Variationally harmonic maps with general boundary conditions: Boundary regularity

Abstract

Let $${\mathcal{M}}$$ and $${\mathcal{N}}$$ be Riemannian manifolds, $${\mathcal{N}}$$ compact without boundary. We develop a definition of a variationally harmonic map $$u\in H^1({\mathcal{M}},{\mathcal{N}})$$ with respect to a general boundary condition of the kind u(x)∊Γ(x) for a.e. $$x\in\partial{\mathcal{M}}$$ , where $$\Gamma(x)\subset{\mathcal{N}}$$ are given submanifolds depending smoothly on x. The given definition of variationally harmonic maps is slightly more restrictive, but also more natural than the usual definition of stationary harmonic maps. After deducing an energy monotonicity formula, it is possible to derive a regularity theory for variationally harmonic maps with general boundary data. The results include full boundary regularity in the Dirichlet boundary case Γ(x) = {g(x)} for $$g\in C^{2,\alpha}(\partial{\mathcal{M}},{\mathcal{N}})$$ if $${\mathcal{N}}$$ does not carry a nonconstant harmonic 2-sphere.