Hölder Continuity and Differentiability Almost Everywhere of ( K 1 , K 2 ) -Quasiregular Mappings
HHolder Continuity and Differentiability Almost Everywhere of (K K )-Quasiregular Mappings GAO HONGYA1 LIU CHA01 LI JUNWEr2,11. College of Mathematics and Computer Science, Hebei University, Baoding, 071002, China2. Information Center, Hebei Normal College for Nationalities, Chengde, 067000, ChinaAbstract. This paper deals with (K K )-quasiregular mappings. It is shown, by Morrey's Lemma and isoperimetric inequality, that every (K K )-quasiregular mapping satisfies a Holder condition with exponent a on compact subsets of its domain, where { l/K1, a = any positive number less than 1 , 1, for K > 1, for K1 = = =
0, for K1 < 1. Differentiability almost every where of (K K )-quasiregular mappings is also derived. AMS Subject Classification: 30C65. Keywords: (K1, K2)-quasiregular mapping, Holder continuity, differentiability almost everywhere, Morrey's Lemma, isoperimetric inequality. §1 Introduction and Statement of Results Let n be an arbitrary open set in R n , n 2: 2. For any point x E n and r >
0, we denote by B(x, r) the ball with radius r centered at x and S(x, r) = oB(x, r) the sphere of B(x,r). Let IB(x,r)I = W n r n be then-dimensional Lebesgue measure of the ball B(x, r), where W n be the volume of the unit ball in R n . Denote by po,(x) = dist(x, an) the distance from X to an, with the subscript n omitted whenever no confusion can result. For p 2: 1, we denote by LP(n) the £P space of functions on n, W ,P(n) will denote the corresponding Sobolev space of functions in LP(n) whose distributional first derivatives belong also to the space LP(n). Similarly, W ,P(n, R n ) will be the space of functions f = (! , j , · · · , J n ) : n ---t R n such that J i E w =
1, 2, · · · , n. For A an n X n matrix, we define the norm of A as IAI = SUP1e1= IAel-
Research supported by National Natural Science Foundation of China (Grant No. 10971224) and Natural Science Foundation of Hebei Province (Grant No. A2011201011).
We say that a function f : n ----+ Rn does not change sign in n if either u(x) u(x) :::;
0 almost everywhere inn. A mapping f : n ----+ Rn is said to be satisfy Holder condition with exponent a on compact subsets of n, where O < a :::;
1, if for every compact set V cc n there is a number M(V), ::S M(V) < +oo, such that for any x1, x2 E V, lf(x1) - f(x2)I ::S M(V)lx1 - x2la. If f satisfies a Holder condition with exponent a = on compact subsets of n, then f is said to satisfy a Lipschitz condition on compact subsets of n. Let f = (!1,/2,··· , r ) : n----+ Rn be a mapping in W 1,n(n,Rn).
The linear mapping
Df(x) = ( l) axj l'.Si,j'.Sn is defined for almost all x En. Its determinate detDf(x) is called the Jacobian off at the point x, and is denoted by :1 ( x, f). In [1], Zheng and Fang gave the definition for (K1,
K2)-quasiregular and quasicon formal mappings.
Definition A mapping f = (! J2, · · · , r) : f2----+ Rn is called (K1,
K2)-quasiregular with O < K1 < +oo, 0 ::S K2 < +oo, if it satisfies the following conditions: (i) f belongs to the class W (ii) the Jacobian :J(x, f) does not change sign inn, and IDf(x)ln ::S K1l:J(x,f)I +K2 (1.1) for almost all x E n. A mapping f = (!1, f 2, · · · ,Jn) : n ----+ Rn is said to be (K1, K2)-quasiconformal if it satisfies {i}, {ii}, and (iii) f is a homeomorphism.
The estimate of the modulus of continuity of (K1, [2].
The Holder property was first proved for a (K1,
This esti mate has important applications to elliptic equations with two variables. In [7], Gilbarg and Trudinger obtained an a priori C estimate for quasilinear elliptic equations with two variables by using the Holder continuity method established in the study of plane (K1,
K2)-quasiregular mappings, and then established the existence theorem of Dirich let problem. Many results on quasiregular mappings and their applications to nonlinear PDEs and elasticity theory have been established recently, see [8-10] and the references therein. Because of the importance of plane (K1,
K2)-quasiregular mappings to the a pri ori estimates in nonlinear PDE theory, Zheng and Fang [1] developed the theory of (K K 2)-quasiregular mappings in by using the theory of outer differential forms and Grassman algebra, and obtained an LP-integrability (p > n) result for space (K1, K2)-quasiregular mappings. For some other developments on (K1,
K2) quasiregular mapping theory, see [11-15]. It is a typical situation in quasiconformal analysis that one wants to build up the Holder continuity theory for (K1, K2)-quasiregular mappings. In this paper, we gener alize the results of [1,11,13], and the following Holder continuity result is obtained.
Theorem
Let f: n-+ Rn be a (K1, K2)-quasiregular mapping. Assume that k IDf(x)lndx = M < +oo. Then the function f satisfies a Holder condition with exponent a on compact subsets of n, where l/K1, for K1 > a= any positive number less than 1 , for K1 = and K2 > for K1 = and K2 = (1.2) 1, for K1 < Further, if V is contained strictly inside n, then for any x, y E V lf(x) - f(y)I :S Llx - Yla, where the constant L depends only on V, the constants K1 and
K2, the dimension n, the distance from V to the boundary of n, and the constant M. A counterexample
The mapping f with f (0) = and f : x f----t xlxla-l for x where a= K1, shows that the exponent K1 in Theorem is optimal. For this f we have lf(x) - f(0)I = lxla. The following corollary is a direct consequence of Theorem
Corollary
Let n be an open domain in Rn, and F(n,
K1,
K2,
M) the collection of all (K1, K2)-quasiregular mappings f on n such that k IDf(x)lndx :SM.
Then the set of functions f is equi-uniformly continuous on every compact subset of n. Definition 1.2.
The mapping f is said to have property N if the image of every set E c n of measure zero is a set of measure zero. Corollary 1.2.
Let f : n--+ Rn be a (K1, K2)-quasiregular mapping with O < K1 < or K1 = and K2 = then f has property N. Proof. [16,
Theorem states that every locally Lipschitz mapping has property N, which together with Theorem yields the desired result. (cid:143) Definition 1.3.
A mapping f : n --+ Rn is said to be differentiable at a point a
E n if there exists a linear mapping L : Rn --+ Rn such that f(x) = f(a) + L(x - a)+ ,8(x)lx - al for all x En, where ,8(x) --+ as x--+ a. The mapping L is called the differential off at the point a. The following theorem states that any (K1,
K2)-quasiregular mapping f is differen tial almost everywhere. Theorem 1.2.
Let f be a (K1, K2)-quasiregular mapping. Then for almost all x En the linear mapping D f ( x) is the differential off at the point x. Proof. The proof of Theorem is almost line by line of the proof of [16,
Theorem by using Corollary
We omit the details. (cid:143) §2 Preliminary Lemmas
The proof of Theorem is based on two facts. The first is a lemma due to Morrey. The second is an isoperimetric inequality due to Reshetnyak.
Lemma 2.1. (Morrey's Lemma [l7 l) Let n C Rn be an open subset, and f: n--+
Rk a function of the class w ,m(n, Rk), where ~ m ~ n. Assume that there exist numbers a(O such that r IDf(x)lmdx ~ Mrn-m+ma (2.1) J B(a,r) for every ball B(a, r) c n with radius at most b. Then there exists a continuous function J such that f(x) = f(x) almost everywhere, and the oscillation of J on any ball B(x, r) c n with r ~ b/3 and r < p(x)/3 does not exceed CM1fmra, where C < +oo is a constant. Lemma 2.2. (Isoperimetric Inequality [ l) Suppose that n c Rn and the mapping f: [2-----,
Rn is in the class W ,n(n,Rn). Thenfor any a EU and almost allt E (0,p(a)) f :J(x, f)dx ::; !__ f ID f(x) lndo-(x), (2.2) S(a,t) n S(a,t) where do- is the area element of the sphere S(a, t).
With the Morrey's Lemma and isoperimetric inequality in hands, we can now prove the following two lemmas, which will be used in the proof of Theorem 1.1.
Lemma 2.3.
Suppose that n C Rn is an open set and f a (K1,
K2)-quasiregular mapping. For x En and r < p(x) let where w(r) = r IDf(x)lndx. S(x,r) for K1 i- for K1 = l, Then the function r f----+ v(x,K1,K2,n,r) is nondecreasing.
Proof.
For r < p(x), (1.1) leads to f IDJ(x)lndx ::; K1 f l:J(x, f)ldx + K2 f dx ls~~ ls~~ h~~ = K1 f :J(x, f)dx + K2IB(x, r)I S(x,r) because :J(x, f) does not change sign inn. On the basis of Lemma 2.2 r :J(x, f) ::; '!:._ r ID f (x) lndo-(x) S(x,r) n S(x,r) for almost all r E (0,p(x)). From (2.4) and (2.5) we get f IDf(x)lndx::; Kir f IDJ(x)lndo-(x) + K2wnrn. S(x,r) n S(x,r)
Let r IDJ(x)lndo-(x) = s(r). S(x,r) (2.3) (2.4) (2.5) (2.6) Applying Fubini's theorem, we get that w(r) = J; s(t)dt for all r E (0, p(x)). This leads us to conclude that the function w is absolutely continuous and w'(r) = s(r) for almost all r E (0, p(x)). From (2.6) we have that
K1rw'(r) w(r) ::; - - - + K2wnrn n for almost all r. This is equivalent to
K1rw'(r) - - - - - w(r) + K2Wn'rn
2 0. n Multiplying both sides of this inequality by ,,.-(n/Ki)-l yields
K1w'(r) w(r) K n-n/K -I O I I + > · nrn K1 ,,.n K 1 +1 -
We get after obvious transformations that
K1, K2, n, r) > - ' where v(x, K1, K2, n, r) is defined by (2.3). Consequently, the function r f------7 v(x, K1, K2, n, r) is nondecreasing, as desired. (cid:143)
Lemma 2.4.
Suppose that n C Rn is an open set and f a (K1,
K2)-quasiregular mapping. Let J O ID f ( x) In dx = Mn. Then the vector-valued function f is equivalent, in the sense of the theory of integral, to some continuous function f. Further, for every set V lying strictly inside n the oscillation of J on any ball B(a, r) of radius r < about an a E V does not exceed Gr°', where d = dist(V, Proof.
Let a E V and w(a,r) = r IDf(x)lndx ~ Mn. J B(a,r)
According to Lemma 2.3, the function r f----t v(a, K1, K2, n, r) is nondecreasing. We now divide the proof into four cases.
Case 1 K1 > In this case, ( K K ) _ w(a, r) K2wn n-n/Ki v a,
1, 2, n, r - /K + K r 'rn l
1 - < v(a, K1, K2, n, ~ Mn(2d/3)-n/Ki + ; ~nl (2d/3t-n/Ki (2.7) ·- C1(K1, K2, M, d, n). for all r E (0, w(a r) < C ,,.n/K1 - K2Wn rn < C ,,.n/K1. ' - i K1 -1 - i Case K = I and K > It is no loss of generality to assume that d < w(a, r) v(a, K1, K2, n, r) = ,,.n + K2nwn ln r ( < v(a, K1, K2, n, ~ Mn(2d/3)-n + K2nwn ln(2d/3).
This implies, for any O < a < w(a, r) :S [Mn(2d/3)-n + K2nwn ln(2d/3)]rn -
K2nwnrn ln r = Mn(2d/3)-nrn + K2nWn7'n ln(2d/3r) (2.9) = [
Mn(2d/3)-nrn(l-a) + K2nWn7'n(l-a) ln(2d/3r)] rn°'. since limr--->O+ rn(l-a) ln(2d/3r) =
0 for any O < a <
1, then we take such that 5n(l-a) ln(25/3d) :S
1. When O < r :S we have from (2.9) that w(a,r) :S [ (
3: ) n d-na + rn°' := C2rn°'.
Case 3 K1 = =
0. (2.8) implies w(a, r) :S Mn(2d/3)-nrn := C3rn.
Case K1 <
1. In this case, (2.7) also holds for all r E (0, 2d/3), thus W(a' "') < C n/K1 + K2wn n = [c n/K1-n+ K2wn] n ' - l7' 1-K/ l7' l-K1 7' :S [ C1 (2d/3)n(l-K1)/K1 + l~~J rn := C4rn
In all the cases we have derived that for O < r :S r IDJ(x)lndx::; Gr°', } B(a,r) where a is defined as (1.2) and C depends only on K1, K2, M, d, n.
The required result follows directly from Lemma 2.1. (cid:143) §3 Proof of Theorem 1.1
Proof.
Let V be a compact subset of n, and let I be the smaller of the numbers /3 and d/3, where is the constant in Lemma 2.1 and d = dist(V, an). We consider the function h defined as follows on the product V x V: h(x, y) = lf(x)- J(y)l/lx -yl°' for x y, and h(x, x) =
0. Let H be the set of pairs (x, y) E V x V such that Ix - YI and let G = (V x V) \ H. The set H is compact, and thus h is bounded on H by continuity. The conclusion of Lemma 2.4 enables us to deduce that h is bounded also on G. Consequently, his bounded on V x V, and thus lf(x) - J(y)I :S Llx - YI°' for any x,y E V. (cid:143) References [1] Zheng S.Z., Fang A.N., £P-integrability of (K1, K2)-quasiregular mappings, Acta Math. Sin., 1998, 41(5), 1019-1024. (In Chinese) [2] M.Kreines, Sur une calsse de fonctions de plusierus variables, Mat. Sb., 1941, 51(9), 713-720. [3] Yu.G.Reshetnyak, On a sufficient condition for Holder continuity of a mapping, Dok!. Akad. Nauk SSSR, 1960, 130, 507-509. [4] Yu.G.Reshetnyak, Estimates of the modulus of continuity for certain mappings, Sibirsk Math. Zh., 1966, 7, 1106-1114. [5] E.D.Callender, Holder continuity of n-dimensional quasiconformal mappings, Pacific J. Math., 1960, 10, 499-515. [6] L.Simon, A Holder estimate for quasiconformal maps between surfaces in Euclidean space, Acta Math., 1977, 139: 19-51. [7] D.Gilbarg, N.S.Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1983. [8] T.Iwaniec, G.Martin, Geometric function and non-linear analysis, Clarendon Press, Oxford, 2001. [9] O.Martio, V.Ryazanov, U.Srebro, E.Yakubov, Moduli in modern mapping theory, Springer-Verlag, 2008. [10] K.Astala, T.Iwaniec, G.Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, 2009. [11] Gao H.Y., Regularity for weakly (K1, K2)-quasiregular mappings, Sci. in China, Series A., 2003, 46(4): 499-505. [12] Gao H.Y., Huang, Q.H., Qian F., Regularity for weakly (K1, K2(x))-quasiregular mappings of several n-dimensional variables, Front. Math. China, 2011, 6(2), 241-251. [13] Gao H.Y., Zhou S.Q., Meng Y.Q., A new inequality for weakly (K1, K2)-quasiregular mappings, Acta Math. Sin., English Series, 2007, 23(12): 2241-2246. [14] Gao H.Y., Li T., On degenerate weakly (K1,K2)-quasiregular mappings, Acta Math. Sci., 2008, 28B(l): 163-170. [15] Gao H.Y., Liu H.H., Zhou S.Q., Higher integrablity for weakly (K1, K2