Boundary regularity for polyharmonic maps in the critical dimension

Abstract

Abstract. We consider the Dirichlet problem for intrinsic and extrinsic k-polyharmonic maps from a bounded, smooth domain R 2k to a compact, smooth Riemannian manifold N R l without boundary. For any smooth boundary data, we show that any k-polyharmonic map u 2 W k;2 .; N / is smooth near the boundary @.

Boundary partial regularity for a class of biharmonic maps

“…This is a continuation of our previous study in [13]. Here we consider the Dirichlet problem for (extrinsic) biharmonic maps into Riemannian manifolds in dimension at least 5 and address the issue of boundary regularity for a class of stationary biharmonic maps.…”

“…where ν is the unit outward normal of ∂Ω. For dimension of Ω, n = 4, the complete boundary smoothness of biharmonic maps has been proved by Ku [11] for N = S L−1 and Lamm-Wang [13] for any compact Riemannian manifold N 1 . For dimensions n ≥ 5, as in the interior case, it seems necessary to require a boundary monotonicity inequality analogous to (1.4) in order to obtain possible boundary regularity.…”

Regularity and uniqueness of a class of biharmonic map heat flows

“…It asserts smoothness of biharmonic maps when the dimension n = 4, and the partial regularity of stationary biharmonic maps when n ≥ 5. Here we mention in passing the interesting works on biharmonic maps by Angelsberg [1], Strzelecki [31], Hong-Wang [17], Lamm-Rivière [24], Struwe [40], Ku [20], Gastel-Scheven [10], Scheven [34,35], Lamm-Wang [25], Moser [28,29], Gastel-Zorn [11], Hong-Yin [18], and Gong-Lamm-Wang [12]. Now we describe the initial and boundary value problem for the heat flow of biharmonic maps.…”

Boundary regularity for minimizing biharmonic maps

“…Let us mention here two inconclusive results in the direction of boundary regularity. Firstly, it was shown in [14] by Lamm and C. Wang that polyharmonic maps, in the conformal case m = 2k, enjoy the property of being continuous in a neighborhood of the boundary. The proof is strongly dependent on the relation m = 2k and one might not extend this method to the case m > 2k.…”