Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.

[1]  An optimal estimate for the singular set of a harmonic map in the free boundary. , 1989 .

[2]  Richard S. Hamilton,et al.  Harmonic Maps of Manifolds with Boundary , 1975 .

[3]  T. Rivière,et al.  Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps , 2010, 1012.2438.

[4]  Harmonic mappings with partially free boundary , 1982 .

[5]  Tristan Rivière,et al.  Three-term commutator estimates and the regularity of $\half$-harmonic maps into spheres , 2011 .

[6]  PABLO RAÚL STINGA,et al.  Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation , 2015, SIAM J. Math. Anal..

[7]  F. Lin,et al.  The analysis of harmonic maps and their heat flows , 2008 .

[8]  Ma Li.  Harmonic map heat flow with free boundary , 1991 .

[9]  L. Caffarelli,et al.  An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.

[10]  Jerome Wettstein,et al.  Existence, Uniqueness and Regularity of the Fractional Harmonic Gradient Flow in General Target Manifolds , 2021, 2109.11458.

[11]  F. Lin,et al.  Evolution equations with a free boundary condition , 1998 .

[12]  Michael Struwe,et al.  Existence and partial regularity results for the heat flow for harmonic maps , 1989 .

[13]  Nicola Garofalo,et al.  Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations , 2017, Advances in Mathematics.

[14]  M. Struwe.  The evolution of harmonic mappings with free boundaries , 1991 .

[15]  Mazzeo Rafe.  Elliptic theory of differential edge operators I , 1991 .

[16]  Aleš Nekvinda,et al.  Characterization of traces of the weighted Sobolev space $W^{1,p}(\Omega,d_M^\epsilon)$ on $M$ , 1993 .

[17]  Juncheng Wei,et al.  Infinite time blow-up for half-harmonic map flow from R into S1 , 2017, American Journal of Mathematics.

[18]  J. Wettstein.  Uniqueness and regularity of the fractional harmonic gradient flow in Sn−1 , 2021, Nonlinear Analysis.

[19]  K. Steffen,et al.  A partial regularity theorem for harmonic maps at a free boundary , 1989 .

[20]  Yunmei Chen,et al.  The weak solutions to the evolution problems of harmonic maps , 1989 .

[21]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[22]  X. Cheng.  Estimate of the singular set of the evolution problem for harmonic maps , 1991 .

[23]  James Roberts.  A regularity theory for intrinsic minimising fractional harmonic maps , 2018, Calculus of Variations and Partial Differential Equations.

[24]  K. Nyström,et al.  Extension properties and boundary estimates for a fractional heat operator , 2015, 1511.02893.

[25]  J. Jost,et al.  Harmonic maps which solve a free-boundary problem. , 1987 .

[26]  V. Millot,et al.  On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres , 2013, 1307.7015.

[27]  D. Duffy.  Second‐Order Parabolic Differential Equations , 2013 .

[28]  R. Moser.  Intrinsic Semiharmonic Maps , 2011 .

[29]  Changyou Wang,et al.  Weak solutions of geometric flows associated to integro-differential harmonic maps , 2015, 1512.08353.

[30]  F. Lin,et al.  Partially constrained boundary conditions with energy minimizing mappings , 1989 .