Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity

Andrea C. G. Mennucci

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Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity (errata)

Andrea C. G. Mennucci (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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This errata corrects one error in the 2004 version of this paper [Mennucci, (2004) 426–451].

Regularity of $C^{1}$ solutions of the Hamilton-Jacobi equation

Albert Fathi (2003)

Annales de la Faculté des sciences de Toulouse : Mathématiques

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Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations

Alessandra Cutrì, Francesca Da Lio (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper we prove a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form $u_{t} + H (x, D u) = 0$ in $I R^{n} \times (0, T)$ where the Hamiltonian may be noncoercive in the gradient As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.

Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations

Guy Barles, Alessio Porretta (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider a class of stationary viscous Hamilton-Jacobi equations aswhere $λ \geq 0$ , $A (x)$ is a bounded and uniformly elliptic matrix and $H (x, ξ)$ is convex in $ξ$ and grows at most like ${| ξ |}^{q} + f (x)$ , with $1 < q < 2$ and $f \in L^{N / q^{'}} (Ω)$ . Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate, ${(1 + | u |)}^{\bar{q} - 1} u \in H_{0}^{1} (Ω)$ , for a certain (optimal) exponent $\bar{q}$ . This completes the...

Sharp Domains of Determinacy and Hamilton-Jacobi Equations

Jean-Luc Joly, Guy Métivier, Jeffrey Rauch (2004-2005)

Séminaire Équations aux dérivées partielles

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If $L (t, x, \partial_{t}, \partial_{x})$ is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets $Ω_{0} \subset {t = 0}$ . The frozen constant coefficient operators $L (\underset{̲}{t}, \underset{̲}{x}, \partial_{t}, \partial_{x})$ determine local convex propagation cones, $Γ^{+} (\underset{̲}{t}, \underset{̲}{x})$ . Influence curves are curves whose tangent always lies in these cones. We prove that the set of points $Ω$ which cannot be reached by influence curves beginning in the exterior of $Ω_{0}$ is...

A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable

Pierre Cardaliaguet (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Hölder continuous with Hölder exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse Hölder inequality.

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