English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Given a continuous Hamiltonian $H : (x,p,u) \mapsto H(x,p,u)$ defined on $T^*M \times \mathbb R $, where $M$ is a closed connected manifold, we studyviscosity solutions, $u_\lambda : M\to \mathbb R$, of discounted equations: $H(x, d_x u_\lambda, \lambda u_\lambda(x))=c$ in $M$, where $\lambda >0$ iscalled a discount factor and $c$ is the critical value of $H(\cdot, \cdot ,0)$.  When $H$ is convex and superlinear in $p$ and non--decreasing in $u$, underan additional non--degeneracy condition, we obtain existence and uniqueness(with comparison principles) results of solutions and we prove that the familyof solutions $(u_\lambda)_{\lambda >0}$ converges to a specific solution $u_0$of $ H(x, d_x u_0, 0)=c$ in $M$. Our degeneracy condition requires $H$ to beincreasing (in $u$) on localized regions linked to the support of Mathermeasures, whereas usual similar results are obtained for Hamiltonians that areeverywhere increasing in $u$.

Convergence of the solutions of the nonlinear discounted Hamilton-Jacobi equation: The central role of Mather measures