A survey of partial differential equations methods in weak KAM theory

Evans, Lawrence C.

doi:10.1002/cpa.20009

Cited by 38 publications

(19 citation statements)

References 41 publications

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“…General weak KAM theory (as recounted for instance in [5]) then implies that the measure ν has the form ν = δ { p=Dû} σ . This by the way already follows from our estimate (3.22).…”

Section: Now Multiply (34) By W ε and Integratementioning

confidence: 99%

“…This section is motivated by the classical observation (see for instance [5]) that if u = u(P, x) is a smooth solution of (1.1) and if we can solve the expressions…”

Section: Canonical Change Of Variablesmentioning

confidence: 99%

“…The PDE approach to weak KAM theory (see [5,6] and also Fathi [7][8][9][10]) focusses upon two fundamental equations, the Hamilton-Jacobi type equation…”

Section: Introductionmentioning

confidence: 99%

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Further PDE methods for weak KAM theory

Evans

2008

Calc. Var.

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We introduce and make estimates for several new approximations that in appropriate asymptotic limits yield the key PDE for weak KAM theory, namely a HamiltonJacobi type equation for a potential u and a coupled transport equation for a measure σ.We revisit as well a singular variational approximation introduced in [E1], and demonstrate "approximate integrability" of certain phase space dynamics related to the Hamiltonian flow. Other examples include a pair of strongly coupled PDE suggested by the Lions-Lasry theory [L-L1] of mean field games and a new and extremely singular elliptic equation suggested by sup-norm variational theory.

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“…General weak KAM theory (as recounted for instance in [5]) then implies that the measure ν has the form ν = δ { p=Dû} σ . This by the way already follows from our estimate (3.22).…”

Section: Now Multiply (34) By W ε and Integratementioning

confidence: 99%

“…This section is motivated by the classical observation (see for instance [5]) that if u = u(P, x) is a smooth solution of (1.1) and if we can solve the expressions…”

Section: Canonical Change Of Variablesmentioning

confidence: 99%

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Further PDE methods for weak KAM theory

Evans

2008

Calc. Var.

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“…Introducing the adjoint PDE (1.3) is inspired by the first author's recent paper [6] on nonconvex Hamilton-Jacobi equations and also by various techniques for the PDE approach to weak KAM theory (see [7]). Savin [12] proved for n = 2 dimensions that the viscosity solution u of (1.1) is in fact C 1 .…”

Section: Basic Equationsmentioning

confidence: 99%

Adjoint Methods for the Infinity Laplacian Partial Differential Equation

Evans

Smart

2011

Arch Rational Mech Anal

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To study fine properties of certain smooth approximations u ε to a viscosity solution u of the infinity Laplacian partial differential equation (PDE), we introduce Green's function σ ε for the linearization. We can then integrate by parts with respect to σ ε and derive various useful integral estimates. We are, in particular, able to use these estimates (i) to prove the everywhere differentiability of u and (ii) to rigorously justify interpreting the infinity Laplacian equation as a parabolic PDE.

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“…The ground energy level E, common to (2.2, 2.4), admits several equivalent definitions. Evans and Gomes ( [11] [13] [14]) defined E as the effective hamiltonian value…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for optimal mass transportation

Wolansky

2011

Calc. Var.

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The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations obtained by different Lagrangian actions on Riemannian manifolds. As a special case, for any pair of non-negative measures λ + , λ − of equal masswhere W p , p ≥ 1 is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.

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A survey of partial differential equations methods in weak KAM theory

Cited by 38 publications

References 41 publications

Further PDE methods for weak KAM theory

Further PDE methods for weak KAM theory

Adjoint Methods for the Infinity Laplacian Partial Differential Equation

Limit theorems for optimal mass transportation

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