Extra Form
강연자 Marshall Slemrod
소속 Univ. of Wisconsin
date 2010-10-14
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as isometric immersions into R3. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptichyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in R3. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in R3. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a C1,1 isometric immersion of the two-dimensional manifold in R3 satisfying our prescribed initial conditions. To achieve this, we introduce a vanishing viscosity method depending on the features of initial value problems for isometric immersions and present a technique to make the apriori estimates including the L∞ control and H?1?compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of an isometric immersion of the manifold into R3 satisfying our initial conditions.
첨부 '1'
List of Articles
카테고리 제목 소속 강연자
수학강연회 Creation of concepts for prediction models and quantitative trading file Haafor 이승환
수학강연회 Counting number fields and its applications file UNIST 조재현
수학강연회 Counting circles in Apollonian circle packings and beyond file Brown Univ. 오희
수학강연회 Convex and non-convex optimization methods in image processing file Hong Kong Baptist University Michael Ng
수학강연회 Contact topology of singularities and symplectic fillings file 순천대학교 권명기
특별강연 Contact topology and the three-body problem file 서울대학교 Otto van Koert
수학강연회 Contact instantons and entanglement of Legendrian links file IBS-CGP /POSTECH 오용근
수학강연회 Contact Homology and Constructions of Contact Manifolds file 서울대 Otto van Koert
수학강연회 Conservation laws and differential geometry file Univ. of Wisconsin Marshall Slemrod
수학강연회 Connes's Embedding Conjecture and its equivalent file RIMS Narutaka Ozawa
수학강연회 Connectedness of a zero-level set as a geometric estimate for parabolic PDEs file KAIST 김용정
수학강연회 Congruences between modular forms file 서울대 유화종
수학강연회 Conformal field theory in mathematics file 고등과학원 강남규
수학강연회 Conformal field theory and noncommutative geometry file 동경대학교 Kawahigashi
수학강연회 Compressible viscous Navier-Stokes flows: Corner singularity, regularity file POSTECH 권재룡
특별강연 Combinatorics and Hodge theory file 미국 프린스턴대 교수, 한국 고등과학원 석학교수 허준이
수학강연회 Combinatorial Laplacians on Acyclic Complexes file 서울대학교 국웅
수학강연회 Codimension Three Conjecture file 교토대학교/서울대학교 Masaki Kashiwara
수학강연회 Cloaking via Change of Variables file KAIST 임미경
Classification of simple amenable operator algebras file Lakehead University Grazia Viola
Board Pagination Prev 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Next
/ 15