Patrick Bernard

Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France
Universite Paris-Dauphine – CEREMADE (UMR 7534)


Bernard P.
The Siegel – Bruno Linearization Theorem
2023, vol. 28, nos. 4-5, pp.  756-762
The purpose of this paper is a pedagogical one. We provide a short and selfcontained account of Siegel’s theorem, as improved by Bruno, which states that a holomorphic map of the complex plane can be locally linearized near a fixed point under certain conditions on the multiplier. The main proof is adapted from Bruno’s work.
Keywords: linearization, normal forms
Citation: Bernard P.,  The Siegel – Bruno Linearization Theorem, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 756-762
Bernard P.
Semi-concave Singularities and the Hamilton–Jacobi Equation
2013, vol. 18, no. 6, pp.  674-685
We study the Cauchy problem for the Hamilton–Jacobi equation with a semiconcave initial condition.We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity solution).We also give conditions for an explicit semi-concave function to be a viscosity solution. These conditions generalize the entropy inequality characterizing piecewise smooth solutions of scalar conservation laws in dimension one.
Keywords: Hamilton–Jacobi equations, viscosity solutions, variational solutions, calculus of variations
Citation: Bernard P.,  Semi-concave Singularities and the Hamilton–Jacobi Equation, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 674-685

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