Evolution equations with a free boundary condition

Yunmei Chen, Fang-Hua Lin

Research output: Contribution to journalArticle

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 ∂tu - Δu = Γ(u)∇u, ∇u) ⊥ TuN, on M × [0, ∞) , u(t, x) ∈ Σ, for x ∈ ∂M, t > 0 , ∂u/∂n(t, x) ⊥ 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.

Original languageEnglish (US)
Pages (from-to)196-197
Number of pages2
JournalJournal of Geometric Analysis
Volume8
Issue number2
StatePublished - 1998

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Regular Solution
Free Boundary
Evolution Equation
Boundary conditions
Leray-Schauder Fixed Point Theorem
Monotonicity Formula
Unit normal vector
Blow-up Time
Totally Geodesic
Local Existence
Harmonic Maps
Heat Flow
A Priori Estimates
Energy
Theorem
Compact Manifold
Submanifolds
Existence Theorem
Global Existence
Initial-boundary-value Problem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Evolution equations with a free boundary condition. / Chen, Yunmei; Lin, Fang-Hua.

In: Journal of Geometric Analysis, Vol. 8, No. 2, 1998, p. 196-197.

Research output: Contribution to journalArticle

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