GRADIENT ESTIMATION OF A p-HARMONIC MAP

This article presents Lp estimates for the gradient of p-harmonic maps. Since the system satisfies a natural growth condition, it is difficult to use standard elliptic estimates. We use spherical coordinates to convert the system into another system with angle functions. The new system can be estimate by the standard elliptic technique. 1. Results Let G ⊂ R (n ∈ {2, 3}) be a bounded and simply connected domain with smooth boundary ∂G. Denote Sn−1 = {x ∈ R : x1 + x2 + · · ·+ xn = 1}. Let g be a smooth map from ∂G into Sn−1 satisfying deg(g, ∂G) = d = 0. Denote by {ei}i=1 an orthogonal basis of R. We are concerned with the estimate of the gradient of p-harmonic maps on G, where p > 2. We call u ∈ W 1,p(G,Sn−1) a p-harmonic map on G, if it is a weak solution of (cf. [4]) −div |∇u|p−2∇u) = u|∇u|. (1.1) The L estimate of the gradient of the weak solutions of p-Laplace system is essential for the better regularity (cf. [3, 4, 5, 6, 7, 11, 12]). Thus, in this paper we prove the following theorem. Theorem 1.1. If u is a p-harmonic map on G and u = g on ∂G, then there exists a constant C > 0 which only depends on G, g, p, n, such that ‖∇u‖Lp(G) ≤ C. Different from [12], it is not easy to estimate the weak solution since (1.1) satisfies the natural growth condition. In [11], a sharp Gagliardo-Nirenberg inequality is used for obtaining regularity of theW -solution. For theW 1,p weak solution, this estimate can not be used. To prove the main theorem, we should list some preliminaries. Proposition 1.2. The p-harmonic map u on G satisfies ∫ G |∇u|p−2(u ∧∇u)∇ζdx = 0, ∀ζ ∈W 1,p 0 (G). (1.2) 2000 Mathematics Subject Classification. 35J70, 49J20, 58G18.