Displaying similar documents to “Graph selectors and viscosity solutions on Lagrangian manifolds”

Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

Gianni Dal Maso, Hélène Frankowska (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We investigate the value function of the Bolza problem of the Calculus of Variations
 V ( t , x ) = inf ∫ 0 t L ( y ( s ) , y ' ( s ) ) d s + ϕ ( y ( t ) ) : y ∈ W 1 , 1 ( 0 , t ; ℝ n ) , y ( 0 ) = x , with a lower semicontinuous Lagrangian and a final cost ϕ , and show that it is locally Lipschitz for whenever is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize...

Geometric constraints on the domain for a class of minimum problems

Graziano Crasta, Annalisa Malusa (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider minimization problems of the form min u ∈ ϕ + W 0 1 , 1 ( Ω ) ∫ Ω [ f ( D u ( x ) ) - u ( x ) ] d x where Ω ⊆ ℝ N is a bounded convex open set, and the Borel function f : ℝ N → [ 0 , + ∞ ] is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of and the zero level set of , we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

Approximation Algorithms for the Traveling Salesman Problem with Range Condition

D. Arun Kumar, C. Pandu Rangan (2010)

RAIRO - Theoretical Informatics and Applications

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We prove that the Christofides algorithm gives a 4 3 approximation ratio for the special case of traveling salesman problem (TSP) in which the maximum weight in the given graph is at most twice the minimum weight for the graphs. A graph is if the number of odd degree vertices in any minimum spanning tree of the given graph is less than 1 4 times the number of vertices in the graph. We prove that the Christofides algorithm is more efficient (in terms of runtime) than the previous...