On uniqueness sets of additive eigenvalue problems and applications

Mitake, Hiroyoshi; Tran, Hung V.

doi:10.1090/proc/14152

Cited by 11 publications

(17 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The corollary above claims that M is an uniqueness set of (EP), that is, if v, w are two weak KAM solutions of (EP) and v = w in M, then v = w in R N . see [12], and [15], [14] for related results.…”

mentioning

confidence: 99%

The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space

Ishii

Siconolfi

2020

Communications in Partial Differential Equations

View full text Add to dashboard Cite

We study the asymptotic behavior of the solutions to a family of discounted Hamilton-Jacobi equations, posed in R N , when the discount factor goes to zero. The ambient space being noncompact, we introduce an assumption implying that the Aubry set is compact and there is no degeneracy at infinity. Our approach is to deal not with a single Hamiltonian and Lagrangian but with the whole space of generalized Lagrangians, and then to define via duality minimizing measures associated to both the corresponding ergodic and discounted equations. The asymptotic result follows from convergence properties of these measures with respect to the narrow topology. We use as duality tool a separation theorem in locally convex Hausdorff spaces, we use the strict topology in the space of the bounded generalized Lagrangians as well.

show abstract

mentioning

confidence: 99%

The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space

Ishii

Siconolfi

2020

Communications in Partial Differential Equations

View full text Add to dashboard Cite

show abstract

“…This is done in Section 5. We refer the readers to [5,20,1,15,16,11] and the references therein for the developments of the nonlinear adjoint method.…”

Section: 2mentioning

confidence: 99%

Generalized ergodic problems: Existence and uniqueness structures of solutions

Jing

Mitake

Tran

2020

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat n-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.2010 Mathematics Subject Classification. 35B10 35B27 35B40 35D40 35F21 49L25 .

show abstract

“…Under the above estimates, we prove the main result. The following argument is introduced in [20] to study comparison principle with respect to Mather measures for a single equation.…”

Section: Proof Of Main Theoremmentioning

confidence: 99%

“…In the single case, one approach, studied in [3] and [4], is to find a uniqueness set; that is, if two solutions coincide on this set, they are totally equal on the domain. Recently, in [20], a new and simple way to find uniqueness sets was studied. In particular, the authors proved new comparison principle with respect to Mather measures in the single case.…”

Section: Introductionmentioning

confidence: 99%

“…

Here, we address a uniqueness structure of viscosity solutions for ergodic problems of weakly coupled Hamilton-Jacobi systems. In particular, we study comparison principle with respect to generalized Mather measures as a generalization of [20], which addressed the case of a single equation. To get the main result, it is important to construct Mather measures effectively.

…”

mentioning

confidence: 99%

See 1 more Smart Citation