The Journal of Geometric Analysis

, Volume 8, Issue 2, pp 179–197 | Cite as

Evolution equations with a free boundary condition

  • Yunmei Chen
  • Fang Hua Lin


In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂1,u −Δu = Γ(u)(∇u, ∇u) [tT Tu uN, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥u Tu(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = uo(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M.

Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝn, in which the nonlinear equation reduces to ∂tu-Δu = 0.

We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than KN <-0 and Σ is totally geodesic in N.


Regular Solution Free Boundary Condition Partial Regularity Global Smooth Solution Smooth Riemannian Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Baldes, A. Harmonic mappings with partially free boundary,Manus. Math.,40, 255–275, (1982).CrossRefMathSciNetMATHGoogle Scholar
  2. [2]
    Chang, K.C., Ding, W.Y., and Ye, R.G. Finite time blowup of harmonic maps heat flow in two dimensions, preprint.Google Scholar
  3. [3]
    Chipot, M., Fila, M., and Quittner, P. Stationary solutions, blowup and convergence,Acta Math. Univ. Comeniance,LXI, 35–103, (1991).MathSciNetGoogle Scholar
  4. [4]
    Chen, Y.M. and Struwe, M. Existence and partial regularity results for the heat flow of harmonic maps,Math. Z.,201, 83–103, (1989).CrossRefMathSciNetMATHGoogle Scholar
  5. [5]
    Duzaar, F. and Steffen, K. A partial regularity for harmonic maps at a free boundary, preprint.Google Scholar
  6. [6]
    Duzaar, F. and Steffen, K. An optimal estimate for the singular set of a harmonic map in the free boundary, preprint.Google Scholar
  7. [7]
    Eells, J. and Sampson, J.H. Harmonic mappings of Riemannian manifolds,Am. J. Math.,86, 109–160, (1964).CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    Gulliver, R. and Jost, J. Harmonic maps which solve a free-boundary problem,J. Reine U. Ang. Math.,381, 61–89, (1987).MathSciNetMATHGoogle Scholar
  9. [9]
    Hamilton, R. Harmonic maps of manifold with boundary,Springer-Verlag, Lecture Notes in Math.,471, (1975).Google Scholar
  10. [10]
    Hardt, R. and Lin, F.H. Partially constrained boundary conditions with energy minimizing mappings,CPAM,XIII, 309–334, (1984).Google Scholar
  11. [11]
    Hardt, R. and Lin, F.H. A remark onH 1-mappings,Manuscripta Math.,56, 1–10, (1986).CrossRefMathSciNetMATHGoogle Scholar
  12. [12]
    Li, Ma. Harmonic map heat flow with free boundary, preprint, Trieste, (1990).Google Scholar
  13. [13]
    Ladyzenskaja, O.A., Solonnikov, V.A., and Ural’ceva, N.N. Linear and quasilinear equations of parabolic type,Am. Math. Soc. Trans. Math. Monographs, 23, (1968).Google Scholar
  14. [14]
    Struwe, M. The evolution of harmonic mappings with free boundaries,Manuscripta Math,70, 373–384, (1991).CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1998

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesville
  2. 2.Courant Institute of Mathematical ScienceNew York UniversityNew York

Personalised recommendations