Abstract
Full Text
MATHEMATICS
Yu. G. Reshetnyak
ON ONE SUFFICIENT CRITERION FOR HÖLDER CONTINUITY OF A MAPPING
(Presented by Academician M. A. Lavrent’ev on 16 X 1959)
Let (\mathbf y(\mathbf x)=(y_1(\mathbf x), y_2(\mathbf x), \ldots, y_m(\mathbf x))) be a continuous mapping of an open domain (M) in (n)-dimensional Euclidean space (E^n) into (m)-dimensional space (E^m), (m \geq n). We say that (\mathbf y(\mathbf x)\in W_p^l(M)) if each of the functions (y_k(\mathbf x)), (k=1,2,\ldots,m), has in (M) generalized partial derivatives of order (l), in the sense of S. L. Sobolev ((^1)), summable on every compact set (F\subset M). We put
[
\lambda[\mathbf x;\mathbf y]=\sum_{i=1}^{m}\sum_{j=1}^{n}
\left[\frac{\partial y_i(\mathbf x)}{\partial x_j}\right]^2 .
]
Let (Q(\mathbf x_0,r)) be the ball (|\mathbf x-\mathbf x_0|<r), and (S(\mathbf x_0,r)) the sphere (|\mathbf x-\mathbf x_0|=r) in (E^n). For an arbitrary measurable function (\lambda(\mathbf x)\geq 0) we put
[
V_\lambda(\mathbf x,r)=
\iint_{Q(\mathbf x,r)} [\lambda(\mathbf x)]^{n/2}\,dx_1\ldots dx_n,
]
[
F_\lambda(\mathbf x,r)=
\int_{S(\mathbf x,r)} [\lambda(\mathbf x)]^{(n-1)/2}\,d\sigma,
]
where (d\sigma) is the element of area of the sphere (S(\mathbf x,r)).
A mapping (\mathbf y(\mathbf x)) of the domain (M\subset E^n) is called a mapping of class (I(k)), (0<k<1), if for every point (\mathbf x\in M), for (0<r\leq r_0(\mathbf x)), where (r_0(\mathbf x)) is the distance from (\mathbf x) to the boundary of (M), the inequality
[
[F_\lambda(\mathbf x,r)]^n \geq (nk)^{n-1}\omega_{n-1}[V_\lambda(\mathbf x,r)]^{n-1},
\tag{1}
]
holds, where (\lambda(\mathbf x)=\lambda(\mathbf x;\mathbf y)), and (\omega_{n-1}) is the area of the sphere (S(0,1)).
Theorem 1. If the mapping (\mathbf y(\mathbf x)\in I(k)), then for every compact set (F\subset M), when (\mathbf x_1\in F), (\mathbf x_2\in F),
[
|\mathbf y(\mathbf x_1)-\mathbf y(\mathbf x_2)|\leq A|\mathbf x_1-\mathbf x_2|^k,
]
where the quantity (A) depends only on (d=\inf_{\mathbf x\in F} r_0(\mathbf x)) and on
[
|\lambda|{L=}(F)
\left(\iint_F [\lambda(\mathbf x)]^{n/2}\,dx_1\ldots dx_n\right)^{2/n}.
]
Proof. Applying Hölder’s inequality, we obtain
[
F_\lambda(\mathbf x,r)\leq
\left(\int_{S(\mathbf x,r)} [\lambda(\mathbf x)]^{n/2}\,d\sigma\right)^{(n-1)/n}
\left(\int_{S(\mathbf x,r)} d\sigma\right)^{1/n}
=
\left(\frac{dV_\lambda(\mathbf x,r)}{dr}\right)^{(n-1)/n}
(\omega_{n-1}r^{\,n-1})^{1/n}.
\tag{2}
]
Hence, substituting (2) into (1), we obtain:
[
r \frac{d V_\lambda (x,r)}{dr} - nk V_\lambda (x,r) \geqslant 0.
\tag{3}
]
Multiplying both sides of (3) by (r^{-1-nk}) and integrating with respect to (r) from (r_1) to (r_2), (0<r_1\leqslant r_2<r_0(x)), we obtain
[
\frac{V_\lambda (x,r_2)}{r_2^{nk}}-\frac{V_\lambda (x,r_1)}{r_1^{nk}}\geqslant 0.
]
It follows that for (0<r<d), where (d=\inf_{x\in F} r_0(x)),
[
V_\lambda (x,r)\leqslant \frac{V_\lambda (x,d)}{d^{nk}} r^{nk}\leqslant A r^{nk},
\tag{4}
]
where (A) depends only on (d) and on (|\lambda|_{L^{n/2}(F)}). On the basis of Morrey’s theorem ((^2)), the assertion of the theorem follows from this.
Let us note some applications of Theorem 1. A. G. Sigalov ((^3)) proved a theorem on the existence of a minimum in the class of continuous surfaces (x(u,v)) belonging to (W_2^1(Q)), for two-dimensional functionals of the calculus of variations of the form
[
J(x,Q)=\iint_Q F(x,x_u \times x_v)\,du\,dv,
\tag{5}
]
where (Q) is the square ((0\leqslant u\leqslant 1,\ 0\leqslant v\leqslant 1)); the function (F) satisfies certain conditions, in particular, for all ((x,a))
[
0<m|a|\leqslant F(x,a)\leqslant M|a|
]
((m) and (M) are constants). In the course of the proof of this theorem, A. G. Sigalov established that the surface realizing the minimum, in our terminology, belongs to the class (I\left(\frac{m}{M}\right)).
Hence, on the basis of Theorem 1, we obtain the following theorem.
Theorem 2. If a function (x(u,v)\in W_2^1(Q)) realizes the minimum of the functional (5) under the conditions of A. G. Sigalov’s theorem, then it satisfies the Hölder condition inside (Q) with exponent (k=m/M).
Let (m=n). A mapping (y(x)) is called quasiconformal if:
a) (y(x)) is a topological mapping;
b) the vector function (y(x)) has continuous first derivatives (\partial y(x)/\partial x^k), (k=1,2,\ldots,n), and the Jacobian of the mapping
[
J(x)=\frac{\partial (y_1,y_2,\ldots,y_n)}{\partial (x_1,x_2,\ldots,x_n)}>0
\tag{6}
]
for all (x\in M);
c) there exists a constant (q\geqslant 1) such that, for each point (x\in M), the mapping (y(x)) transforms an infinitesimal sphere with center (x) into an infinitesimal ellipsoid whose ratio of the greatest and least semiaxes does not exceed (q).
The number (q) is called the coefficient of the quasiconformal mapping. For quasiconformal mappings, as is known, the inequality
[
[\lambda(x;y)]^{n/2}\leqslant [k(q)]^{\,n-1} J(x),
\tag{7}
]
holds, where (k(q)<\infty) depends only on (q).
Lemma. Every quasiconformal mapping belongs to the class (I[k(q)]).
Proof. Introduce in (M) a metric with line element
[
dS^2=dy^2=\sum_{i,j} g_{ij}\,dx_i\,dy_j,\qquad
g_{ij}=\frac{\partial y}{\partial x_i}\frac{\partial y}{\partial x_j};
\tag{8}
]
this metric is Euclidean. The area (F_y(S)) and the volume (V_y(Q)) of the interior domain (Q), measured in the metric (8) for a smooth closed surface (S\subset M), satisfy the isoperimetric inequality
[
[F_y(S)]^n \geq n^{\,n-1}\omega_{n-1}[V_y(Q)]^{n-1}.
\tag{9}
]
Let us note that the volume (V_y(Q)) of the domain (Q) is equal to the integral of the Jacobian (J(x)) over the domain (Q).
Consider in (M) the metric
[
dS_1^2=\lambda(x;y)(dx_1^2+\cdots+dx_n^2).
\tag{10}
]
For all (x) and arbitrary (dx_1,\ldots,dx_n),
[
dS_1^2 \geq dS^2.
]
Consequently, the area of the surface (S) in the metric (10) is not less than its area in the metric (8), i.e.
[
\int_S [\lambda(x)]^{(n-1)/2}\,d\sigma \geq F_y(S),
]
where (d\sigma) is the element of area in the ordinary Euclidean metric. On the basis of (9), it follows from this that
[
\left{\int_S [\lambda(x)]^{(n-1)/2}\,d\sigma\right}^n
\geq
n^{\,n-1}\omega_{n-1}
\left{\iint_Q J(x,y)\,dx\right}^{n-1}
\geq
]
[
\geq
n^{\,n-1}\omega_{n-1}[k(q)]^{n-1}
\left(\iint_Q [\lambda(x)]^{n/2}\,dx\right)^{n-1}.
]
The last inequality is obvious, and means that (y(x)\in I(k)).
From the lemma, on the basis of Theorem 1, the following theorem follows.
Theorem 3. Quasiconformal mappings with quasiconformality coefficient (q) of the ball (Q{|x|<1}) onto itself, on every compact set (F\subset Q(0,1)), satisfy the Hölder condition with an exponent depending only on (q), and with a coefficient depending only on (q,F).
Received
9 X 1959
CITED LITERATURE
- S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
- C. B. Morrey, Multiple Integrale Problems in the Calculus of Variations and Related Topics, Univ. Calif. Publ. Math., 1943.
- A. G. Sigalov, Uspekhi Mat. Nauk, 6, issue 2, 16 (1951).