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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
Article
  • 75 Downloads

Abstract

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.

Keywords

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.

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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

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