Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor
Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor
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DOI：
10.2748/tmj/1178225593
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发表时间：
1995-06
影响因子：
0.5
通讯作者：
H. Ishii;P. Souganidis
中科院分区：
文献类型：
--
作者：
H. Ishii;P. Souganidis
在这篇文章中，我们研究了法向速度取决于法向和曲率张量的非紧超曲面的广义运动。这项工作扩展了 Evans 和 Spruck（针对平均曲率）以及 Chen、Giga 和 Goto（针对具有亚线性曲率依赖性的一般运动）的迄今为止的经典著作，因为它允许对曲率张量的一般依赖性。它还允许对广义演化进行一般处理，包括非紧超曲面。还给出了许多关于无内部、凸性等的结果。介绍。在过去的几年里，在理解表面的演化方面取得了实质性进展，在奇点第一次出现之后，表面的演化速度取决于曲率张量和法线方向。所谓的水平集方法，基于将表面表征为某些完全非线性简并抛物线偏微分方程解的水平集（为了明确性，零水平集），由 Evans 和 Spruck [ES] 成功开发用于平均曲率运动，由 Chen、Giga 和 Goto [CGG] 成功开发用于更一般的演化，然而，其中法向速度最多线性地取决于曲率张量。 [ES]和[CGG]的基本工具是粘度解理论。我们参考 Crandall、Ishii 和 Lions [CIL] 的《用户指南》，对粘度解理论及其范围进行一般性讨论；参考 [ES] 和 [CGG]，了解水平集方法的起源；参考 Soner [Son] 和 Barles、Soner 和 Souganidis [BSS] 的替代公式、扩展、讨论等。超曲面广义运动的一些最引人注目的理由是通过使用它来获得有关超曲面广义运动的严格结果。反应扩散方程的渐近行为（例如，参见 Evans、Soner 和 Souganidis [ESS] 以及 Barles、Soner 和 Souganidis [BSS]），以及最近的 Katsoulakis 和 Souganidis [KS1]、[KS2] 中粒子系统的流体动力学极限（另请参见 Souganidis [Sou]）。本说明的目的是将 [CGG] 的结果扩展到以下情况： * 部分得到日本教育科学文化部科学研究补助金编号 04640189 的支持。 f 部分由 NSF 拨款 DMS-9025617 和 DMS-9296117 (PYI)、ARO 合同 DAAL 03-90-G-0012 ONR 合同 N 00014-93-1-0015 和 Alfred P. Sloan 基金会提供支持。这项工作的一部分是在访问中央大学数学系时完成的。 1991年数学学科分类。初级35K55；中学 35K65。 228 H. ISHΠ 和 P. E. SOUGANIDIS 法向速度是法向矢量和曲率张量的一般连续函数。这种演化在几何中非常自然地出现，因为它们包括，例如，高斯曲率，以及图像处理等应用（参见，例如，Lions [L]和Alvarez，Guichard，Lions和Morel [AGLM]）等。研究这种演化的主要困难是它们产生的偏微分方程的奇点阶数高于[ES]，[CGG]等所考虑的阶数。为了克服这个困难，我们扩展了粘度解定义中允许的测试函数，然后证明比较原理以及此类中的存在性结果。这里水平集方法的一个新特征是，我们关于非线性偏微分方程解的零水平集的唯一性结果足够尖锐，足以处理非紧超曲面的广义演化。因此，我们的论点比 Ilmanen [I] 中关于非紧超曲面的广义演化的论点稍微自然一些。本文的结构如下：在第 1 节中，我们阐述了问题，给出了定义，并回顾了适合我们设置的粘度解决方案理论中的基本事实。我们还回顾了超曲面广义运动的水平集方法的定义。最后，我们提出了一些可以放入我们的框架中的超曲面运动的例子。在第二节中，我们陈述并证明了我们的主要结果，即粘度解的比较原理以及一般存在性结果。最后，在第 3 节中，我们陈述了一些关于广义演化的规律性特性的结果。大约在这项工作完成时，Goto [G] 证明了类似的结果，但在紧凑接口的情况下。后藤的方法与我们的方法不同，它基于引入进化传播的有限速度的概念。作者要感谢审稿人指出定理1.7证明原版中的错误，并为我们提出改进本文英文表达的建议。 1. 提出问题、定义和基本事实。我们考虑β τ = Ωx(0, Л)中的非线性方程(1.1) ut + F{Du,D u) = 0，其中T>0，Ω是R的开子集，ut、Du和Du分别表示未知函数κ: ί2x[0,T]^Λ的时间导数、空间梯度和空间Hessian矩阵，F: RxS^>R是给定函数，S表示空间NxN 对称矩阵。在整篇论文中，我们将假设 (1.2) Fe C(J0)，其中 Jo = (R \{0}) x S , )F 是椭圆形，即，对于所有 pR\{0} 和 X，YeS ，( L 3 ) [ifXF(p, Y)，非紧超曲面的广义运动 229
In this note we study the generalized motion of noncompact hypersurfaces with normal velocity depending on the normal direction and the curvature tensor. This work extends the by-now-classical works of Evans and Spruck (for mean curvature) and Chen, Giga and Goto (for general motions with sublinear curvature dependence), because it allows general dependence on the curvature tensor. It also allows a general treatment of the generalized evolution including noncompact hypersurfaces. A number of results regarding no interior, convexity, etc. are also presented. Introduction. During the past few years there has been a substantial progress in understanding the evolution of surfaces, moving with normal velocity depending on the curvature tensor and the normal direction, past the first time singularities occur. The so-called level set approach, which is based on characterizing the surfaces as a level set (for definiteness the zero level set) of the solution of certain fully nonlinear degenerate parabolic PDE's, was developed successfully by Evans and Spruck [ES] for motions by mean curvature and by Chen, Giga and Goto [CGG] for more general evolutions, in which, however, the normal velocity depends, at most linearly, on the curvature tensor. The basic tool of [ES] and [CGG] is the theory of viscosity solutions. We refer to the User's Guide by Crandall, Ishii and Lions [CIL] for a general discussion of the theory of viscosity solutions and its scope, to [ES] and [CGG] for the origin of the level set approach and to Soner [Son] and Barles, Soner and Souganidis [BSS] for alternative formulations, extensions, discussions, etc. Some of the most striking justifications of the generalized motion of hypersurfaces were provided by its use towards obtaining rigorous results regarding the asymptotic behavior of reaction-diffusion equations (see, for example, Evans, Soner and Souganidis [ESS] and Barles, Soner and Souganidis [BSS]) and, more recently, the hydrodynamic limits of particle systems in Katsoulakis and Souganidis [KS1], [KS2] (see also Souganidis [Sou]). The purpose of this note is to extend the results of [CGG] to cases where the * Partially supported by Grant-in-Aid for Scientific Research No. 04640189, The Ministry of Education, Science and Culture, Japan. f Partially suported by NSF grants DMS-9025617 and DMS-9296117 (PYI), ARO contract DAAL 03-90-G-0012 ONR contract N 00014-93-1-0015 and the Alfred P. Sloan Foundation. Part of this work was done while on visit to the Department of Mathematics, Chuo University. 1991 Mathematics Subject Classification. Primary 35K55; Secondary 35K65. 228 H. ISHΠ AND P. E. SOUGANIDIS normal velocity is a general continuous function of the normal vector and the curvature tensor. Such evolutions arise very naturally in geometry, since they include, for example, the Gaussian curvature, as well as in applications like image processing (see, for example, Lions [L] and Alvarez, Guichard, Lions and Morel [AGLM]), etc. The main difficulty in studying such evolutions is that they give rise to PDE's with singularities of order higher than the one's considered by [ES], [CGG], etc. To overcome this difficulty, we extend the class of admissible test functions in the definition of viscosity solutions and then prove a comparison principle as well as an existence result in this class. A new feature of the level set approach here is that our uniqueness result concerning the zero level sets of solutions of nonlinear PDE's is sharp enough to treat the generalized evolutions of noncompact hypersurfaces. As a result, our arguments are slightly more natural than those in Ilmanen [I] concerning generalized evolutions of noncompact hypersurfaces. The paper is organized as follows: In Section 1 we formulate the problem, give the definitions and recall basic facts from the theory of viscosity solutions adapted to our setting. We also recall the definition of the level set approach to the generalized motion of hypersurfaces. Finally, we present a number of examples of motions of hypersurfaces which can be put in our framework. In Section 2 we state and prove our main results, namely, a comparison principle for viscosity solutions as well as a general existence result. Finally, in Section 3 we state a number of results regarding the regularity properties of the generalized evolution. At about the time when this work was completed, Goto [G] proved similar results but in the case of compact interfaces. Goto's approach, which is different from ours, is based on introduction of a notion of finite speed of propagation for the evolution. The authors would like to thank the referee for pointing out an error in the original version of proof of Theorem 1.7 and for making suggestions for us to improve English expressions in this paper. 1. Formulation of the problem, definitions and basic facts. We consider the nonlinear equation (1.1) ut + F{Du,D u) = 0 in β τ = Ωx(0, Γ), where T>0, Ω is an open subset of R, ut, Du and D u denote the time derivative, the spatial gradient and the spatial Hessian of the unknown function κ: ί2x[0,T]^Λ respectively, F: RxS^>R is a given function and S denotes the space of NxN symmetric matrices. Throughout the paper we will be assuming that (1.2) Fe C(J0), where Jo = (R \{0}) x S , )F is elliptic, i.e., for all peR\{0} and X, YeS , ( L 3 ) [ifXF(p, Y), GENERALIZED MOTION OF NONCOMPACT HYPERSURFACES 229