scholarly journals Uniqueness for first-order Hamilton-Jacobi equations and Hopf formula

1987 ◽  
Vol 69 (3) ◽  
pp. 346-367 ◽  
Author(s):  
G Barles
Keyword(s):  
First Order ◽  
Hopf Formula ◽  
Jacobi Equations

scholarly journals Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula

2019 ◽  
Vol 80 (2) ◽  
pp. 1195-1239 ◽  
Author(s):  
Yat Tin Chow ◽  
Wuchen Li ◽  
Stanley Osher ◽  
Wotao Yin
Keyword(s):  
Hopf Formula ◽  
Jacobi Equations ◽  
Density Space

scholarly journals An Efficient Filtered Scheme for Some First Order Time-Dependent Hamilton--Jacobi Equations

2016 ◽  
Vol 38 (1) ◽  
pp. A171-A195 ◽  
Author(s):  
Olivier Bokanowski ◽  
Maurizio Falcone ◽  
Smita Sahu
Keyword(s):  
Time Dependent ◽  
First Order ◽  
Jacobi Equations

HOMOGENIZATION OF FIRST-ORDER EQUATIONS WITH u/∊-PERIODIC HAMILTONIAN: RATE OF CONVERGENCE AS ∊ → 0 AND NUMERICAL METHODS

2011 ◽  
Vol 21 (06) ◽  
pp. 1317-1353 ◽  
Author(s):  
YVES ACHDOU ◽  
STEFANIA PATRIZI
Keyword(s):  
Numerical Methods ◽  
Rate Of Convergence ◽  
Effective Hamiltonian ◽  
First Order ◽  
Jacobi Equations ◽  
Homogenization Problems

We consider homogenization problems for first-order Hamilton–Jacobi equations with u∊/∊ periodic dependence, recently introduced by Imbert and Monneau, and also studied by Barles: this unusual dependence leads to nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter ∊ tends to 0. We obtain the same rates as those obtained by Capuzzo Dolcetta and Ishii for the more usual homogenization problems without the dependence in u∊/∊. In the second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.

scholarly journals Regularity for Hamilton-Jacobi equations via approximation

1995 ◽  
Vol 51 (2) ◽  
pp. 195-213 ◽  
Author(s):  
Bum Il Hong
Keyword(s):  
Initial Value Problems ◽  
Stability Theorem ◽  
Polynomial Functions ◽  
Moving Grid ◽  
Approximation Properties ◽  
Initial Value ◽  
First Order ◽  
Spatial Dimensions ◽  
Regularity Results ◽  
Jacobi Equations

We prove new regularity results for solutions of first-order partial differential equations of Hamilton-Jacobi type posed as initial value problems on the real line. We show that certain spaces determined by quasinorms related to the solution's approximation properties in C(ℝ) by continuous, piecewise quadratic polynomial functions are invariant under the action of the differential equation. As a result, we show that solutions of Hamilton-Jacobi equations have enough regularity to be approximated well in C(ℝ) by moving-grid finite element methods. The preceding results depend on a new stability theorem for Hamilton-Jacobi equations in any number of spatial dimensions.

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