Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 36, 2011, 531–536
A NOTE ON MAPPINGS OF FINITE DISTORTION: EXAMPLES FOR THE SHARP MODULUS OF CONTINUITY Daniel Campbell and Stanislav Hencl Charles University, Department of Mathematical Analysis Sokolovská 83, 186 00 Prague 8, Czech Republic;
[email protected] Charles University, Department of Mathematical Analysis Sokolovská 83, 186 00 Prague 8, Czech Republic;
[email protected] Abstract. We construct a mapping with exponentially integrable distortion which attains a modulus of continuity by Onninen and Zhong, showing that it is sharp.
1. Introduction Let Ω ⊂ Rn , n ≥ 2, be a connected and open set. A Sobolev mapping f ∈ is said to have finite distortion if the Jacobian Jf (x), i.e., the determinant of the matrix of derivatives Df (x) is locally integrable and there is a measurable function K(x) ≥ 1 finite almost everywhere such that
1,1 Wloc (Ω, Rn )
|Df (x)|n ≤ K(x)Jf (x) a.e. x ∈ Ω. Here we have used the operator norm of the differential matrix with respect to the Euclidean distance. If we, moreover, require that K(x) ∈ L∞ (Ω), we arrive at mappings of bounded distortion also called quasiregular mappings. In [7] Reshetnyak proved among many other things that quasiregular mappings are Hölder continuous with the exponent 1/K, where K is the L∞ norm of the distortion. It has been shown recently that mappings of finite distortion with exponentially integrable distortion exp(λK(x)) ∈ L1 (Ω) share many nice properties of mappings of bounded distortion. We would like to point the reader’s attention to the monographs [1] and [3] for the motivation, applications and the history of the subject. Our aim is to study the modulus of continuity of the mappings of finite distortion with exp(λK) ∈ L1 (Ω). Let us first recall the history of such estimates. First, it was shown by Iwaniec, Koskela and Onninen [2] that mappings in this class are continuous and satisfy C |f (x) − f (y)| ≤ . 1/n e log log(e + 1/|x − y|) doi:10.5186/aasfm.2011.3633 2010 Mathematics Subject Classification: Primary 30C65, 46E35. Key words: Mapping of finite distortion, modulus of continuity. The second author was supported by the RSJ algorithmic trading grant.
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This was later improved by Koskela and Onninen [5] to |f (x) − f (y)| ≤
C logλ/n−ε (1/|x − y|)
and finally using very delicate arguments it has been shown by Onninen and Zhong [6] that (1.1)
|f (x) − f (y)| ≤
C . logλ/n (1/|x − y|)
Extremal mappings for continuity of mappings of finite distortion are usually radial maps and therefore the natural candidate for the extremal map is f0 (x) =
x 1 . λ/n |x| log (1/|x|)
Standard computations (see (2.2) below) give us K(x) = and hence
n 1 log λ |x|
ˆ
ˆ 1 B(0, ) 2
exp(λK(x)) dx =
1 B(0, ) 2
1 dx = ∞. |x|n
This elementary computation suggests that there is some room for improvement in the estimate (1.1) and maybe we can add some supplementary term like log log 1/|x − y| to some negative power to our estimate. We show that, surprisingly, this is not the case and the modulus of continuity (1.1) is already sharp. Theorem 1.1. Given λ > 0, there is a mapping of finite distortion f : B(0, 12 ) → Rn such that ˆ 1 B(0, ) 2
exp(λK(x)) dx < ∞
and C for all x ∈ B(0, 21 ). log (1/|x|) There have also been studies on mappings of subexponentially integrable distortion (see e.g. [4]). One requires that ˆ ¡ ¢ (1.2) exp A (K(x)) dx < ∞ |f (x) − f (0)| ≥
λ/n
B
for some Orlicz function A and the above mentioned example corresponds to the case A (t) = λt. We call an infinitely differentiable and strictly increasing function A : [0, ∞) → [0, ∞) with A (0) = 0 and limt→∞ A (t) = ∞ an Orlicz function. As usual we impose additional condition ˆ ∞ 0 A (s) = ∞. (1.3) s 1
A note on mappings of finite distortion: Examples for the sharp modulus of continuity
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It is easy to see that the critical functions for this condition are t , log(e + t) t A3 (t) = λ and so on. log(e + t) log(e + log(e + t))
A1 (t) = λt, (1.4)
A2 (t) = λ
We will also require that 3
(i) ∃t0 > 0 ∀t > t0 A −1 (nt) < t 2 , (ii) A 0 (t) is non-increasing, (1.5)
t , A (t) (iv) b(0) := lim b(t) is finite and positive.
(iii) b0 (t) is non-increasing for b(t) := t→0+
Let us note that the critical functions from (1.4) satisfy these conditions and therefore these assumptions are not substantially restrictive. It has been shown in [4] that a mapping f is continuous under the assumptions (1.2) and (1.3) and that the assumption (1.3) is sharp. It was proved in [6] that under the assumptions (1.2) and (1.3) we have ´ ³ ˆ R dt |f (x) − f (y)| ≤ C exp − −1 (n log C/t) |x−y| tA for |x−y| sufficiently small and B(x, 80R) ⊂ Ω. Note further that if we put A1 (t) = λt we arrive at the modulus given in (1.1). Our result shows the sharpness of this estimate. Theorem 1.2. Suppose that an Orlicz function A satisfies (1.3) and (1.5). Then there is a ball B := B(0, r) and a mapping of finite distortion f : B → Rn such that ˆ ¡ ¢ exp A (K(x)) dx < ∞ B
and (1.6)
³ ˆ |f (x) − f (0)| ≥ C exp −
1/2 |x|
´ dt for all x ∈ B. tA −1 (n log 1/t)
2. Proofs of the theorems To prove Theorem 1.1 we simply set a
x (log 1/|x|) log 1/|x| f (x) = , |x| logλ/n (1/|x|) where a > 0. The additional term clearly satisfies a
lim (log 1/|x|) log 1/|x| = 1
|x|→0
and thus the modulus of continuity of our f is exactly as required in (1.1). On the other hand, the additional term slightly affects the distortion and the standard
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Daniel Campbell and Stanislav Hencl
computation (see the general case below for details) will give us K(x) ∼ and hence
n 1 n2 a 1 log − 2 log log , λ |x| λ |x|
ˆ 1 B(0, ) 2
exp(λK(x)) dx < ∞
for sufficiently large a. To prove Theorem 1.2, let us put B := B(0, min{exp(−t0 ), e−e }) and choose α > b(0)−1 n−2 . Without loss of generality we can assume that t0 is big enough such that 3 1 t2 (2.1) t2 < for all t > t0 . α(α + 1) log t We define the function f as, ³ ˆ 12 ´ α+2 x 1 −1 log |x|−1 f (x) := exp − dt (log |x| ) for x 6= 0. 1 −1 (n log ) |x| |x| tA t Note that −1
lim (log |x| )
α+2 log |x|−1
|x|→0
= lim exp |x|→0
³ (α + 2) log log |x|−1 ´ log |x|−1
= 1,
which easily gives that f satisfies the condition given in (1.6). Let ρ : (0, ∞) → (0, ∞) be a strictly monotone, differentiable function and let us consider the mapping x f (x) = ρ(|x|), x 6= 0. |x| It can be verified by an elementary computation (see e.g. [1, Chapter 2.6.]) that n ρ(|x|) o |Df (x)| = max , |ρ0 (|x|)| , and thus |x| (2.2) n ρ(|x|) |x| |ρ0 (|x|)| o K(x) = max , . |x| |ρ0 (|x|)| ρ(|x|) It follows that for our mapping we obtain ½ ³ ¾ 1 log log |x|−1 − 1 ´ |f (x)| max 1, + (α + 2) . |Df (x)| = |x| A −1 (n log |x|−1 ) log2 |x|−1 Clearly,
³
1 log log |x|−1 − 1 ´ + (α + 2) = 0, x→0 A −1 (n log |x|−1 ) log2 |x|−1 and therefore the greater element is the first one. From (1.5) (i) and (2.1) we obtain lim
3
A −1 (nt) < t 2
t0 . α(α + 1) log t
This, however, implies that α(α + 1)
A −1 (nt) log t < 1 for all t > t0 . t2
A note on mappings of finite distortion: Examples for the sharp modulus of continuity
Now, by multiplying both sides by get that A −1 (n log |x|−1 )
A −1 (nt) log t t2
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and by substituting t = log |x−1 | we
³ −1 ´2 log log |x|−1 −1 −1 log log |x| > α(α + 1) A (n log |x| ) log2 |x|−1 log2 |x|−1
for all x ∈ B. Using this fact and because log log |x|−1 > 1 for all x ∈ B, we deduce that 1 ´ K(x) = ³ log log |x|−1 −1 1 + (α + 2) log2 |x|−1 A −1 (n log |x|−1 ) ≤
A −1 (n log |x|−1 ) −1
log |x| 1 + (α + 1)A −1 (n log |x|−1 ) log log2 |x|−1 ³ −1 ´ −1 −1 −1 −1 log log |x| ˜ ≤ A (n log |x| ) 1 − αA (n log |x| ) =: K(x). log2 |x|−1
Note that (2.3)
³ A −1 (n log |x|−1 ) ´2 ˜ A −1 (n log |x|−1 ) − K(x) = αn2 log log |x|−1 . n log |x|−1
By (1.5) (iii) we obtain that b(s) − b(0) = b0 (ξ)s ≥ b0 (s)s and therefore (2.4)
³ s ´2 b(s) − sb0 (s) A 0 (s) = b2 (s) ≥ b(0). A (s) b2 (s)
From (1.5) (ii) we know that A 0 (t) is a non-increasing function and hence (2.5)
A (a − d) = A (a) − A 0 (ξ)d ≤ A (a) − A 0 (a)d
for some ξ ∈ (a − d, a). We now use (2.5) putting ³ ³ 1 ´ 1 ´ ˜ −1 −1 a := A n log , d := A n log − K(x) |x| |x| using (2.3) and then (2.4) (where we put s := A −1 (n log |x|−1 )) to get that ˜ A (K(x)) ≤ A (K(x)) i ¢ ¡ −1 ¢h −1 ¡ ¡ −1 ¢ −1 −1 −1 0 ˜ − K(x) n log |x| ≤ A A (n log |x| ) − A A (n log |x| ) A ¡ ¢³ A −1 (n log |x|−1 ) ´2 ≤ n log |x|−1 − αn2 A 0 A −1 (n log |x|−1 ) log log |x|−1 n log |x|−1 ≤ n log |x|−1 − b(0)αn2 log log |x|−1 . But this, for α > b(0)−1 n−2 , yields ˆ ˆ ³ 1 1 ´ exp(A (K(x))) dx ≤ exp n log − b(0)αn2 log log dx |x| |x| B B ˆ 1 ≤ dx < ∞. b(0)αn2 1 n B |x| log |x|
¤
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References [1] Astala, K., T. Iwaniec, and G. Martin: Elliptic partial differential equations and quasiconformal mappings in the plane. - Princeton Math. Ser. 48, Princeton Univ. Press, Princeton, NJ, 2009. [2] Iwaniec, T., P. Koskela, and J. Onninen: Mappings of finite distortion: Monotonicity and continuity. - Invent. Math. 144, 2001, 507–531. [3] Iwaniec, T., and G. Martin: Geometric function theory and nonlinear analysis. - Oxford Math. Monogr., Clarendon Press, Oxford, 2001. [4] Kauhanen, J., P. Koskela, J. Malý, J. Onninen, and X. Zhong: Mappings of finite distortion: Sharp Orlicz-conditions. - Rev. Mat. Iberoamericana 19, 857–872, 2003. [5] Koskela, P., and J. Onninen: Mappings of finite distortion: the sharp modulus of continuity. - Trans. Amer. Math. Soc. 355:5, 2003, 1905–1920. [6] Onninen, J., and X. Zhong: A note on mappings of finite distortion: the sharp modulus of continuity. - Michigan Math. J. 53:2, 2005, 329–335. [7] Reshetnyak, Yu. G.: Space mappings with bounded distortion. - Transl. Math. Monogr. 73, Amer. Math. Soc., 1989.
Received 28 May 2010
