Evolution equations with a free boundary condition (Q1307110)

The authors prove local existence and uniqueness of classical solutions for the heat flow of harmonic maps between compact, Riemannian manifolds \(M\) and \(N\) (\(\partial N\) empty), with a free boundary condition. Techniques include a localized reflection method, and the LeRay-Schauder fixed point theorem. Given an additional geometric restriction on \(N\) and the smooth submanifold of \(N\) containing the image of \(\partial M\) (a ``convexity supporting condition''), they obtain global existence. They also derive the energy monotonicity formula and small energy regularity theorem. The authors further construct examples which show that classical solutions may develop singularities in finite time, and find a lower bound for these blow-up times.