zbMATH — the first resource for mathematics

Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. (English) Zbl 0735.35082
The authors construct unique global continuous viscosity solutions of the initial value problem in \(\mathbb{R}^ n\) for a class of degenerate parabolic equations that they call geometric. A typical example is \[ u_ t- |\nabla u|\text{div}(\nabla u/|\nabla u|)-\nu|\nabla u|=0. (*) \] For a solution \(u\), let \(\Gamma(t)=\{x\in\mathbb{R}^ n: u(x,t)=\gamma\}\) and \(D(t)=\{x\in\mathbb{R}^ n: u(x,t)>\gamma\}\). Then the family \((\Gamma(t),D(t))\), \(t\geq0\), is uniquely determined by \((\Gamma(0),D(0))\) and is independent of \(u\) and \(\gamma\). Moreover \((\Gamma(t),D(t))\) becomes empty in a finite time provided \(\nu\leq0\). This extends a result of G. Huisken [J. Differ. Geom. 20, 237-266 (1984; Zbl 0556.53001)]. The main tool is a comparison principle for viscosity solutions, but proofs are not given.
Reviewer: A.Pryde (Clayton)

35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI
[1] S. Angenent: Parabolic equations for curves on surfaces II, intersections, blowup and generalized solutions. CMS-Technical Summary Report #89-24 University of Wisconsin. · Zbl 0749.58054
[2] K. A. Brakke: The Motion of a Surface by its Mean Curvature. Princeton Univ. Press (1978). · Zbl 0386.53047
[3] M. Gage and R. Hamilton: The shrinking of convex plane curves by the heat equation. J. Diff. Geom., 23, 69-96 (1986). · Zbl 0621.53001
[4] M. Grayson: The heat equation shrinks embedded plane curves to points, ibid., 26,285-314 (1987). · Zbl 0667.53001
[5] G. Huisken: Flow by mean curvature of convex surfaces into spheres, ibid., 20, 237-266 (1984). · Zbl 0556.53001
[6] H. Ishii: On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE’s. Comm. Pure Appl. Math., 42, 15-45 (1989). · Zbl 0645.35025
[7] H. Ishii and P. L. Lions: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Diff. Eq. · Zbl 0708.35031
[8] R. Jensen: The maximum principle for viscosity solutions of fully nonlinear second-order partial differential equations. Arch. Rational Mech. Anal., 101, 1-27 (1988). · Zbl 0708.35019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.