A PRIORI ESTIMATE OF GRADIENT OF A SOLUTION TO CERTAIN DIFFERENTIAL INEQUALITY AND QUASICONFORMAL MAPPINGS

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1 A PRIORI ESTIMATE OF GRADIENT OF A SOLUTION TO CERTAIN DIFFERENTIAL INEQUALITY AND QUASICONFORMAL MAPPINGS DAVID KALAJ Abstract. We will prove a global estimate for the gradient of the solution to the Poisson differential inequality ux a ux 2 + b, x B n, where a, b < and u S n C,α S n, R m. If m = and a n + / u 4n n, then u is a priori bounded. This generalizes some similar results due to E. Heinz [3] and Bernstein [3] for the plane. An application of these results yields the theorem, which is the main result of the paper: A quasiconformal mapping of the unit ball onto a domain with C 2 smooth boundary, satisfying the Poisson differential inequality, is Lipschitz continuous. This extends some results of the author, Mateljević and Pavlović from the complex plane to the space. Contents. Introduction and statement of main results 2. Some lemmas 6 3. A priori estimate for a solution to Poisson differential inequality 3 4. Applications-The proof of Theorem C Bounded curvature and the distance function An open problem 22 References 22. Introduction and statement of main results In the paper B n denotes the unit ball in R n, and S n denotes the unit sphere n > 2. We consider the vector norm x = n i= x2 i /2 and the matrix norm A = max{ Ax : x = }. Let Ω R n and Ω R m be open sets and let u : Ω Ω be a differentiable mapping. By u we denote its derivative, i.e. D u... D n u u = D u m... D n u m 99 Mathematics Subject Classification. Primary 35J05, Secondary 30C65. Key words and phrases. Green formula, PDE, harmonic function, subharmonic function, quasiconformal mappings.

2 2 DAVID KALAJ If n = m, then the Jacobian of u is defined by J u = det u. The Laplacian of a twice differentiable mapping is defined by n u = D ii u. i= The solution of the equation u = g in the sense of distributions see [7] in the unit ball, satisfying the boundary condition u S n = f L S n, is given by ux = P x, ηfηdση Gx, ygydy, x <.. S n B n Here P x, η = x 2 x η n.2 is the Poisson kernel and dσ is the surface n dimensional measure of the Euclidean sphere satisfying the condition: P x, ηdση. The first integral in. S n is called the Poisson integral and is usually denoted by P [f]x. It is a harmonic mapping. The function Gx, y = c n where x y n 2 + x 2 y 2 2 x, y n 2/2 c n =,.3 n 2ω n.4 and ω n is the measure of S n, is the Green function of the unit ball. The Poisson kernel and the Green function are harmonic in x. Definition.. A homeomorfism continuous mapping u : Ω Ω between two open subsets Ω and Ω of the Euclidean space R n will be called a K K quasi-conformal quasi-regular or shortly a q.c. q.r. mapping if: i u is absolutely continuous function in almost every segment parallel to some of the coordinate axes and there exist partial derivatives which are locally L n integrable functions on Ω. We will write u ACL n. ii u satisfies the condition ux n ux J u x Kl ux n n J u x Kl ux n.5 K K for almost every x in Ω where lu x := inf{ uxζ : ζ = } and J u x is the Jacobian determinant of u see [33] or [37]. We refer also to the monographs [34] and [35] for the basic theory of quasiregular mappings. Notice that the condition u ACL n guarantees the existence of the first derivative of u almost everywhere see [33]. Moreover J u x = det ux 0 for a.e. x Ω. For a continuous mapping u, the condition i is equivalent to the fact that u belongs to the Sobolev space Wn,loc Ω. For a function a mapping u defined in a domain Ω we define u = u = sup{ ux : x Ω}. We say that u C k,α Ω, 0 < α, k N, if u l,α := D β u + sup D β ux D β uy x y α <. β l x,y Ω β =l

3 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 3 It follows that for every α 0, ] and l N u l := D β u u l,α..6 We first have that for u C,α Ω β l ux uy u,α x y for every x, y Ω,.7 and for real u C,α Ω u 2 2 u 0 u,α..8 More generally, for every real differentiable function τ and real u C,α Ω, we have τu τ u 0 u,α..9 Let Ω have a C k,α boundary Ω. The norms on the space C k,α Ω can be defined as follows. If u 0 C k,α Ω then it has a C k,α extension u to the domain Ω. The norm in C k,α Ω is defined by: u 0 k,α := inf{ u Ω,k,α : u Ω = u 0 }. Equipped with this norm the space C k,α Ω becomes a Banach space. See also [6, p. 42] for the definition of an equivalent norm in C k,α Ω, by using the partition of unity. One of the starting points of this paper is the following theorem which was one of the main tools in proving some recent results of the author and Mateljevic see [2] and [20]. Proposition.2. Heinz-Bernstein, see [3] and [3]. Let s : U R s : U B m be a continuous function from the closed unit disc U into the real line closed unit ball satisfying the conditions: s is C 2 on U, 2 s b θ = se iθ is C 2 with K = max ϕ [0,2π 2 s ϕ e iϕ, and 2 3 s a s 2 + b on U for some constants a < a < /2 respectively and b <. Then the function s is bounded on U by a constant ca, b, K. The Heinz-Bernstein theorem appeared on 90 in the Bernstein s paper [3] and was reproved by E. Heinz on 956 in [3]. This theorem is important in connection with the Dirichlet problem for the system u u = Q, u, u, x, x 2 ; u = u x, x 2,..., u m x, x 2,.0 x x 2 where Q = Q,..., Q m, and Q j are quadratic polynomials in the quantities ui x k, i =,..., m, k =, 2 with the coefficients depending on u and x, x 2 Ω. An example of the system.0 is the system of differential equations that present a regular surface S with fixed mean curvature H with respect to isothermal parameters x, x 2. Also it is important in the connection with minimal surfaces and the Monge-Ampere equation. We will consider the n dimensional generalization of the system.0. Indeed, we will consider a bit more general situation. Assume that a twice differentiable mapping u = ux,..., x n satisfies the following differential inequality, which will be the main subject of the paper:

4 4 DAVID KALAJ u a u 2 + b where a, b > 0.. The inequality. will be called the Poisson differential inequality. Recall that the harmonic mapping equations for u = u,... u n : N M of the Riemann manifold N = B n, h jk into a Riemann manifold M = Ω, g jk Ω R n are n n h /2 α h /2 h αβ β u i + Γ i klud α u k D β u l, i =,..., n,.2 α,β= α,β,k,l= where Γ i kl are Christoffel Symbols of the metric tensor g jk in the target space M: Γ i kl = gmk 2 gim x l + g ml x k g kl x m = 2 gim g mk,l + g ml,k g kl,m, the matrix g jk h jk is an inverse of the metric tensor g jk h jk, and h = deth jk. See e.g. [8] for this definition. Remark.3. If Christoffel Symbols of the metric tensor g jk are bounded in M, and if the metric in N is conformal and bounded i.e. if h jk x = ρxδ jk, such that ρ is bounded in N then u satisfies.. Note that the Poisson differential inequality is related to the problem diva, u u = f, u, u,.3 where x B n r := rb n, ux R m and each Ax, u, for x R n and u R m is an endomorphism on HomR n, R m satisfying uniformly strongly elliptic and uniformly continuous conditions; moreover f satisfies the following so called natural growth condition see [8] fx, u, p ar p 2 + b p HomR n, R m..4 The problem of interior and boundary regularity of solutions to.3 has been treated by many authors, and among many results, it is proved that, every solution of.3, under some structural conditions and some conditions on the domains and initial data, has Hölder continuous extension to the boundary and is C,α inside. See [2] for some recent results on this topic and also [8] and [5] for earlier references. In this paper we generalize Proposition.2 for the space. Namely we prove the theorems: Theorem A. Let s : B n R m be a continuous function from the closed unit ball B n into R m satisfying the conditions: s is C 2 on B n, 2 s : S n R m is C,α and 3 s a s 2 +b on B n for some constants b and a satisfying the condition 2a s := max{ sx : x B n }. Then the function s is bounded on B n. If a C n / s where C n = n + πγn + /2 8nΓn/2 +.5

5 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 5 then s is bounded by a constant Ca, b, n, M, where M = s S n,α. Theorem B. Let s : B n R be a continuous function from the closed unit ball B n into R satisfying the conditions: s is C 2 on B n, 2 s : S n R is C,α and 3 s a s 2 + b on B n for some constants a and b. Then the function s is bounded on B n. If a C n / s then s is bounded by a constant Ca, b, n, u S n,α. According Remark.3 it follows that Theorem A is a partial extension of [9, Theorem 4, ii] where it is proved a similar result, for the family of harmonic mappings u : N M, which map the manifold N into a regular ball B r Q M. Theorem A and Theorem B roughly speaking, assert that every solution to Poisson differential inequality has a Lipschitz continuous extension to the boundary, if the boundary data is C,α. An application of Theorem B yields the following theorem which is a generalization of analogous theorems for plane domains due to the author and Mateljevic see [23] and [20]. Theorem C. Let u : B n Ω be a twice differentiable quasiconformal mapping of the unit ball onto the bounded domain Ω with C 2 boundary, satisfying the Poisson differential inequality. Then u is bounded and u is Lipschitz continuous. Remark.4. Every C 2 mapping satisfies locally the Poisson differential inequality and is locally quasiconformal provided that the Jacobian is non vanishing. Thus the family of mappings satisfying the conditions of Theorem C is quite large. According to Fefferman theorem [5] every biholomorphism of the unit ball onto a domain with smooth boundary is smooth up to the boundary and therefore quasiconformal. This fact together with the fact that every holomorphic mapping is harmonic, it follows that Theorem C can be also considered as a partial extension of Fefferman theorem. The paper contains this introduction and three other sections. In section 2 we prove some important lemmas. In section 3, by using Lemma 2. and some a priori estimates for Poisson equation proved in [0], we prove Theorem A and Theorem B, which are a priori estimates and global estimates for the the solution to Poisson differential inequality on the space, which are generalization of analogous classic Heinz-Bernstein theorem for the plane. In the final section, we previously show that, if u : B n Ω is q.c. and satisfies Pisson differential inequality, then the function χx = dux, where du = distu, Ω, satisfies as well Pisson differential inequality in some neighborhood of the boundary. By using this fact and Theorem B we prove Theorem C. This extends some results of the author, Mateljevic and Pavlovic [23], [26], [20], [2] and [32] from the plane to the space. It is important to notice that, the conformal mappings and decomposition of planar harmonic mappings as the sum of an analytic and an anti-analytic function played important role in establishing some regularity boundary behaviors of q.c. harmonic mappings in the plane [32] and [23]. This cannot be done for harmonic mappings

6 6 DAVID KALAJ defined in the space see [], [22] and [36] for some results on the topic of Euclidean and hyperbolic q.c. harmonic mappings in the space. The theorem presented here Theorem B made it possible to work on the problem of q.c. harmonic mappings on the space without employing the conformal and analytic functions. Notice that, the family of conformal mappings on the space coincides with Möbius transformations. Therefore this family is smaller than the family of conformal mappings in the plane. 2. Some lemmas We will follow the approach used in [3]. The following lemma will be an essential tool in proving the main results of Section 3. It depends upon three lemmas. It is an extension of a similar result for complex plane treated in [3]. Lemma 2. The main lemma. Assumptions: A The mapping u : D B m is C 2, D B n satisfies for x D the differential inequality u a u 2 + b 2. where 0 < a, b <. A2 There exists a real valued function Gu of class C 2 for u + ɛ ɛ > 0 such that G α 2.2 and the function φx = Gux satisfies the differential inequality φ β u 2 γ 2.3 where α, β and γ are positive constants. Conclusions: C There exists a fixed positive number c = c a, b, α, β, γ, n, u such that for x 0 D and r 0 = distx 0, D, the following inequality holds: ux 0 c + max x x 0 r 0 ux ux 0 r If a is small enough a satisfies the inequality a C n i.e then c can be chosen independently of u. Remark 2.2. After writing this paper, the author learned that a similar result has been obtained in the paper [9], for the class of harmonic mappings. By using the fact that the family of bounded harmonic mappings is Hölder continuous in compacts for some Hölder exponent σ < a result obtained in [4], Jost and Karcher proved that c can be chosen independently of u without the previous restriction [9, Theorem 3.]. However, it seems that our results are new for the class of solutions to Poisson differential inequality and the author believes that c can be chosen independently of u without the restriction a C n = n+ πγn+/2 8nΓn/2+. However to prove the main result Theorem C we only need an estimate that is not necessarily a priori see Theorem B. We will prove the lemma 2. by using the following three lemmas:

7 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 7 Lemma 2.3. Let u satisfy the hypotheses of the lemma 2. and let the ball y x ρ be contained in D. Then we have for 0 < ρ < ρ the inequality c n u 2 dy y x ρ 2.5 γρ 2 ρ n 2 ρ n 2 ρ n 2 ρ n 2 2nβ + α β Proof. By using. we obtain for x < ρ ux + ρη uxdση S n = y x ρ c n y x n 2 c n ρ n 2 max uy ux. y x =ρ gydy. 2.6 If we now apply the identity 2.6 to the mapping φx = Gux, by using φ a, we obtain the inequality: c n y x ρ y x n 2 c n ρ n 2 φdy 2.7 α ux + ρη ux dση α max uy ux. S n x y =ρ On the other hand from 2.3 we deduce c n y x ρ y x n 2 c n ρ n 2 φdy c n β y x ρ y x n 2 c n ρ n 2 c n γ y x ρ y x n 2 c n ρ n 2 c n β y x n 2 c n ρ n 2 y x ρ y x ρ γρ2 2n. Combining this inequality with 2.7 we obtain c n y x n 2 c n ρ n 2 u 2 dy Now let 0 < ρ < ρ. From 2.9 we get ρ n 2 ρ n 2 c n u 2 dy y x ρ y x ρ u 2 dy dy u 2 dy γρ2 2βn + α β max uy ux. y x =ρ c n y x n 2 c n ρ n 2 γρ2 2βn + α β max uy ux y x =ρ u 2 dy

8 8 DAVID KALAJ and therefore c n u 2 dy ρ n 2 ρ n 2 γρ 2 y x ρ ρ n 2 ρ n 2 2βn + α β max uy ux. y x =ρ Lemma 2.4. Let Y : D B m be a C 2 mapping of a domain D B n. Let B n x 0, ρ D and let Z R m be any constant vector n 3, m N. Then we have the estimate: and Y x 0 n ρ n + ω n y x 0 =ρ y x 0 ρ Y x 0 γ n ρ + ω n y x 0 ρ where γ n is defined in 2.9. Y y Z dσy y x 0 n y x 0 ρ n Y dy, y x 0 n y x 0 ρ n 2.0 Y dy, 2. Proof. Assume that v C 2 B n. From vx = Hx + Kx := P x, ηvηdση S n Gx, y vydy B n 2.2 where H is a harmonic function, it follows that v xh = P x x, ηh vηdση S n G x x, yh vydy. B n 2.3 By differentiating.2 and.3 we obtain and Hence and Therefore and P x x, ηh = 2 x, h x η n n x 2 x η, h x η n+2 G x x, yh = c n n 2 x y, h x y n c n n 2 y 2 x, h n 2 y, h + x 2 y 2 2 x, y n/2. P x 0, ηh = n η, h = n η, h η n+2 G x 0, yh = ω n y, h y n + ω n y, h. P x 0, ηh P x 0, η h = n h 2.4 G x 0, yh G x 0, y h = ω n y n y h. 2.5

9 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 9 By using 2.3, 2.4 and 2.5 we obtain v0h P x 0, η h vη dση + S n G x 0, y h vy dy. B n Hence we have v0 n vη dση S n + y n y vy dy. ω n B n 2.6 Let vx = Y x 0 + ρx Z. Then v0 = Y x 0 Z, v0 = ρ Y x 0 and vy = ρ 2 Y x 0 + ρy. Inserting this into 2.6 we obtain ρ Y x 0 G0 + K0 n Y x 0 + ρη Z dση S n ρ 2 y n y Y x 0 + ρy dy. ω n B n Introducing the change of variables ζ = x 0 +ρη in the first integral and w = x 0 +ρy in the second integral of 2.7 we obtain Y x 0 n ρ n Y ζ Z dσζ + ω n ζ x 0 =ρ w x 0 ρ w x 0 n w x 0 ρ n Y dw 2.8 which is identical with 2.0. To get 2. we do as follows. We again start by 2.2. Let vx = Y x 0 + ρx. Then H is defined by Hx = P x, ηy x 0 + ρηdση. S n Applying the Schwartz lemma see [2, Theorem 6.26] to the harmonic function H h : x Hx, h, where h is a unit vector in R m, we obtain that Since, H0 = max H0k, h = max H h0 k S n,h Sm h = γ n := 2n ω n 2 2Γ + n/2 = < n. nω n πγn + /2 Y x 0 + ρx = Hx + Kx 2.9 and v0 = ρ Y x 0, by using inequality 2.9 together with the previous estimate for K0, it follows 2.. Lemma 2.5. Let u : B n R m be a continuous mapping. Then there exists a positive function δ u = δ u ε, ε 0, 2, such that if x, y B n, and x y < δ u ε then ux uy ε. In [4, Theorem 3] is proved that the family of harmonic mappings u : N M, which map the monifold N into the regular ball B r Q M is uniformly continuous in compact subsets of N. This implies that the function δ u ε can be chosen independently on u. This fact can improve the conclusion of Lemma 2. as it is done in [9, Theorem 3.].

10 0 DAVID KALAJ Proof of Lemma 2.. In order to estimate the function u 2 in the ball x x 0 < r 0 we introduce the quantity M = max r 0 x x 0 ux x x 0 <r 0 Obviously there exists a point x : x x 0 < r 0 such that M = r 0 x x 0 ux. 2.2 Let d = r 0 x x 0 and θ 0,. If we apply Lemma 2.4 to the case where Y x = ux and Z = ux, x = x and ρ = dθ, and use 2.2, we obtain M d min{γ n dθ, n d n θ n uy ux dσy} y x =dθ + ω n y x n y x d n θ n u dy. Using now 2. we obtain M d min{γ n dθ, n d n θ n + n 2ac n + b ω n y x dθ y x dθ y x =dθ y x dθ uy ux dσy} y x n y x d n θ n y x n y x d n θ n dy. u 2 dy 2.22 We shall now estimate the right hand side of First of all, according to Lemma 2.5, we have for every ε > 0 and dθ < δ u ε the inequality: n d n θ n uy ux dσy nε dθ On the other hand b ω n y x dθ y x =dθ y x n y x n d n θ n dy = b dθ n + Next let λ be a real number such that 0 < λ < θ. Then we have the inequality a ω n y x dθ y x n y x d n θ n u 2 dy a ω n y x dλ y x n y x d n θ n u 2 dy n 2adλc n d n λ n d n θ n u 2 dy. y x dθ In order to estimate the right hand side of this inequality we first observe that, on account of 2.2 we have for x x dλ the estimate u 2 M 2 d 2 λ 2,

11 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS and therefore a ω n y x dλ y x n y x d n θ n u 2 dy a M 2 ω n d 2 λ 2 λd λn+ d n + θ n ω n M 2 = a d 2 λ 2 λd λn+ d n + θ n. Moreover from Lemma 2.3, we conclude that a u 2 dy ω n y x dθ ω n n 2a dn 2 θ n 2 θ n 2 n 2a dn 2 θ n 2 θ n 2 Where K := max x x0 r 0 ux ux 0. Inserting now 2.26 and 2.27 in 2.25 we obtain a y x n y x d n θ n a y x dθ M 2 λn+ λ d λ 2 n + θ n dλ n θ n 2 dθ n θ n 2 + n 2aλ γρ 2 2βn + α β max uy ux y x =d γρ 2 2βn + 2Kα. β u 2 dy γρ 2 2βn + 2Kα. β Combining 2.23 for θ < δ u ε/r 0, 2.24 and 2.28 we conclude from 2.22 that the following inequality holds: M d min{nε, γ n} dθ n 2a λ + d θ n 2 θ 2 Myltiplying by d we get: M min{nε, γ n} θ n 2a λ + θ n 2 θ 2 n + b n + dθ + a M 2 λn+ λ d λ 2 n + θ n n θ γρ 2 λ 2βn + 2Kα. β n + b n + d2 θ + a θ λ M 2 λn+ λ λ 2 n + θ n n γρ 2 2βn + 2Kα β Remember that λ and θ are arbitrary numbers satisfying 0 < λ < θ <. The inequality 2.30 can be written in the form where and AM 2 M + B λn+ A = a λ λ 2 n + θ n

12 2 DAVID KALAJ B = min{nε, γ n} n + b θ n + d2 θ n 2a λ θ + θ n 2 θ 2 λ Taking λ = sin θ we obtain that Hence n γθ 2 d 2 2βn + 2Kα β lim 4AB min{4aεn2 θ 0 n +, 4anγ n n + }. 4AB < for ε = n + 4an 2 +, whenever θ θ 0, where θ 0 is small enough. Observe that in the case 4anγ n n +. <, 2.32 θ 0 can be chosen independently of ε i.e. independently of u. The inequality 2.3 is equivalent with M 4AB 2A = M θ M + 4AB 2A = M + θ for θ θ From 2.33 it follows that only one of the following three cases occur: M M θ, for θ 0, θ 0 ; 2 M M + θ, for θ 0, θ 0 ; 3 there exist θ, θ 2 0, θ 0 say θ < θ 2, such that M < M θ and M > M + θ 2 > M θ 2. + As lim 4AB θ 0 2A = +, the case 2 is not possible. Since M + and M are continuous, the case 3 implies that there exists θ 3 θ, θ 2 such that M θ 3 < M < M + θ 3. Thus the case 3 is also excluded. The conclusion is that only the case is true and henceforth M 4AB 2A Since d < r 0 < it follows that d 2 r 0. Therefore = 2B + 4AB = C 2K, θ 0, a, b, α, β, γ, r 0, n. M 2B C a, b, α, β, γ, n, uk + r From r 0 ux 0 M it follows the desired inequality. The following two lemmas, roughly speaking assert that the boundary behavior of any solution of the Poisson differential inequality is approximately the same as the boundary behavior of the set of two harmonic mappings. They are n dimensional generalizations of [3, Lemma 9] and [3, Lemma 9 ]. Since the proofs in [3] only rely on the maximum principle, the proofs of these lemmas clearly apply to n > 2 as well with very small modifications. Lemma 2.6. Let u : B n B m be a C 2 mapping defined on the unit ball and satisfying the inequality u a u 2 + b, 2.35

13 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 3 where 0 < a < 2 and 0 < b <. Furthermore let ux be continuous for x. Then we have for x B n and t S n the estimate ux ut a Y x ut 2a 2.36 a + 2 2a F x b ut 2 + 2n 2a x 2, where and F x = P x, η uη 2 dση, x < S n 2.37 Y x = P x, ηuηdση, x <. S n 2.38 Lemma 2.7. Let χ : B n [, ] be a mapping of the class C 2 B n CB n satisfying the differential inequality: χ a χ 2 + b 2.39 where a and b are finite constants. Then we have for x B n and t S n the estimate [ χx χt ea h p x e aχt + h m x e aχt + 2ab ] a n ea x 2.40 where and h m x = P x, ηe aχη dση S n 2.4 h p x = P x, ηe aχη dση. S n A priori estimate for a solution to Poisson differential inequality Theorem 3.. Let u : D B m be a C 2 mapping, satisfying the differential inequality: u a u 2 + b 3. where 0 < a < and 0 < b <. Then there exists a constant c 2 = c 2 a, b, n, u such that for x 0 D and r 0 = distx 0, D there holds ux 0 c 2 + max x x 0 r 0 ux ux r 0 If in addition a C n then c 2 can be chosen independent of u and 3.2 is an a priori estimate. Proof. Let us consider the function Gu = u 2 and φx = Gux. Evidently we have Gu = 2u 2 if u 3.3 and m φ = D 2 Gu uxe i, uxe i + Gx, ux 3.4 i= = 2 u u, u.

14 4 DAVID KALAJ From 3. we conclude φ 2 a u 2 2b. 3.5 The conditions of Lemma 2. are therefore satisfied by taking α = 2, β = 2 a and γ = 2b. 3.2 follows with c 2 a, b, n, u = c a, b, α, β, γ, n, u. Theorem 3.2. Let u : B n B m be continuous in B n, u B n C 2, u S n C,α and satisfy the inequalities u a u 2 + b, x B n, 3.6 u S n,α K, 3.7 where 0 < a < /2 and 0 < b, K <. Then there exists a fixed positive number c 4 a, b, n, K, u such that ux c 4 a, b, n, K, u, x B n. 3.8 If in addition a C n then c 4 a, b, n, K, u can be chosen independently of u and 3.8 is an a priori estimate. Proof. Let x 0 = rt B m, t S m. From Theorem 3. we conclude that the inequality ux 0 c 2 a, b, n, u + max x x 0 r ux ux r holds. We shall estimate the quantity First of all we have Q = max ux ux 0. x x 0 r ux ux 0 ux ut + ux 0 ut for x <. Applying now Lemma 2.6 we obtain ux ux 0 a 2a Y x ut + Y x 0 ut a + 2 2a F x ut 2 + F x 0 ut 2 + b 2n 2a [ x 2 + x 0 2 ], 3.0 where the harmonic functions Y and F are defined by 2.37 and To continue, we use the following result due to Gilbarg and Hörmander see [0, Theorem 6. and Lemma 2.], Proposition 3.3. The Dirichlet problem u = f in Ω, u = u 0 on Ω C has a unique solution u C,α, for every f C 0,α, and u 0 C,α, and we have where C is a constant. u,α C u 0,α, Ω + f 0,α 3.

15 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 5 Applying 3. on harmonic functions Y and F, according to.6,.7 and.8, we first have Y x ut + Y x 0 ut 2CK r 3.2 and F x ut 2 + F x 0 ut 2 4CK r. 3.3 Combining 3.0, 3.2 and 3.3 we obtain [ ux ux 0 2CK a 2a + 4CKa ] 2 2a + b r. 3.4 n 2a Thus for c 3 a, b, K, n = 2CK a 2a + 4CKa 2 2a + b n 2a we have Q c 3 a, b, K, n r. Inserting this into 3.9 we obtain ux 0 c 2 a, b, n, u + c 3 a, b, K, n = c 4 a, b, K, n, u. Since x 0 is arbitrary point of the unit ball the inequality 3.8 is established. Whether Theorem 3.2 holds replacing the condition 0 < a < /2 by 0 < a <, is not known by the author. However adding the condition of quasiregularity we obtain the following extension of Theorem 3.2. Corollary 3.4. Assume that u : B n R n is a K quasiregular, twice differentiable mapping, continuous on B n, and u S n C,α. If in addition it satisfies the differential inequality then u a u 2 + b for some constants a, b > u C 8 K, a, b, u. Proof. From.5 we obtain for i =,... n and hence Thus for every i =,..., n K l un J u u i n 2 J u The conclusion follows according to Theorem 3.6. u n J u K 3.6 u K 4/n u i u i ak 2/n u i b. 3.8 In the rest of the paper we will prove an analogous result for arbitrary a and b. The only restriction is u being a real function, i.e. m =. Theorem 3.5. Let B n x 0, r 0 D B n and let χ : D B n [, ] be a mapping of the class C 2 D satisfying the differential inequality: χ a χ 2 + b 3.9 where a and b are finite constants. Then we have the estimate χx 0 c 5 a, b, n, χ + max x x 0 r 0 χx χx r 0

16 6 DAVID KALAJ If a C n then c 5 a, b, n, χ can be chosen independently of χ and 3.20 is an a priori estimate. Proof. Let us consider a twice differentiable function φt, t and ϕx = φχx. The function ϕ satisfies the differential equation Using 3.9 we obtain Taking φt = e 2at we obtain ϕ = φ χ 2 + φ χ. 3.2 ϕ φ a φ χ 2 b φ ϕ 2a 2 e 2a χ 2 2abe 2a The conditions of Lemma 2. are therefore satisfied by taking α = 2ae 2a, β = 2a 2 e 2a, and γ = 2abe 2a. Hence we conclude that 3.20 holds for c 5 a, b, n, χ = c a, b, α, β, γ, n, χ. Theorem 3.6. Let χ : B n R be continuous in B n, χ B n C 2, χ S n C,α and satisfy the inequalities χ a χ 2 + b, x B n, 3.24 χ S n,α K 3.25 where 0 < a, b, K. Then there exists a fixed positive number c 6 = c 6 a, b, K, n, χ, which do not depends on χ for a C n / χ such that χx c 6, x B n Remark 3.7. The condition χ S n,α K of Theorem 3.6 is the best possible, i.e. we cannot replace it by χ S n K. For example O. Martio in [28] gave an example of a harmonic diffeomorphism w = P [f], of the unit disk onto itself such that f C S and w is unbounded. This example can be easily modified for the space. For example we can simply take ux, x 2,..., x n = P [f]x, x 2. Then u S n C but u is not bounded. Proof. The proof follows the same lines as the proof of Theorem 3.2. The only difference is applying Theorem 3.5 instead of Theorem 3. and Lemma 2.7 instead of Lemma 2.6 to the function χ 0 = χx/m, where M = max{ χx : x B n }. Let x 0 = rt B m, t S m. From Theorem 3.5 we conclude that the inequality χ 0 x 0 c 5 a, b, n, χ + max x x 0 r 0 χ 0 x χ 0 x r 0 holds. We shall estimate the quantity First of all we have Q = max χ 0x χ 0 x 0. x x 0 r χ 0 x χ 0 x 0 χ 0 x χ 0 t + χ 0 x 0 χ 0 t for x <. 3.28

17 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 7 Applying now Lemma 2.7 we obtain [ ] χ 0 x χ 0 x 0 ea h p x e aχ0t + h p x 0 e aχ0t a [ ] + ea h m x e aχ0t + h m x 0 e aχ0t a ab n ea x + x 0, where the harmonic functions h p and h m are defined by 2.4 and Applying 3. on harmonic functions h p and h m, according to.6,.7 and.9 for τ = e at and τ 2 = e at, we first have and h p x e aχ0t + h p x 0 e aχ0t 2ae a CK r 3.30 h m x e aχ0t + h m x 0 e aχ0t 2ae a CK r. 3.3 Combining we obtain [ ] χ 0 x χ 0 x 0 4CKe 2a + 4a2 b n ea r Thus for we have Inserting this into 3.27 we obtain c 7 a, b, K, n = 4CKe 2a + 4a2 b n ea Q C 7 a, b, K r. χ 0 x 0 c 5 a, b, n, χ + c 7 a, b, K, n = c 6a, b, K, n, χ. Since x 0 is an arbitrary point of the unit ball the inequality 3.26 is valid for c 6 a, b, K, n, χ = c 6a, b, K, n, χ M. 4. Applications-The proof of Theorem C 4.. Bounded curvature and the distance function. Let Ω be a domain in R n having a non-empty boundary Ω. The distance function is defined by dx = dist x, Ω. 4. The function d is uniformly Lipschitz continuous and there holds the inequality dx dy x y. 4.2 Now let Ω C 2. For y Ω, let νy and T y denote respectively the unit inner normal to Ω at y and the tangent hyperplane to Ω at y. The curvature of Ω at a fixed point y 0 Ω is determined as follows. By the rotation of coordinates we can assume that x n coordinate axis lies in the direction νy 0. In some neighborhood N y 0 of y 0, Ω is given by x n = ϕx, where x = x,..., x n, ϕ C 2 T y 0 N y 0 and ϕy 0 = 0. The curvature of Ω at y 0 is then described by orthogonal invariants of the Hessian matrix D 2 ϕ evaluated at y 0. The eigenvalues of D 2 ϕy 0, κ,..., κ n are called the principal

18 8 DAVID KALAJ curvatures of Ω at y 0 and the corresponding eigenvectors are called the principal directions of Ω at y 0. The mean curvature of Ω at y 0 is given by Hy 0 = n κ i = ϕy n i= By a further rotation of coordinates we can assume that the x,..., x n axes lie along principal directions corresponding to κ,..., κ n at y 0. The Hessian matrix D 2 ϕy 0 with respect to the principal coordinate system at y 0 described above is given by D 2 ϕy 0 = diagκ,..., κ n. Proposition 4.. [] Let Ω be bounded domain of class C k for k 2. Then there exists a positive constant µ depending on Ω such that d C k Γ µ, where Γ µ = {x Ω : dx < µ} and for x Γ µ there exists yx Ω such that dx = νyx. 4.4 Proposition 4.2. [] Let Ω be of class C k for k 2. Let x 0 Γ µ, y 0 Ω be such that x 0 y 0 = dx 0. Then in terms of a principal coordinate system at y 0, we have D 2 κ dx 0 = diag κ d,..., κ n, κ n d Lemma 4.3. Let Ω C 2. For x Γ µ and yx Ω there holds the equation dx = n i= κ i yx κ i yxd. 4.6 If for some x 0 Ω, the mean curvature of y 0 = yx 0 Ω, is positive: then dx is subharmonic in some neighborhood of y 0. In particular if Ω is convex then the function Γ µ x dx is subharmonic. Proof. The equation 4.6 follows from 4.5 and 4.9 it is a special case of the relation 4. taking ux = x. If Ω is convex then for every i κ i 0. Hence dx 0 and thus dx is subharmonic. Lemma 4.4. Let u : Ω Ω be a K q.r. and χ = dux. Then χ u K 2/n χ 4.7 in u Γ µ for µ > 0 such that /µ > κ 0 = max{ κ i x : x Ω, i =,..., n }. Proof. Observe first that d is a unit vector. From χ = d u it follows that χ d u = u. To continue we need the following observation. For a non-singular matrix A we have inf x = Ax 2 = inf Ax, Ax = inf A t Ax, x x = x = = inf{λ : x 0, A t Ax = λx} = inf{λ : x 0, AA t Ax = λax} = inf{λ : y 0, AA t y = λy} = inf x = At x

19 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 9 Next we have χ = u t d and therefore for x u Γ µ, we obtain χ inf e = ut e = inf e = u e = lu K 2/n u. The proof of 4.7 is completed. Lemma 4.5. Let D and Ω R n be open domains, and Ω be a C 2 hypersurface homeomorphic to S n. Let u : D Ω be a twice differentiable K quasiregular surjective mapping satisfying the Poisson differential inequality. Let in addition χx = dux. Then there exists a constant a = Ca, b, K, Ω such that χx a χx 2 + b in u Γ µ for some µ > 0 with /µ > κ 0 = max{ κ i y : y Ω, i =,..., n }. If in addition u is harmonic and Ω is convex then χ is subharmonic for x u Γ µ. Proof. Let y Ω. By the considerations taken in the begin of this section we can choose an orthogonal transformation O y so that the vectors O y e i, i =,..., n make the principal coordinate system in the tangent hyperplane T y of Ω, that determine the principal curvatures of Ω and O y e n = νy. Let x 0 B n. Choose y 0 Ω so that dux 0 = dist ux 0, y 0. Take Ω := O y0 Ω. Let d be the distance function with respect to Ω. Then du = do y0 u and χx = do y0 ux. Thus n χx = D 2 doy0 uxo y0 uxe i, O y0 uxe i 4.9 Next we have i= dux, ux. dux, ux u a u 2 + b. 4.0 Applying 4.5 n D 2 doy0 ux 0 O y0 ux 0 e i, O y0 ux 0 e i i= = = = n n i= j,k= n j,k= n i= D j,k doy0 ux 0 D i O y0 u j x 0 D i O y0 u k x 0 D j,k doy0 ux 0 O y0 ux 0 t e j, O y0 ux 0 t e k κ i κ i d O y0 ux 0 t e i Since the principal curvatures κ i = κ i are bounded by κ 0, combining 4.9, 4.0, 4. and 4.7, and using the relations we obtain for x u Γ µ O y0 ux 0 t e i 2 = ux 0 t O t y 0 e i 2 ux 0 2, χ K 4/n a + nκ 0 µκ 0 χ 2 + b, 4.2

20 20 DAVID KALAJ which is the desired inequality. A K q.c. self-mapping f of the unit ball B n need not be Lipschitz continuous. It is holder continuous i.e. there hold the inequality fx fy M n, K x y K/ n. 4.3 See [6] for the details. See [27] for the extension of the Mori s theorem for domains satisfying the quasihyperbolic boundary conditions as well as for quasiconvex domains. Under some additional conditions on interior regularity we obtain that a q.c. mapping is Lipschitz continuous. Theorem 4.6 The main result. Let u : B n Ω be a twice differentiable quasiconformal mapping of the unit ball onto the bounded domain Ω with C 2 boundary satisfying the Poisson differential inequality. Then u is bounded and u is Lipschitz continuous. Proof. From Lemma 4.5 χ a χ 2 + b for x Γ µ. On the other hand, by a theorem of Martio and Nyakki [30] u has a continuous extension to the boundary. Therefore for every x S n, lim y x χy = χx = 0. Let χ be an C 2 extension of the function χ x u Γ µ in B n by Whitney theorem it exists [39]. Let b 0 = max{ χx : x B n \ u Γ µ/2 }. Then χ a χ 2 + b + b 0. Thus the conditions of Theorem 3.6 are satisfied. The conclusion is that χ is bounded. According to 4.7 u is bounded in u Γ µ and hence in B n as well. The conclusion of the theorem now easily follows. Let u = P [f]. Let S = Sr, θ = Sr, ϕ, θ,..., θ n 2, θ [0, 2π] [0, π] [0, π] be the spherical coordinates and let T θ = S, θ. Let in addition x = ft θ and n x be the normal on Ω defined by the formula n x = x ϕ x θ x θn 2. Since fs n = Ω it follows that n x = n x ν x = D x ν x 4.4 where ν x is the unit inner normal vector that defines the tangent hyperplane of Ω at x = ft θ T P n x = {y : x y, ν x = 0}. Since Ω is convex it follows that x y, ν x 0 for every x Ω and y Ω. 4.5 Let in addition usr, θ = y, y,..., y n. following corollary. Then in these terms we have the Corollary 4.7. If u is a q.c. harmonic mapping of the unit ball onto a convex domain Ω with C 2 boundary, then: J u L B n, 4.6 J b ut := lim r J u rt L S n, 4.7

21 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 2 and there hold the inequality J b ut where K is the quasiconformality constant. We need the following lemma. distu0, Ωn, 4.8 2K n2 n Lemma 4.8. Let A : R n R n be a linear operator such that A = [a ij ] i,j=,...,n. If A is K quasiconformal, then there hold the following double inequality K n A n x x n Ax Ax n A n x x n. 4.9 Here denotes the vectorial product. Both inequalities in 4.9 are sharp. The author believes that the Lemma 4.8 is well-known, and its proof is given in the forthcoming author s paper [24]. Proof. Since in view of Theorem 4.6 u = D i u j n i,j= is bounded, every harmonic mapping D i u j is bounded. Therefore there exists v i,j L S n such that D i u j = P [v i,j ]. Thus lim r D i u j rt = v i,j t for every i, j. The relations 4.6 and 4.7 are therefore proved. On the other hand since y j = u j S, j =,..., n, we have for a.e. t = S, θ S n the relations: lim y iϕr, θ = x iϕ θ, i {,..., n}, 4.20 r and lim y iθ r j r, θ = x iθj θ, i {,..., n}, j {,..., n 2}, 4.2 lim y x i θ y i r, θ irr, θ = lim, i {,..., n} r r r From 4.20, 4.2, 4.22 and. we obtain for a.e. t = S, θ S n : lim J u Sr, θ = lim r r x y r x 2 y 2 x n y n r r... x ϕ x 2ϕ... x nϕ x θ x 2θ... x nθ x θn 2 x 2θn 2... x nθn 2 + r = lim r S η x n n = lim r S n Using 4.5 and the inequality x f η... x n f n η x ϕ... x nϕ x θ... x nθ dση x θn 2... x nθn 2 + r η Sr, θ n ft θ fη, n f T T θ dση. lim r + r η Sr, θ n 2 n

22 22 DAVID KALAJ we obtain lim J u Sr, θ D xθ r 2 n ft θ fη, ν x dση S n = D xθ 2 n ft θ, ν x u0, ν x = D xθ 2 n ft θ u0, ν x = D xθ 2 n dist T P n fs,θ, u0 D xθ distu0, Ω. 2n Thus for a.e. t = S, θ S n, we have JuS, b θ = J u Sθ D T θ D xθ distu0, Ω D T θ 2 n From the left side of 4.9, using the inequality u n J u t we obtain D x θ D T θ K n ut n K n J u t n /n. Combining the last inequality and 4.23 we obtain An open problem. It remains an open problem whether every q.c. harmonic mapping of the unit ball onto a domain with C 2 boundary is bi-lipschitz continuous. This question has affirmative answer for the plane case see [25]. References [] M. Arsenović; V. Kojić; M. Mateljević: On Lipschitz continuity of harmonic quasiregular maps on the unit ball in R n. Ann. Acad. Sci. Fenn. Math , no., [2] S. Axler; P. Bourdon; W. Ramey: Harmonic function theory, Springer Verlag New York [3] S. Bernstein: Sur la généralisation du problém de Dirichlet, Math. Ann. 6990, pp document,.2, [4] M. Bonk; J. Heinonen: Smooth quasiregular mappings with branching. Publ. Math. Inst. Hautes Études Sci. No , [5] C. Fefferman: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, [6] R. Fehlmann; M. Vuorinen: Mori s theorem for n-dimensional quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math , no., [7] F. W. Gehring: Rings and quasiconformal mappings in space. Trans. Amer. Math. Soc. 03, 962, [8] M. Giaquinta: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 05. Princeton University Press, Princeton, NJ, 983. vii+297 pp., [9] M. Giaquinta; S. Hildebrandt: A priori estimates for harmonic mappings. J. Reine Angew. Math , [0] D. Gilbarg; L. Hörmander: Intermediate Schauder estimates. Arch. Rational Mech. Anal. 74, , 3 [] D. Gilbarg; N. Trudinger: Elliptic Partial Differential Equations of Second Order, Vol. 224, 2 Edition, Springer 977, , 4.2

23 A PRIORI ESTIMATE AND QUASICONFORMAL MAPPINGS 23 [2] J. Grotowski:Boundary regularity for quasilinear elliptic systems. Comm. Partial Differential Equations , no. -2, [3] E. Heinz: On certain nonlinear elliptic differential equations and univalent mappings, J. d Anal. Math.5, 956/57, document,.2,, 2, 2 [4] S. Hildebrandt; J. Jost; K.-O. Widman: Harmonic mappings and minimal submanifolds. Invent. Math /8, no. 2, , 2 [5] S. Hildebrandt; K. O. Widman: On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order. Ann. Scuola Norm. Sup. Pisa Cl. Sci , no., [6] L. Hörmander: The boundary problems of physical geodesy. Arch. Rational Mech. Anal , -52. [7] L. Hörmander: Notions of convexity, Progress in Mathematics, vol. 27, Birkhäuser Boston Inc., Boston, MA, 994. [8] J. Jost: Harmonic maps between surfaces. Lecture Notes in Mathematics, 062. Springer- Verlag, Berlin, 984. x+33 pp. [9] J. Jost; H. Karcher: Geometrische Methoden zur Gewinnung von a-priori-schranken fr harmonische Abbildungen. German Manuscripta Math , no., [20] D. Kalaj; M. Mateljević: Inner estimate and quasiconformal harmonic maps between smooth domains, Journal d Analise Math , , 2,, [2] D. Kalaj; M. Mateljević: On certain nonlinear elliptic pde and quasiconfomal mapps between euclidean surfaces, arxiv: [22] D. Kalaj: On harmonic quasiconformal self-mappings of the unit ball, Ann. Acad. Sci. Fenn., Math. Vol 33, -, 2008., [23] D. Kalaj: Quasiconformal harmonic mapping between Jordan domains Math. Z. Volume 260, Number 2, , 2008., [24] D. Kalaj: On the quasiconformal self-mappings of the unit ball satisfying the PDE u = g, to appear. 4. [25] D. Kalaj: Lipschitz spaces and harmonic mappings, Ann. Acad. Sci. Fenn., Math. 34, [26] D. Kalaj; M. Pavlovic: On quasiconformal self-mappings of the unit disk satisfying the Poisson equation, Transactions of AMS in press. [27] P. Koskela; J. Onninen; J. T. Tyson: Quasihyperbolic boundary conditions and capacity: Holder continuity of quasiconformal mappings Comment. Math. Helv [28] O. Martio: On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn., Ser. A I , [29] O. Martio; S.Rickman; J. Väisälä: Definition for quasiregular mappings, Ann. Acad. Sci. Fenn., Vol Ser AI, 969, N 448, p [30] O. Martio; R. Nyakki: Continuation of quasiconformal mappings. Russian Translated from the English by N. S. Dairbekov. Sibirsk. Mat. Zh , no. 4, [3] O. Martio; S. Rickman; J. Väisälä J.: Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn., Vol Ser AI, 97, N 465, p. -3. [32] M. Pavlović: Boundary correspondence under harmonic quasiconformal homeomorfisms of the unit disc, Ann. Acad. Sci. Fenn., Vol 27, [33] J. Rešetnjak: Generalized derivatives and differentiability almost everywhere. Russian Mat. Sb. N.S , , [34] J. Rešetnjak: Space mappings with bounded distortion, Translations of Mathematical Monographs, Vol. 73, American Mathematical Society, Providence, RI 989. [35] S. Rickman, Quasiregular Mappings, Springer, Berlin 993. [36] L.-F. Tam; T. Wan: On quasiconformal harmonic maps, Pacific J. Math , no. 2, [37] M. Vuorinen: Conformal geometry and quasiregular mappings. Lecture Notes in Mathematics, 39. Springer-Verlag, Berlin, 988. xx+209 pp.. [38] H. Werner: Das Problem von Douglas fuer Flaechen konstanter mittlerer Kruemmung, Math. Ann., , p [39] H. Whitney: Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc , no.,

24 24 DAVID KALAJ University of Montenegro, Faculty of Natural Sciences and Mathematics, Cetinjski put b.b. 8000, Podgorica, Montenegro. address: