A Regularity Of Weakly Quasiregular Mappings And A Compactness Of Q-harmonic Maps In Carnot Group

ABSTRACT:Carnot group is a kind of nilpotent layered Lie groups. The non exchangeability of vector field in Carnot group make it present different properties from Euclidean space. Many of conclusions and better properties of partial differ-ential equations have been already proofed in the European space. How to extend these conclusions and properties to Carnot group has been a hot-button issue and attented by many scholars, now Fourier integral operator, paradifferential operator, pseudo-differential operator and other harmonic analysis and partial differential e-quation problems in Heisenberg group and Carnot group has already become the main object of study. This paper mainly consider two analytical problems in Carnot group.A self-improving integrability of weakly quasiregular mappings defined in Heisen-berg group is first studied. Li(Bergen University,2011) used the theory of stopping time to get the Caccioppoli inequalities when p<n. In this paper we establish a reverse Holder inequality of the horizontal derivatives due to the relationship be-tween Jacobian determinant of contact maps f*and the horizontal derivatives Hf*so that the integrability of horizontal derivatives has been gotten a self-improving, and eventually increased to greater than the first layer space dimension2n.On the other hand, Q-harmonic map in Carnot group has been studied. In this regard, based on Green estimation of elliptic equation, Jost-Xu(Trans. AMS) got the smoothness result of subelliptic harmonic maps into a sphere. Later, Wang Changyou(CVPDE,2004) demonstrated the regularity of subelliptic harmonic maps from a Carnot group into a compact Riemannian manifold without boundary. In this paper, for a homogeneous dimension Q let Q(?)RQ be a bounded smooth domain and N-(?)RL be a compact smooth Riemannian manifold without boundary. Sup-pose that{uk}∈HW1,Q(Ω,N-) are weak solutions in the critical dimension to the perturbed Q-harmonic maps satisfying Φkâ†'0in Sobolev space (HW1,Q(Q,N-))*, and Ukâ†'u weakly in HW1,Q(Ω,N-). Then u is a Q-harmonic map. In particular, the space of Q-harmonic maps is sequentially compact for the weak-HW1,Q topology.