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Hypercontractivity Of HamiltonJacobi Equations
 J. Math. Pures Appl
, 2000
"... .  Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of HamiltonJacobi equations. By the infimumconvolution description of the Hamilt ..."
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Cited by 124 (19 self)
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.  Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of HamiltonJacobi equations. By the infimumconvolution description
Minimal entropy conditions for Burgers equation
 Quarterly Appl. Math. (2004
"... Abstract. We consider strictly convex, 1d scalar conservation laws. We show that a single strictly convex entropy is sufficient to characterize a Kruzhkov solution. The proof uses the concept of viscosity solution for the related HamiltonJacobi equation. 1. ..."
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Cited by 16 (1 self)
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Abstract. We consider strictly convex, 1d scalar conservation laws. We show that a single strictly convex entropy is sufficient to characterize a Kruzhkov solution. The proof uses the concept of viscosity solution for the related HamiltonJacobi equation. 1.
Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations
, 1988
"... We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvaturedependent speed. The speed may be an arbitrary function of curvature, and the front also can be passively advected by an underlying flow. These algorithms approximate the equations of motion, w ..."
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Cited by 1183 (60 self)
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, which resemble HamiltonJacobi equations with parabolic righthand sides, by using techniques from hyperbolic conservation laws. Nonoscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps
The HamiltonJacobi Difference Equation
"... . We study a system of difference equations which, like Hamilton's equations, preserves the standard symplectic structure on R 2m . In particular, we construct a differentialdifference equation which we call the HamiltonJacobi difference equation, the analog of the HamiltonJacobi equation ..."
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Cited by 6 (0 self)
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of the HamiltonJacobi difference equation is equivalent to the general solution of the canonical system of difference equations to which it is related. We consider a sequence (X (0) ; Y (0) ); (X (1) ; Y (1) ); : : : ; (X (n) ; Y (n) ); : : : in R m \Theta R m , generated from the initial point
Solutions of the HamiltonJacobi Equation ∗
, 1999
"... We discuss a new relation between the low lying Schroedinger wave function of a particle in a onedimentional potential V and the solution of the corresponding HamiltonJacobi equation with −V as its potential. The function V is ≥ 0, and can have several minina (V = 0). We assume the problem to be c ..."
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We discuss a new relation between the low lying Schroedinger wave function of a particle in a onedimentional potential V and the solution of the corresponding HamiltonJacobi equation with −V as its potential. The function V is ≥ 0, and can have several minina (V = 0). We assume the problem
HamiltonJacobi equations with obstacles
 ARCH. RATIONAL MECH. ANAL
"... We consider a problem in the theory of optimal control proposed for the first time by Bressan. We characterize the associated minimum time function using tools from geometric measure theory and we obtain as a corollary an existence theorem for a related variational problem. ..."
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Cited by 5 (0 self)
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We consider a problem in the theory of optimal control proposed for the first time by Bressan. We characterize the associated minimum time function using tools from geometric measure theory and we obtain as a corollary an existence theorem for a related variational problem.
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 270 (26 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di
HOMOGENIZATION OF METRIC HAMILTONJACOBI EQUATIONS
"... Abstract. In this work we provide a novel approach to homogenization for a class of static HamiltonJacobi (HJ) equations, which we call metric HJ equations. We relate the solutions of the HJ equations to the distance function in a corresponding Riemannian or Finslerian metric. The metric approach a ..."
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Cited by 12 (1 self)
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Abstract. In this work we provide a novel approach to homogenization for a class of static HamiltonJacobi (HJ) equations, which we call metric HJ equations. We relate the solutions of the HJ equations to the distance function in a corresponding Riemannian or Finslerian metric. The metric approach
Let Ω be a bounded convex open subset of RN, N 1, and let J be the integral functional
, 2002
"... Abstract. We consider minimization problems of the form min u∈ϕ+W1,10 (Ω) Z Ω [f(Du(x)) − u(x)] dx where Ω RN is a bounded convex open set, and the Borel function f: RN! [0, +1] is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of Ω and the zero level s ..."
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set of f, we prove that the viscosity solution of a related Hamilton{Jacobi equation provides a
Fast sweeping methods for static hamiltonjacobi equations
 Society for Industrial and Applied Mathematics
, 2005
"... Abstract. We propose a new sweeping algorithm which discretizes the Legendre transform of the numerical Hamiltonian using an explicit formula. This formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically. The minimiz ..."
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Cited by 55 (5 self)
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. The minimization that is related to the Legendre transform in our sweeping scheme can either be solved analytically or numerically. We illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions.
Results 1  10
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