Related papers: Global generalized characteristics for the Dirichl…

We investigate the properties of the set of singularities of semiconcave solutions of Hamilton-Jacobi equations of the form \begin{equation*} u_t(t,x)+H(\nabla u(t,x))=0, \qquad\text{a.e. }(t,x)\in…

The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation \begin{equation}\label{eq:discount 1}\tag{HJ$_\lambda$} \lambda v(x)+H( x, Dv(x) )=0 , \quad…

In this article, we are interested in the Dirichlet problem for parabolic viscous Hamilton-Jacobi Equations. It is well-known that the gradient of the solution may blow up in finite time on the boundary of the domain, preventing a classical…

The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the…

We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-\Delta u + |Du|^m = f\hbox{in }\Omega\times (0,+\infty)$ with inhomogeneous Dirichlet boundary conditions, where $\Omega$ is a…

We focus on the global semiconcavity of solutions to first-order Hamilton--Jacobi equations with state constraints, especially for the Hamiltonian $H(x, \beta):=|\beta|^p-f(x)$ with $p \in (1, 2]$. We first show that the solution is locally…

Global classical solutions to the viscous Hamilton-Jacobi equation with homogenious Dirichlet boundary conditions are shown to converge to zero at the same speed as the linear heat semigroup when p > 1. For p = 1, an exponential decay to…

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish…

We show that if a Hamilton-Jacobi equation admits a differentiable solution whose gradient is Lipschitz, then this solution is the unique semi-concave weak solution. Our result does not rely on any convexity (nor concavity) assumptions on…

In this paper we deal with the well-posedness of Dirichlet problems associated to nonlocal Hamilton-Jacobi parabolic equations in a bounded, smooth domain $\Omega$, in the case when the classical boundary condition may be lost. We address…

This paper studies the structure of the singular set (points of nondifferentiability) of viscosity solutions to Hamilton-Jacobi equations associated with general mechanical systems on the n-torus. First, using the level set method, we…

We prove that singularities propagate globally for viscosity solutions of Hamilton-Jacobi equations related to magnetic mechanical systems on closed Riemannian manifolds. Our main result shows that for any weak KAM solution $u$, the…

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical…

For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of…

We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define…

We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity…

We examine Hamilton-Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii-Jensen inequality, we prove that viscosity solutions are locally…

We consider a path-dependent Hamilton--Jacobi equation with coinvariant derivatives over the space of continuous functions. We prove two uniqueness results for viscosity (generalized) solutions defined in terms of coinvariantly smooth test…

We consider the simplest example of a time-dependent first order Hamilton-Jacobi equation, in one space dimension and with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. We show that if the…

We perform a complete analysis of the limiting behaviour of a class of quasilinear problems with Dirichlet boundary data g. We show that the Lipschitz constant of g plays a role in controlling the Gamma-convergence of the natural energies.…