Geometry of some simple nonlinear differential operators.

*(English)*Zbl 0648.47048The geometry of the maps \(A:f\to Df+f\) 2 on the space of functions of period 1 and B:f\(\to D\) \(2f+f\) 2/2 on the space of functions vanishing at \(x=0\) and \(x=1\) is studied. A is a fold, i.e. there are coordinates on the domain and on the range, so that A is expressed as \((x_ 1,x_ 2,x_ 3,...)\to (x\) \(2_ 1,x_ 2,x_ 3,...)\). B is not so simple: indeed it presents local folds in codimension 1, cusps in co-dimension 2, and a whole series of higher singularities, though, being an analytic map, the degree of the singularity is always finite. The number of preimages of a point is finite too. The singular set of B is comprised of sheets \(M_ n=\{f:\lambda_ n(f)=0\}\) in which \(\lambda_ 1(f)<\lambda_ 2(f)\), etc. is the spectrum of \(F=-D\) \(2+f\) subject to Dirichlet boundary conditions. The first sheet is a convex surface. The others lie one below the other and have each one more principal direction of negative curvature relative to the ambient space.

Reviewer: Fuhua Ling

##### MSC:

47H99 | Nonlinear operators and their properties |

47E05 | General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX) |

58C25 | Differentiable maps on manifolds |

58K99 | Theory of singularities and catastrophe theory |

##### Keywords:

nonlinear differential operator; local folds; cusps; higher singularities; degree of the singularity; Dirichlet boundary conditions; convex surface; principal direction of negative curvature relative##### References:

[1] | A. Ambrosetti - G. Prodi , On the inversion of some differentiable maps between Banach spaces with singularities , Ann. Mat. , 93 ( 1972 ), 231 - 246 . MR 320844 | Zbl 0288.35020 · Zbl 0288.35020 · doi:10.1007/BF02412022 |

[2] | M.S. Berger , Non-linearity and Functional Analysis , Academic Press , N.Y.C. , 1977 . |

[3] | M.S. Berger - P.T. Church , Complete, integrability and perturbation of a nonlinear Dirichlet problem (I) , Indiana Univ. Math. J. , 28 ( 1979 ) 935 - 952 . MR 551157 | Zbl 0425.35090 · Zbl 0425.35090 · doi:10.1512/iumj.1979.28.28066 |

[4] | R. Courant - D. Hilbert , Methods of Mathematical Physics , vol. 1 , Interscience Publishers , New York , 1953 . MR 65391 | Zbl 0051.28802 · Zbl 0051.28802 |

[5] | M. Golubitski - V. Guillemin , Stable Mappings and their Singularities , Springer-Verlag , N.Y.C. , 1973 . MR 341518 | Zbl 0294.58004 · Zbl 0294.58004 |

[6] | A.C. Lazer - P.J. McKenna , On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Roy. Soc . Edinburgh , to appear. Zbl 0533.35037 · Zbl 0533.35037 · doi:10.1017/S0308210500012993 |

[7] | H.P. McKean , Curvature of an \infty -dimensional manifold related to Hill’s equation , J. Differential Geom ., 17 ( 1982 ), 523 - 529 . Zbl 0481.58013 · Zbl 0481.58013 |

[8] | J.C. Scovel , Geometry of some nonliner differential operators , Ph.D. Thesis, New York University , 1983 . |

[9] | J.C. Scovel , Geometry of some simple nonlinear differential operators, part II, sub. to J.D.E . Mar 1985 [10] H. Whitney , On singularities of mappings , Ann. of Math . ( 1955 ), 374 - 416 . MR 73980 |

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