of quasiconformal mappings, each of which is a homeomorphism of the domain \(D\subset E^2\) onto the strip \(\Delta=\{y:0<y_2<\delta\}\). Suppose that on the boundaries of the domains \(D\) and \(\Delta\) three distinct prime ends \(e_i\) and \(e_i^*\) \((i=1,2,3)\) are fixed, the bodies of the prime ends \(e_1^*\) and \(e_2^*\) being \(\infty\), and the body of the prime end \(e_3^*\) being a certain point \(a\in\partial\Delta\). Suppose that \(f(e_i)=e_i^*\) \((i=1,2,3)\) for all \(f\in\{f\}\). Suppose that for all \(f\in\{f\}\) the inequality

\[ I(f^{-1},\Delta,\Phi)\leqslant K, \]

is satisfied, where \(\Phi(y,x,Z)=h^2(x)\|Z\|^2\), and \(h(x)\) is a continuous positive function in the domain \(D\), and moreover such that in certain subdomains \(g_i\) \((i=1,2,3)\) of \(D\), into each of which one prime end \(e_i\) \((i=1,2,3)\) enters, this function generates a metric \(\rho_h\) topologically equivalent to the spherical metric. Consider a domain \(G\subset D\) having the property that the set \(D\setminus G\) consists of two subdomains, with the prime end \(e_1\) entering one of them and \(e_2\) the other, while the prime end \(e_3\) enters the domain \(G\). Then

\[ \sup_{x\in G}|f(x)-a|\leqslant \delta\exp\left[2\pi K\beta^{-2}(G)\right]. \tag{5} \]

Here \(\beta(F)=\inf d_h(l)\), and the infimum is taken over all curves \(l\subset D\) such that \(l\) separates \(e_3\) from \(e_1\) or from \(e_2\), and the set \(l\cap G\) is nonempty.

Let us note that, when the hypotheses of this theorem are satisfied, \(\beta(G)>0\), and inequality (5) gives a nontrivial estimate for the order of growth up to the boundary for the family of mappings \(\{f\}\).

1. Theorem 3. Suppose that \(y=f(x)\) is a homeomorphic mapping of a simply connected domain \(D\subset E^2\) onto a domain \(\Delta\). Let the boundary of the domain \(\Delta\) contain a rectilinear segment \(\gamma^*\) (finite or infinite), lying on a certain line \(l\) parallel to the axis \(Oy_1\), and let the domain \(\Delta\) lie on one side of the line \(l\). Let some arc of prime ends \(\gamma\)

of the domain \(D\) under the mapping \(y=f(x)\) passes into \(\gamma^*\), and let the integral be bounded
\[ \int_D |\nabla f_2|^2\,dx \leqslant K, \]
where the derivatives are understood in the sense of S. L. Sobolev. Consider a family \(\{S_r\}\) of concentric circles \(S_r\) of radii \(r\) such that the set \(S'_r=S_r\cap D\) is nonempty for \(r\in(0,r_2)\). Let a component \(K_r\) be chosen from each set \(S'_r\), in such a way that the ends of the arc \(K_r\) lie on the arc \(\gamma\), and the family \(\{K_r\}\) determines some prime end \(e\in\gamma^*\). Denote by
\[ \alpha(r)=\sup_{y\in f(K_r)} \rho(y,\gamma^*) \]
the deviation of the set \(f(K_r)\) from the set \(\gamma^*\), and suppose that \(\alpha(r)\) is a measurable function.

Then the inequality
\[ \int_{r_1}^{r_2}\frac{\alpha^2(r)}{r}\,dr \leqslant 2\pi K. \tag{6} \]
holds.

5. Theorem 4. Let \(y=f(x)\) be a homeomorphic mapping of a simply connected domain \(D\subset E^2\) onto a domain \(\Delta\) lying in the strip \(G=\{y:\delta_1<y_2<\delta_2\}\) \((\delta_i=\mathrm{const})\). Let two prime ends \(e_1\) and \(e_2\), dividing the boundary of the domain \(D\) into two open arcs of prime ends \(\gamma_1\) and \(\gamma_2\), be taken on the boundary of the domain \(D\). Denote \(\gamma_i^*=\{y:y_2=\delta_i\}\) \((i=1,2)\), and suppose that for every prime end \(e\in\gamma_i\), \(\rho(f(x),\gamma_i^*)\to0\) as \(x\to e\). Let the integral
\[ \int_g \frac{1}{(1+|f|^2)^2}\,|\nabla f_1|^2\,dx, \tag{7} \]
be bounded, where \(g\) is some subdomain of \(D\), into which the prime end \(e_1\) enters, and \(q=\partial g\setminus \partial D\) is a section of the domain \(D\). Denote
\[ G_{-\delta}=\{y\in G:y_2\leqslant-\delta\},\quad G_\delta=\{y\in G:y_2\geqslant\delta\},\quad R_\delta=\{y\in G:y_1=\delta\}. \]
Then two cases are possible: 1) there exists a number \(E>0\) such that either \(G_{-E}\subset\Delta\), or \(G_E\subset\Delta\); 2) there exists a number \(E\) such that either \(G_{-E}\subset G\setminus\Delta\) and \(R_{-E}\subset\partial\Delta\), or \(G_E\subset G\setminus\Delta\) and \(R_E\subset\partial\Delta\).

Suppose now that, instead of boundedness of the integral (7), boundedness of the integral
\[ \int_g \frac{1}{(1+|f|^2)^2}\,|\nabla f_2|^2\,dx \]
holds.

Denote by \(G_1\) the component of the set \(G\setminus f(q)\) containing the set \(f(g)\). Then the set \(f(g)\) may be obtained from the set \(G_1\) by making a finite or countable number of cuts in \(G_1\), going from infinity parallel to the axis \(Oy_1\).

Remark. For a conformal mapping \(y=f(x)\), under the hypotheses of Theorem 4 the domain \(\Delta\) will always coincide with the strip \(G\). The following example shows that case 2) in Theorem 4 can actually occur. Let
\[ D=\{x\in E^2:0<x_2<1\},\quad \Delta=\{y\in E^2:0<y_1,\ 0<y_2<1\}, \]
\[ y_1=f_1(x)\equiv e^{x_1},\quad y_2=f_2(x)\equiv x_2. \]
In this case all the hypotheses of Theorem 4 are fulfilled, and the integral (7) is bounded for \(g=D_{-E}\), \(E=1\).

Theorem 5. Suppose that \(y=f(x)\) is a homeomorphic mapping of a simply connected domain \(D\subset E^2\) onto a domain \(\Delta\). Let \(\{S_r\}\) be a family of concentric circles such that the set \(S'_r=S_r\cap D\) is not

* A family of sections \(\{q_\tau\}\), \(\tau\in(\tau_1,\tau_2)\), determines a prime end \(e\) of the domain \(D\) if, for every sequence of numbers \(\{\tau^{(n)}\}\) \((n=1,2,\ldots)\), strictly monotonically converging to \(\tau_i\) \((i=1\) or \(i=2)\), the sequence of sections \(\{q_{\tau^{(n)}}\}\) determines the prime end \(e\).

empty for \(r \in (\tau_1,\tau_2)\), with \(\tau_1=0\) if \(\tau_2 \ne +\infty\). Suppose that \(\{K_r\}\) is a family of sections defining some simple end \(e\) of the domain \(D\), where \(K_r\) is a component of the set \(S'_r\). Let the function \(d(f(K_r))\) be a measurable function of the variable \(r\), and let the set \(D_1=\bigcup K_r\) be measurable. Let the integral \(I(f,D,F)\) be finite, where
\[ F(x,y,Z)=u^2(x)\,(1+|y|^2)^{-2}\|Z\|^2, \]
\(u(x)\) is a continuous positive function in \(D\) such that \(u(x)\to 0\) as \(x\to e\), \(\inf\limits_{x\in g/g_1} u(x)>0\), where \(g\) is some subdomain into which the simple end \(e\) enters, and \(g_1\subset g\) is any subdomain into which the simple end \(e\) enters.

Then for every segment \([r_1,r_2]\subset(\tau_1,\tau_2)\) there exists an \(\bar r \in [r_1,r_2]\) such that
\[ d_\rho(f(K_{\bar r})) \le [2\pi \varkappa(r_1,r_2) I(f,D_1,F)]^{1/2}\ln^{-1/2} r_2/r_1, \tag{8} \]
where
\[ \varkappa(r_1,r_2)=\sup_{[r_1,r_2]}\beta(r)\big/ \inf_{x\in D_1} u^2(x), \]
\(\beta(r)=l(K_r)/\pi r\), and \(l(K_r)\) is the length of the arc \(K_r\).

Remark. Inequality (8) makes it possible to draw conclusions about the behavior of the mapping \(y=f(x)\) in a neighborhood of the simple end \(e\), when this neighborhood has the form of a “zero angle,” whose “vertex” is the body of the simple end \(e\).

Example. Suppose that for the mapping \(y=f(x)\) the conditions of Theorem 5 are satisfied; let \(u(x)=x_2^\alpha\), \(-1\le \alpha<0\), and let the domain \(D\) lie in the upper half-plane. Suppose that the simple end \(e\) has \(\infty\) as its body and that there exists a subdomain \(g\), into which the simple end \(e\) enters, such that \(g\) is contained between two parallel straight lines. Then, using inequality (8), one can show that under the mapping \(y=f(x)\) the simple end \(e\) passes into some simple end \(e^*\) of the domain \(\Delta\).

Donetsk Computing Center
Academy of Sciences of the Ukrainian SSR

Received
22 V 1969

REFERENCES

1. I. S. Ovchinnikov, DAN, 187, No. 1 (1969).
2. C. Caratheodory, Math. Ann., 73, 323 (1913).