相关论文: Herglotz' variational principle and Lax-Oleinik ev…

We establish new results for path-dependent Hamilton-Jacobi equations with nonlinear monotone, and coercive operators on Hilbert space, which were initially studied in Bayraktar and Keller [J. Funct. Anal., 275 (8) (2018), pp. 2096-2161].…

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 study the following Cauchy problem for the linear wave equation with both time-dependent friction and time-dependent viscoelastic damping: \begin{equation} \label{EqAbstract}\tag{$\ast$} \begin{cases} u_{tt}- \Delta u + b(t)u_t -…

Based on a fixed point argument, we give a {\it dynamical representation} of the viscosity solution to Cauchy problem of certain weakly coupled systems of Hamilton-Jacobi equations with continuous initial datum. Using this formula, we…

We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler-Lagrange equations are derived. The method is to first prove the existence of…

The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the…

In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smooth, time-dependent, elliptic part having a variational…

We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and defined on bounded-time intervals. Under assumptions of regularity, convexity and…

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions…

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…

Assume $M$ 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)+H(x,u(x,t),\partial_xu(x,t))=0,\quad (x,t)\in…

In this article, we are interested in the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi Equations. In the superquadratic case, the third author has proved that these solutions can have…

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…

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 deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to…

We derive the existence and uniqueness of the generalized backward doubly stochastic differential equation with sub-differential of a lower semi-continuous convex function under a non Lipschitz condition. This study allows us give a…

Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of…

A highly nonlinear eigenvalue problem is studied in a Sobolev space with variable exponent. The Euler-Lagrange equation for the minimization of a Rayleigh quotient of two Luxemburg norms is derived. The asymptotic case with a "variable…

We collect examples of boundary-value problems of Dirichlet and Dirichlet-Neumann type which we found instructive when designing and analysing numerical methods for fully nonlinear elliptic partial differential equations. In particular, our…

The author presented a stochastic and variational approach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear…