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On the Euler-Lagrange equation for a variational problem
1. | S.I.S.S.A. (I.S.A.S.), Via Beirut 2/4, 34013 Trieste |
inf$\bar u + W^{1,\infty}_0(\Omega)$$ \int_{\Omega} (\I_D(\nabla u) + g(u)) dx,\ \ \ $ (0.1)
with $D$ convex closed subset of $\R^n$ with non empty interior. We next show that if D* is strictly convex, then the Euler-Lagrange equation can be reduced to an ODE along characteristics, and we deduce that the solution to Euler-Lagrange is different from $0$ a.e. and satisfies a uniqueness property. Using these results, we prove a conjecture on the existence of variations on vector fields [6].
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Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure and Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51 |
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Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial and Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161 |
[18] |
Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369 |
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Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3767-3793. doi: 10.3934/cpaa.2021130 |
[20] |
Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223 |
2021 Impact Factor: 1.588
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