We prove that every K-quasiconformal mapping w of the unit ball Bn onto a C2-Jordan domain Ω is H¨older continuous with constant = 2 − np , provided that its weak Laplacean ∆w is in Lp(Bn) for some n=2 < p < n. In particular it is H¨older continuous for every 0 < < 1 provided that ∆w ∈ Ln(Bn).
Bn denotes the unit ball in Rn, n ≥ 2 and Sn−1 denotes the unit sphere. Also we will assume that n > 2
A homeomorphism u : Ω → Ω′ between two open subsets Ω and Ω′ of Euclid space Rn will be called a K (K ≥ 1) quasi-conformal or shortly a q.c mapping if
(i) u is absolutely continuous function in almost every segment parallel to some of the coordinate axes and there exist the partial derivatives which are locally Ln integrable functions on Ω. We will write u ∈ ACLn and (ii) u satisfies the condition at almost everywhere x in Ω where
G(x; y) = cn { (
1 1 ) |x−y|n 2 − (| x|y|−y=|y| |)n 2 ; if n ≥ 3; log |x−y| ; |1−xy¯| if n = 2 and x; y ∈ C =∼ R2.
(1) 1 where cn = (n−2)Ωn 1 , and Ωn−1 is the measure of Sn−1. Both P and G are harmonic for |x| < 1, x ̸= y .
Let f : Sn−1 → Rn be a Lp, p > 1 integrable function on the unit sphere Sn−1 and let g : Bn 7→ Rn be continuous. The weak solution of the equation (in the sense of distributions) ∆u = g in the unit ball satisfying the boundary condition u|Sn 1 = f ∈ L1(Sn−1) is given by u(x) = P [f ](x) − G[g](x) :=
P (x; )f ( )d ( ) −
G(x; y)g(y)dy; ∫
Sn 1 ∫
Bn |x| < 1. Here d is Lebesgue n − 1 dimensional measure of Euclid sphere satisfying the condition: P [1](x) ≡ 1. It is well known that if f and g are continuous in Sn−1 and in Bn respectively, then the mapping u = P [f ] − G[g] has a continuous extension u˜ to the boundary and u˜ = f on Sn−1. If g ∈ L∞ then G[g] ∈ C1; (Bn). See [Gilbarg & Trudinger, 1983, Theorem 8.33] for this argument.
We will consider those solutions of the PDE ∆u = g that are quasiconformal as well and investigate their Lipschitz character.
A mapping f of a set A in Euclidean n-space Rn into Rn, n ≥ 2, is said to belong to the H¨older class Lip (A), > 0, if there exists a constant M > 0 such that
|f (x) − f (y)| ≤ M |x − y| for all x and y in A. If D is a bounded domain in Rn and if f is quasiconformal in D with f (D) ⊂ Rn, then f is in Lip (A) for each compact A ⊂ D, where = KI (f )1=(1−n) and KI (f ) is the inner dilatation of f . Simple examples show that f need not be in Lip (D) even when f is continuous in D.
However O. Martio and R. N¨akki in [Martio & N¨akki, 1991] showed that if f induces a boundary mapping which belongs to Lip (@D), then f is in Lip (D), where (2) (3) = min( ; KI (f )1=(1−n)); the exponent is sharp.
In a recent paper of the first author and Saksman [Kalaj & Saksman, 2014] it is proved the following result, if f is quasiconformal mapping of the unit disk onto a Jordan domain with C2 boundary such that its weak Laplacean ∆f ∈ Lp(B2), for p > 2, then f is Lipschitz continuous. The condition p > 2 is necessary also. Further in the same paper they proved that if p = 1, then f is absolutely continuous on the boundary of @B2.
We are interested in the condition under which the quasiconformal mapping is in Lip (Bn), for every < 1. It follows form our results that the condition that u is quasiconformal and |∆u| ∈ Lp, such that p > n=2 guaranty that the selfmapping of the unit ball is in Lip (Bn), where = 2 − np . In particular if p = n, then f ∈ Lip (Bn) for < 1.
It should be noted that the topic is very active area of research in geometric function theory, and the
following people have obtained some substantial results in this area: Pavlovi´c and Kalaj, Mateljevi´c, Partyka,
Sakan
Our new result in several-dimensional case is the following: Theorem 1 Let n ≥ 2 and let p > n=2 and assume that g ∈ Lp(Bn). Assume that w is a K-quasiconformal solution of ∆w = g; that maps the unit ball onto a bounded Jordan domain Ω ⊂ Rn with C2-boundary. • If p < n, then w is H¨older continuous with the H¨older constant n = 2 − p . • If p = n, then w is H¨older continuous for every
∈ (0; 1).
The sketch of the proof is given in the next section. 2
In what follows, we say that a bounded Jordan domain Ω ⊂ Rn has C2-boundary if it is the image of the unit ball Bn under a C2-diffeomorphism of the whole Euclidean space. For Jordan domains Ω ⊂ Rn this is well-known to be equivalent to the more standard definition, that requires the boundary to be locally isometric to the graph of a C2-function on Rn−1. In what follows, ∆ refers to the distributional Laplacian. We shall make use of the following well-known facts. Proposition 2.1 (Morrey's inequality) Assume that n < p ≤ ∞ and assume that U is a domain in Rn with C1 boundary. Then there exists a constant C depending only on n, p and U so that for every u ∈ C1(U ) ∩ Lp(U ), where ∥u∥C0; (U) ≤ C∥u∥W 1;p(U)
n = 1 − p : Lemma 1 See e.g.[Astala & Manojlovic, 2015]. Suppose that w ∈ Wl2o;c1(Bn) ∩ C(Bn ), that h ∈ Lp(Bn) for some 1 < p < ∞ and that ∆w = h in Bn; with w Sn 1 = 0; (4) a) If 1 < p < n, then b) If p = n and 1 < q < ∞ then c) if p > n, then ∥∇w∥Lq(Bn) ≤ c(p; n)∥h∥Lp(Bn); q =
pn n − p
: ∥∇w∥Lq(Bn) ≤ c(q; n)∥h∥Ln(Bn): ∥∇w∥L1(Bn) ≤ c(p; n)∥h∥Ln(Bn):
Now we formulate the following fundamental result of Gehring Proposition 2.2 [Gehring, 1973] Let f be a quasiconformal mapping of the unit ball Bn onto a Jordan domain Ω with C2 boundary. Then there is a constant p = p(K; n) > n so that ∫
Bn
|Df |p < C(n; K; f (0); Ω): Now we formulate two lemmas, whose proofs are easily, but the details will be printed elsewhere np .
Lemma 2 If ∆u = g ∈ Lp and r < 1, then Du ∈ Lq(rB) for q ≤ n−p
To prove Theorem 1, we need as well this lemma.
Lemma 3 If H : Rn → R and w = (w1; : : : ; wn) : A → B (where A; B are open subsets in Rn) are functions from C2 class, then:
∑ 1≤i<j≤n
Sketch of the proof of Theorem 1.
In addition to the previous propositions, the proof depends on an approach discovered in [Kalaj & Saksman, 2014]. Details will be published elsewhere. [Fefferman et al., 1991] R.A. Fefferman, C.E. Kenig and J. Pipher: The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 134 (1991), 65–124.
Mateljevic: (K; K′)-quasiconformal harmonic mappings. Potential