A regularity theory of biharmonic maps

Abstract

In this article we prove the regularity of weakly biharmonic maps of domains in Euclidean four space into spheres, as well as the corresponding partial regularity result of stationary biharmonic maps of higher-dimensional domains into spheres.

Boundary regularity for polyharmonic maps in the critical dimension

“…The reader can refer to [4] for k D 1, [2,9,6] for k D 2, and [3] for k 3. Note that by the absolute continuity, the condition (2.7) holds at any x 2 provided r 0 > 0 is chosen to be sufficiently small.…”

“…Bereitgestellt von | MPI fuer Gravitationsphysik Angemeldet Heruntergeladen am | 01.02.17 12:02 N D S l 1 R l has been obtained by Chang, Wang & Yang [2]; and for general target manifolds N R l , the second author [9] has proved the interior regularity for both extrinsic and intrinsic biharmonic maps (see Lamm & Rivière [6] for a new proof).…”

“…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.…”

Harmonic and Biharmonic Maps at Iaşi

“…non-harmonic biharmonic, maps are studied. The theory of biharmonic maps has two main directions: one concerns the analytic properties (see [20,39,45,64]) and the other deals with geometric aspects. In the following we shall only refer to the latter.…”

A regularity result for polyharmonic maps with higher integrability

“…Chang et al [5] showed smoothness for weakly biharmonic maps into the sphere and m ≤ 4 (see also Strzelecki [28]), and asserted partial regularity for stationary biharmonic maps into the sphere and m ≥ 5. Wang generalized these results for arbitrary target manifolds in [31] and [32].…”