On polyharmonic maps into spheres in the critical dimension

  • Paweł Goldstein

    Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
  • Paweł Strzelecki

    Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
  • Anna Zatorska-Goldstein

    Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Abstract

We prove that every polyharmonic map is smooth in the critical dimension . Moreover, in every dimension , a weak limit of a sequence of polyharmonic maps is also polyharmonic.

The proofs are based on the equivalence of the polyharmonic map equations with a system of lower order conservation laws in divergence-like form. The proof of regularity in dimension uses estimates by Riesz potentials and Sobolev inequalities; it can be generalized to a wide class of nonlinear elliptic systems of order .

Cite this article

Paweł Goldstein, Paweł Strzelecki, Anna Zatorska-Goldstein, On polyharmonic maps into spheres in the critical dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, pp. 1387–1405

DOI 10.1016/J.ANIHPC.2008.10.008