Univalent solutions of elliptic systems of Heinz-Lewy type

Theorem 2, states that cp (x) = o ( x ") for some n E ~l implies the existence of Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

Theorem 1 *, which applies for differentiable coefficients. We rely on the classical ideas for the rather involved proof. In the third section we show the non-vanishing of the Jacobian of homeomorphic solutions of the Heinz-Lewy system (1) and give an indirect a priori estimate from below by appropriately modifying the proofs of Heinz in ([11], [15]). This is accomplished if a (u) is Holder continuous and if the ht (u)’s are Lipschitz continuous. We only have to establish the connection between

THEOREM 1. - The first derivatives of u (x) are Hölder continuous in the interior of 03A9 with exponent N., Moreover, the Holder norm on any compact subset Q’ of o can be estimated in the form where C depends only on the parameters ?~, A, L, a, b, M, N and dist (SZ’, The proof of this result will be accomplished in four steps. First we establish a modulus of continuity for u (x) in the interior of Q, essentially applying the Courant-Lebesque lemma [6], Lemma 3. 1. In the following, we shall work

LEMMA 1. - The modulus of continuity of u (x) can be estimated in the form

LEMMA 2. - For all (?, 0 cs l, there is a radius Ro, which depends only on the available parameters, such that for all R, 0 R _ Ro, The Dirichlet growth lemma therefore implies Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

LEMMA 3. - For all cy, 0«7min{ ~ 1/2 }, there are radii R o = R o ( a), such that for all R, 0 R _ Ro, COROLLARY. - 0 a min ~ ~,1., 1/2}. This follows from Campanato’s characterization of the Holder classes. In particular Du (x) is bounded.

LEMMA 4. - There exists a radius R o such that for all R, 0 R R o, Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

Theorem 1 follows as a corollary to Lemma 4. 2. LOCAL BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL INEQUALITIES Let 03C6(x) ~C1 {SZ) be a real-valued function satisfying the differential inequality in a domain Q in the x = (xl, x2)-plane which contains the origin, i. e., in the weak sense. The coefficients depend only on x and are Holder continuous satisfying Assumption (A2) part (i). Without loss of generality, we make the normalization a11 (0) = a22 (0), (0) = a21 (0) = 0. The crucial result of this

THEOREM 2. - If cp (x) = o ( for some n then exists. In order to prove this result, we first note that only differential inequali- ties of the form have to be considered. We approximate a by a differentiable function a for x ~ 0, and then we modify Hartman-Wintner’s proof of the case a (x) -1 [7], Theorem 1.

LEMMA 5. - Let some w 0 ~. l, which satisfies Vol. 6, ir 5-1989.

Lemma 5. Then Now and therefore Multiplying by and integrating over DE (~)), it follows that If then we can to obtain For fixed n >_ 1, it will be shown by induction over k, 1 _ k ~ n, that (10) holds. First, (10) is true for k =1. Suppose now that (10) holds for a k, 1 k ~ n. By Lemma 5, all integrals are absolutely convergent. In particu- lar, there is a constant C, which depends only on the data, such that Vol. 6, nO 5-1989.

Theorem 2* of [7]:

COROLLARY 1. - Let the assumptions of Theorem 2 be satisfied. T’hen either c~ (x) - 0 or there exists a non-negative integer m such that We shall need this theorem in the case n >_ 1, where then obviously m > 1. The following corollaries are essentially identical with [15], Hilfssatz 1 and [11), Hilfssatz 2:

COROLLARY 2. - If n >_ 1 and cp (x) ~ 0, then (p (z) has an asymptotic expansion of the form where A ~ 0 and m >_ 1.

COROLLARY 3. - Let {03C6(k)(x) }~k=1 be a family of C1-solutions to the differential inequality (8), where C is independent of k. Assume that uniformly in DR (k ~ oo). Let cp (x) = o ( I x I ) as x I --~ 0 and assume that (x) ~ 0 in DR for all Then cp (x) _-- 0.

PROPOSITION 1. - There exists a in the u-plane, S = ~ (Mo), such that for each ~ _ (~1, ~2) E ~2, I ~ I =1, there is a function 03A6 E CZ (Ds) with 03A6 (0) = 0, (0) = 03BE, ~ 03A6 ~C2 (D03B4) C and cp (x)=03A6 (u (x)) satisfies a differential inequality of the form in Ixl R, for any solving (19) with u (o) = o, b, -_K.

Corollary 3 of Theorem 2 of the previous section. Let us finally remark that the Lipschitz condition on the coefficients ht can be weakened to the effect that only certain combinations of the need to be Lipschitz continuous. REFERENCES [1] P. W. BERG, On Univalent Mappings by Solutions of Linear Elliptic Partial Differential Equations, Trans. Amer. Math. Soc., Vol. 84, 1957, pp. 310-318. [2] S. CAMPANATO, Equazioni Ellittiche del II0 Ordine e Spazi L(2,03BB), Ann. Mat. Pura Appl., IV. Ser., Vol.