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Global solutions to the heat flow for \(m\)-harmonic maps and regularity. (English) Zbl 1295.58006

This interesting paper is concerned with the heat flow for \(m\)-harmonic maps from a compact \(m\)-dimensional Riemannian manifold with boundary into a compact Riemannian manifold. The \(m\)-harmonic map heat flow is the gradient flow of \(\int|Du|^m\,dx\); more generally, the authors study the heat flow for \(\int(s^2+|Du|^2)^{m/2}\,dx\) for any \(s\in[0,1]\). Cauchy-Dirichlet data are prescribed on the parabolic boundary, which is one of the essential new features here. The authors establish the existence of global weak solutions in \(C^0([0,\infty);L^2)\cap L^\infty(0,\infty;W^{1,m})\). There is a sequence of times \(t_j\to\infty\) for which the time slices of \(u\) converge to a solution of the stationary problem. Moreover, for the case of the target having non-positive sectional curvature, a solution is constructed which is \(C^{1,\alpha}\) in the space variables.
The weak solution is constructed by a time discretization argument inspired by J.-i. Haga et al. [Comput. Vis. Sci. 7, No. 1, 53–59 (2004; Zbl 1120.53304)]. On equidistant time slices of step size \(h\), an approximative solution is constructed successively by letting \(u^{(j)}\) be the minimizer of the functional \(F^{(j)}(v):={1\over m}\int(s^2+|Dv|^2)^{m/2}\,dx+{1\over2h}\int|v-u^{(j-1)}|^2\,dx.\) Even in the degenerate case \(s=0\), the procedure is applied with some nonvanishing \(s_\ell\), letting \(s_\ell\searrow0\) only after \(h\searrow 0\) has been performed. A striking technical feature is the use of estimates in some fractional Sobolev space, which helps to achieve strong convergence of the approximating sequence as \(h\searrow 0\) away from finitely many points.

MSC:

58E20 Harmonic maps, etc.
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K51 Initial-boundary value problems for second-order parabolic systems
35B40 Asymptotic behavior of solutions to PDEs
35D30 Weak solutions to PDEs

Citations:

Zbl 1120.53304
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