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Geometry of some simple nonlinear differential operators. (English) Zbl 0648.47048
The 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
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