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Selection of critical solutions in non-monotone vanishing discount problems
Panrui Ni, Jun Yan, Maxime Zavidovique
We study a generalized vanishing discount problem for Hamilton--Jacobi equations without assuming monotonicity either pointwise or in the averaged sense with respect to all Mather…
Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets
Panrui Ni
Let be a Tonelli Hamiltonian on a closed connected manifold and let solve \[ λu_λ+H(x,Du_λ)=c(H)\qquad\text{in }M. \] We study the convergence rate of …
A PDE formulation of Lyapunov stability for contact-type Hamilton-Jacobi equations
Panrui Ni, Jun Yan
We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume Tonelli Hamiltonians…
Static class-guided selection of elementary solutions in non-monotone vanishing discount problems
Panrui Ni, Jun Yan, Maxime Zavidovique
We study a generalized vanishing discount problem for Hamilton--Jacobi equations, removing the standard monotonicity assumption, either in a global sense or when integrated against…
Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations
Andrea Davini, Panrui Ni, Jun Yan +1
We study the asymptotic behavior of solutions of an equation of the form \begin{equation}\label{abs}\tag{*} G\big(x, D_x u,λu(x)\big) = c_0\qquad\hbox{in } \end{equation} on a c…
A nonlinear semigroup approach to Hamilton-Jacobi equations--revisited
Panrui Ni, Lin Wang
We consider the Hamilton-Jacobi equation \[{H}(x,Du)+λ(x)u=c,\quad x\in M, \] where is a connected, closed and smooth Riemannian manifold. The functions and a…
Weakly coupled Hamilton-Jacobi systems without monotonicity condition: A first step
Panrui Ni
In this paper, we mainly focus on the existence of the viscosity solutions of \begin{equation*} \left\{ \begin{aligned} &H_1(x,Du_1(x),u_1(x),u_2(x))=0,\\ &H_2(x,Du_2(x),u_2(x),u_1…
A representation formula of the viscosity solution of the contact Hamilton-Jacobi equation and its applications
Panrui Ni, Lin Wang, Jun Yan
Assume is a closed, connected and smooth Riemannian manifold. We consider the evolutionary Hamilton-Jacobi equation \begin{equation*} \left\{ \begin{aligned} &\partial_t u(x,t)…