Abstract
We prove existence, locally in time, of a solution of the following Hele-Shaw problem: Given a simply connected curve contained in a smooth bounded domainΩ, find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve inΩ and whose boundary value on the curve equals its curvature. We show that this motion is a curve-shortening motion which does not change the area of the region enclosed by the curve. In caseΩ is the whole plane ℛ2, we also show that if the initial curve is close to an equilibrium curve, i.e., to a circle, then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity.
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References
N. Alikakos &G. Fusco,Slow dynamics of spherical fronts for the Cahn-Hilliard equation, preprint.
F. J. Almgren &L. Wang,Mathematical Existence of crystal growth with Gibbs-Thomson curvature effects, Informal Lecture notes for the workshop on Evolving Phase Boundaries, at Carnegie Mellon University, March 1991.
G. Caginalp,Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A39 (1989), 5887–5896.
Y. G. Chen, Y. Giga &S. Goto,Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom.33 (1991), 749–786.
J. Duchon &R. Robert,Évolution d'une interface par capillarité et diffusion de volume I. Existence locale en temps, Ann. Inst. H. Poincaré, Analyse non linéaire1 (1984), 361–378.
C. M. Elliott &J. Ockendon,Weak and Variational Methods for Moving Boundary Problems, Pitman, Boston, 1982.
L. C. Evans &J. Spruck,Motion of level set by mean curvature I, J. Diff. Geom.33 (1991), 635–681.
M. Gage &R. Hamilton,The shrinking of convex plane curves by heat equation, J. Diff. Geom.23 (1986), 69–96.
G. Gilbarg &N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, 1983.
M. Grayson,The heat equation shrinks embedded plane curves to points, J. Diff. Geom.26 (1987), 285–314.
R. S. Hamilton,Three manifolds with positive Ricci curvature, J. Diff. Geom.17 (1982), 255–306.
G. Huisken,Flow by mean curvature of convex surface into point. J. Diff. Geom.20 (1984), 237–266.
D. S. Jerison &C. E. Kenig,The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc.4 (1981), 203–207.
C. E. Kenig,Elliptic boundary value problems on Lipschitz domains, in Beijing Lectures in Harmonic Analysis (E. M. Stein, ed.) Princeton, New Jersey, 1986, 131–184.
J. L. Lions &E. Magenes,Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, 1972.
S. Luckhaus,Solutions for the two-phase Stefan problem with Gibbs-Thomson law for the melting temperature, Euro. J. Appl. Math.1 (1990), 101–111.
L. E. Payne &H. F. Weinberger,New bounds in harmonic and biharmonic problems, J. Math. Phys.33 (1954), 291–307.
R. L. Pego,Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Soc. Lond. A422 (1989), 261–278.
E. Radkevitch,The Gibbs-Thomson correction and conditions for the classical solution of the modified Stefan problem, Sov. Math. Doklady43 (1991), 274–278.
H. M. Soner,Motion of a set by the curvature of its boundary, Preprint.
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Communicated by D.Kinderlehrer
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Chen, X. The Hele-Shaw problem and area-preserving curve-shortening motions. Arch. Rational Mech. Anal. 123, 117–151 (1993). https://doi.org/10.1007/BF00695274
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DOI: https://doi.org/10.1007/BF00695274