Boundary regularity for minimizing biharmonic maps

Abstract

We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven's interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong-Lamm-Wang.

Regularity of solutions for a fourth‐order elliptic system via Conservation law

“… to both extrinsic and intrinsic biharmonic mappings into general closed Riemannian manifolds in a series of pioneer works via the moving frame method of Helein . Parallel to interior regularity, boundary regularity of biharmonic mappings has also been studied in, for example, . In particular, Lamm and Wang obtained, among other results, smoothness of biharmonic mappings near the boundary in dimension four.…”

Blow-up analysis and boundary regularity for variationally biharmonic maps

“…we obtain the same estimate by applying (13) with R 0 in place of R and then enlarging the domain of integration on the right-hand side. Finally, the estimate ( 13) is immediate in the case R 0 ≤ s ≤ R, since R 0 is a universal constant.…”

“…In the boundary situation, however, the question for the corresponding monotonicity formula remained open for some time. In fact, since such a formula was unknown, the first results on partial boundary regularity [7] and full boundary regularity for minimizers [13] had to impose this monotonicity property as an additional assumption. This gap in the theory has been closed by the first author [1], who provided a suitable boundary monotonicity formula and thereby completed the mentioned results from [7,13].…”

“…In fact, since such a formula was unknown, the first results on partial boundary regularity [7] and full boundary regularity for minimizers [13] had to impose this monotonicity property as an additional assumption. This gap in the theory has been closed by the first author [1], who provided a suitable boundary monotonicity formula and thereby completed the mentioned results from [7,13]. The present article now is concerned with the question whether full boundary regularity can also be derived for biharmonic maps that are not minimizing, but only critical points of the bi-energy.…”

“…We use the notation (y ′ , y m ) := y ∈ R m−1 × R and let R 1 := max{y m , ρ}. Then we use first the interior estimate ( 14) and then the boundary version (13) to deduce…”