We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in super-critical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps. *
We consider critical points u : Ω → N of the bi-energywhere Ω ⊂ R m is a bounded smooth domain of dimension m ≥ 5 and N ⊂ R L a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps u ∈ W 2,2 (Ω, N ), which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the presence of certain non-constant biharmonic 4-spheres or 4-halfspheres in the target manifold. As an application, we deduce full boundary regularity of variationally biharmonic maps provided such spheres do not exist.
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