Regularity of polyharmonic maps in the critical dimension

Abstract

We prove regularity of weakly m-polyharmonic maps (extrinsic or intrinsic) from domains in IR n of dimension n = 2m ≥ 4 to compact Riemannian manifolds, thus extending a previous result by Wang for the case m = 2. Moreover, we prove smoothness of Hölder continuous weakly polyharmonic maps for domains in IR n of dimension n ≥ 2m.

Boundary regularity for polyharmonic maps in the critical dimension

“…In this section, we first recall the Euler-Langrange equation for k-polyharmonic maps derived in [3], the interior regularity for k-polyharmonic maps due to [4,9,3], and then prove a Courant-Lebesgue type lemma for W k;2 -maps.…”

“…Very recently, there have been works on the interior regularity of both extrinsic and intrinsic k-polyharmonic maps in the critical dimension by Gastel & Scheven [3] for k 3. See also Angelsberg & Pumberger [1] for further results on k-polyharmonic maps.…”

“…The first step to prove Theorem 1.1 is the boundary Hölder regularity, which is based on the interior regularity estimates (see [4] for k D 1, [2,9] for k D 2, [3] for k 3), and a boundary maximum principle for k-polyharmonic maps with small k-polyenergies. An immediate consequence of the maximum principle is an L 1 -estimate of u ˆ, which yields that u is continuous up to the boundary.…”

“…By choosing u ˆas a test function to (2.4) and utilizing the growth condition (2.5), we then show that u is Hölder continuous near the boundary. Once we have Hölder continuity of u, we can modify the induction argument of Gastel & Scheven [3] to obtain Hölder continuity of r k u near the boundary. Finally, the smoothness of u near the boundary follows from the classical Schauder theory for linear uniformly elliptic equations.…”

Neck analysis of extrinsic polyharmonic maps

“…For general target manifold, Moser [17] obtained the same result for energy minimizers using a flow method. It is completely solved in full generality by Gastel and Scheven [6]. Hence, in the critical dimension, any weakly polyharmonic map is smooth.…”