#Hyperplan equation full#

A Nash-Kuiper theorem for C 1, 1/5– δ immersions of surfaces in 3 dimensions. Berlin-Heidelberg: Springer, 2012, 83–116ĭe Lellis C, Inauen D, Székelyhidi Jr L. In: Nonlinear Partial Differential Equations. h-principle and rigidity for C 1, α isometric embeddings. Comm Math Phys, 1994, 165: 207–209Ĭonti S, De Lellis C, Székelyhidi Jr L. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm Partial Differential Equations, 2019, in pressĬonstantin P E W, Titi E S. Comm Pure Appl Math, 2019, in pressĬao W, Székelyhidi Jr L. Nonuniqueness of weak solutions to the SQG equation.

Comm Pure Appl Math, 2019, in pressīuckmaster T, Shkoller S, Vicol V. Onsager’s conjecture for admissible weak solutions. Dokl Akad Nauk SSSR (NS), 1965, 163: 11–13īuckmaster T, De Lellis C, Székelyhidi Jr L, et al. C 1, α-isometric immersions of Riemannian spaces. Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications. Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups.Ambrosio L. Spivak, M.: A Comprehensive Introduction to Differential Geometry. Klingenberg, W.: A Course in Differential Geometry. Harris, J.: Algebraic Geometry: A First Course. Goldschmidt, H.: Integrability criteria for systems of non-linear partial differential equations. Prentice-Hall, Englewood Cliffs (1976)įisher, R.J., Laquer, H.T.: Generalized immersions and the rank of the second fundamental form. Academic Press, New York (1969, 1970, 1972, 1977, 1974)ĭo Carmo, M.P.: Differential Geometry of Curves and Surfaces. Thesis, Idaho State University (1995)ĭieudonné, J.: Treatise on Analysis, vols. Birkhäuser, Boston (1985)īrunette, J.: The Clairaut equation: a study in the geometry of partial differential equations. Kluwer Academic, Dordrecht (1990)Īrnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Springer, New York (1991)Īrnold, V.I.: Singularities of Caustics and Wave Fronts. Encyclopaedia of Mathematical Sciences, vol. Springer, New York (1988)Īlekseevskij, D.V., Vinogradov, A.M., Lychagin, V.V.: Geometry I: Basic Ideas and Concepts of Differential Geometry. With one mild assumption, namely that the associated family of linear hyperplanes is immersed, it is proven that a family of affine hyperplanes always has an envelope, and that envelope is essentially unique.Ībraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. The beauty of this approach becomes apparent in the “Envelope Theorem”.

#Hyperplan equation full#

These have been described elsewhere but, briefly, every classical immersion defines a generalized immersion in a canonical way so that generalized immersions can be understood as ordinary immersions “with singularities.” Next, the concept of an envelope is given a modern definition, namely, an envelope is a generalized immersion solving the family that has a universal mapping property relative to all other full rank solutions. In the process, the classical results are generalized and unified.Ī key step in this process is the use of “generalized immersions”. This paper brings a modern perspective to the classical problem of envelopes of families of affine hyperplanes. The subject of envelopes has been part of differential geometry from the beginning.