#### DMCA

## Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations (1989)

Venue: | Proc. Japan Acad. Ser. A 65 |

Citations: | 362 - 16 self |

### Citations

1181 | Fronts propagating with curvature dependant speed: algorithms based on Hamilton-Jacobi formulations
- Osher, Sethian
- 1988
(Show Context)
Citation Context ...d that Γ(t) is determined only by Γ(0), which is not expected for general geometric equations having first-order terms such as (1.5) with v φ 0. We also learned of works of S. Osher and J. A. Sethian =-=[25]-=- and Sethian [26] giving numerical algorithms for evolutions of surfaces with curvature-dependent speed. Their viewpoint of an evolution of surfaces is the same as ours. They regarded them as level su... |

362 |
The heat equation shrinking convex plane curves
- Gage, Hamilton
- 1986
(Show Context)
Citation Context ...the surface. We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R n by its mean curvature has been studied by many authors [1], [3], [4], [8], =-=[10]-=-, [12], [14], [15]. Such a motion is also important in the singular perturbation theory related to Received August 1, 1989 and, in revised form, March 5, 1990. The first author is on leave from and wa... |

321 |
The heat equation shrinks embedded plane curves to round points
- Grayson
- 1987
(Show Context)
Citation Context ...rface. We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R n by its mean curvature has been studied by many authors [1], [3], [4], [8], [10], =-=[12]-=-, [14], [15]. Such a motion is also important in the singular perturbation theory related to Received August 1, 1989 and, in revised form, March 5, 1990. The first author is on leave from and was part... |

295 | Flow by the mean curvature of convex surfaces into spheres
- Huisken
- 1984
(Show Context)
Citation Context ... We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R n by its mean curvature has been studied by many authors [1], [3], [4], [8], [10], [12], =-=[14]-=-, [15]. Such a motion is also important in the singular perturbation theory related to Received August 1, 1989 and, in revised form, March 5, 1990. The first author is on leave from and was partially ... |

253 |
The motion of a surface by its mean curvature
- Brakke
- 1978
(Show Context)
Citation Context ...nishes on the surface. We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R n by its mean curvature has been studied by many authors [1], [3], =-=[4]-=-, [8], [10], [12], [14], [15]. Such a motion is also important in the singular perturbation theory related to Received August 1, 1989 and, in revised form, March 5, 1990. The first author is on leave ... |

175 |
Viscosity solutions of fully nonlinear second-order elliptic partial di erential equations
- Ishii, Lions
- 1990
(Show Context)
Citation Context ..., and ideas of the proof are scattered in various literature, so we include the proof both for completeness and for the reader's convenience. In §3 we state a parabolic version of Ishii's lemma [17], =-=[18]-=-, which is a key to establishing the comparison principle for viscosity solutions. Our results extend Proposition IV. 1 in [18]. §4 establishes comparison results on viscosity sub- and supersolutions ... |

117 |
Perron’s method for Hamilton-Jacobi equations
- Ishii
- 1987
(Show Context)
Citation Context ...scosity solutions does not follow directly from results in [17], [19]. We are forced to extend their theory to our situation. Existence of viscosity solutions is based on Perron's method discussed in =-=[16]-=-, [17]. We construct viscosity sub- and supersolutions of (1.1) and (1.4) and obtain the viscosity solution ua . We now turn to the study of level surfaces of a viscosity solution ua of (1.1) and (1.4... |

91 |
Optimal Control of Diffusion Processes and Hamilton-JacobiBellman Equations, Part II: Viscosity Solutions and Uniqueness,
- Lions
- 1983
(Show Context)
Citation Context ...arabolic equation (1.1) with (1.4) u(0,x) = a(x)eC a(R n ) for some constant α, where Ca(A) is the set of continuous functions a in A such that a - a is compactly supported in A. Recently P. L. Lions =-=[22]-=- introduced a class of weak solutions for degenerate elliptic equations of second order so that a comparison principle holds. Such solutions are called viscosity solutions, and a general theory is est... |

89 |
Convergence problems for functionals and operators.
- Giorgi
- 1979
(Show Context)
Citation Context ...is easy to see h (x) = lim inf h(y), x € L. The wpp^r semicontinuous (u.s.c.) relaxation of Λ to L is defined by A* = -(-Λ)* The concept of Γ~-limit and the relaxations was introduced by E. De Giorgi =-=[11]-=- and it is important, for example in the calculus of variation. For A c R N we consider a dense subset W of J(A) = A xRxR N x S NxN , where S NxN denotes the space of NxN real symmetric matrices. Let ... |

84 |
The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations
- Jensen
- 1988
(Show Context)
Citation Context ... X) may not be continuous at p = 0. A comparison theorem on viscosity solutions was first proved by P. L. Lions [22] for some special degenerate elliptic equations. Later his results were extended in =-=[19]-=-, [17] for general degenerate elliptic equations E(u(y), Du{y), D 2 u{y)) = 0, where E is assumed to be continuous in its variables (see also [6], [20]). A parabolic comparison theorem is also discuss... |

80 |
The normalized curve shortening flow and homothetic solutions
- Abresch, Langer
- 1986
(Show Context)
Citation Context ...less Vu vanishes on the surface. We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R n by its mean curvature has been studied by many authors =-=[1]-=-, [3], [4], [8], [10], [12], [14], [15]. Such a motion is also important in the singular perturbation theory related to Received August 1, 1989 and, in revised form, March 5, 1990. The first author is... |

63 |
A remark on regularization in Hilbert spaces
- Lasry, Lions
- 1986
(Show Context)
Citation Context ... z and k k have the second differential at zk . To prove Proposition 3.2 we approximate u and v by its sup and inf convolutions as in [18], [20]. We recall properties of these convolutions. Lemma 3.5 =-=[21]-=-. Let D be a bounded domain in R N and f,g:D->R be bounded u.s.c. and l.s.c. functions, respectively. For ε > 0, we define f(y) = sup{f(z)-ε- ι \y-z\ 2 } foryeD, z€D ε {g() + ε" 1 |y-z| 2 } foryeD, an... |

53 |
Numerical algorithm for propagating interfaces: HamiltonJacobi equations and conservation
- Sethian
- 1990
(Show Context)
Citation Context ...termined only by Γ(0), which is not expected for general geometric equations having first-order terms such as (1.5) with v φ 0. We also learned of works of S. Osher and J. A. Sethian [25] and Sethian =-=[26]-=- giving numerical algorithms for evolutions of surfaces with curvature-dependent speed. Their viewpoint of an evolution of surfaces is the same as ours. They regarded them as level surfaces of solutio... |

32 |
A uniqueness result for viscosity solutions of second-order fully nonlinear pde’s,
- Jensen, Lions, et al.
- 1988
(Show Context)
Citation Context ...such that the functions z H-> w(z)+p - z attain a maximum at z and k k have the second differential at zk . To prove Proposition 3.2 we approximate u and v by its sup and inf convolutions as in [18], =-=[20]-=-. We recall properties of these convolutions. Lemma 3.5 [21]. Let D be a bounded domain in R N and f,g:D->R be bounded u.s.c. and l.s.c. functions, respectively. For ε > 0, we define f(y) = sup{f(z)-ε... |

26 |
On oblique derivative problems for fully nonlinear second-order elliptic partial differential equations on nonsmooth domains, Nonlinear Anal
- Dupuis, Ishii
- 1990
(Show Context)
Citation Context ...osity supersolution. A function u = u(y) is called a viscosity solution of (2.1) if it is both a viscosity sub- and supersolution of (2.1). Our definition has a meaning for a wider class of E than in =-=[7]-=-, [17], since we do not assume W = J(A) here. We often suppress the word "viscosity", except in statements of theorems, since all solutions in this paper are considered in the viscosity sense. In what... |

24 |
Parabolic equations for curves on surfaces. II. intersections, blow-up and generalized
- Angenent
- 1991
(Show Context)
Citation Context ...Vu vanishes on the surface. We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R n by its mean curvature has been studied by many authors [1], =-=[3]-=-, [4], [8], [10], [12], [14], [15]. Such a motion is also important in the singular perturbation theory related to Received August 1, 1989 and, in revised form, March 5, 1990. The first author is on l... |

20 |
Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with noncontinuous time-dependence
- Nunziante
(Show Context)
Citation Context ...0), D(0)). The main body of this paper consists of §§4-7 preceded by preliminary §§2 and 3. This paper is written so that no previous knowledge of viscosity solutions in [16], [17], [18], [19], [22], =-=[24]-=- is required. The results in this paper have been announced in [5]. After this work was completed, we were informed of a recent work of L. C. Evans and J. Spruck [9] closely related to ours. They also... |

16 |
Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ
- Alexandroff
- 1939
(Show Context)
Citation Context ...Y.-G. CHEN, YOSHIKAZU GIGA & SHUNΊCHI GOTO As is observed in [17, Lemma 5.2], Lemma 3.3 yields the following result since a convex function is almost everywhere differentiable by AlexandrofFs theorem =-=[2]-=-. Lemma 3.4. Suppose that w is as in Lemma 3.3. Let z e D be a maximum point of w . Then there are sequences {z } c D (k = 1,2, --) k satisfying z -• z as k -• oc and {p } c R k k d satisfying \p \ < ... |

13 |
A stable manifold theorem for the curve shortening equation
- Epstein, Weinstein
- 1987
(Show Context)
Citation Context ...s on the surface. We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R n by its mean curvature has been studied by many authors [1], [3], [4], =-=[8]-=-, [10], [12], [14], [15]. Such a motion is also important in the singular perturbation theory related to Received August 1, 1989 and, in revised form, March 5, 1990. The first author is on leave from ... |

7 |
Toward a nonequilibrium thermodynamics of two{phase materials
- Gurtin
- 1988
(Show Context)
Citation Context ...y the Japan Ministry of Education, Science and Culture through grants No. 01740076 and 01540092 for scientific research.750 Y.-G. CHEN, YOSHIKAZU GIGA & SHUNΊCHI GOTO phase transition phenomena (see =-=[13]-=-, [23] and references therein). However, so far whole unique evolution families of surfaces were only constructed under geometric restrictions on initial surfaces such as convexity [10], [14], except ... |

3 |
quadratic forms and fully nonlinear elliptic equations of second order
- Semidifferentials
- 1989
(Show Context)
Citation Context ...liptic equations of second order so that a comparison principle holds. Such solutions are called viscosity solutions, and a general theory is established by R. Jensen [19] and H. Ishii [17] (see also =-=[6]-=- for simplification). For a large class of geometric, degenerate parabolic equations including (1.3) we construct a unique global viscosity solution ua in CQ([0,Γ]xE w ) of (1.1) and (1.4) for every T... |

1 |
Motion of level sets by mean curvature., I, this issue
- Evans, Spruck
(Show Context)
Citation Context ... in [16], [17], [18], [19], [22], [24] is required. The results in this paper have been announced in [5]. After this work was completed, we were informed of a recent work of L. C. Evans and J. Spruck =-=[9]-=- closely related to ours. They also proved the existence of a unique viscosity solution and studied various properties of the level surfaces Γ(/) of the solution, but only for the mean curvature flow ... |

1 |
Evolution gέometric d'interfaces
- Mottoni, Schatzman
- 1989
(Show Context)
Citation Context ...Japan Ministry of Education, Science and Culture through grants No. 01740076 and 01540092 for scientific research.750 Y.-G. CHEN, YOSHIKAZU GIGA & SHUNΊCHI GOTO phase transition phenomena (see [13], =-=[23]-=- and references therein). However, so far whole unique evolution families of surfaces were only constructed under geometric restrictions on initial surfaces such as convexity [10], [14], except n = 2 ... |