53 research outputs found
Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian
We study the long-time behavior of the unique viscosity solution of the
viscous Hamilton-Jacobi Equation with inhomogeneous Dirichlet boundary conditions,
where is a bounded domain of . We mainly focus on the
superquadratic case () and consider the Dirichlet conditions in the
generalized viscosity sense. Under rather natural assumptions on the
initial and boundary data, we connect the problem studied to its associated
stationary generalized Dirichlet problem on one hand and to a stationary
problem with a state constraint boundary condition on the other hand
A new method for large time behavior of degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians
We investigate large-time asymptotics for viscous Hamilton--Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems
Viscosity solutions of general viscous Hamilton-Jacobi equations
We present comparison principles, Lipschitz estimates and study state
constraints problems for degenerate, second-order Hamilton-Jacobi equations.Comment: 35 pages, minor revisio
A short proof of the --regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications
Recently I. Capuzzo Dolcetta, F. Leoni and A. Porretta obtain a very
surprising regularity result for fully nonlinear, superquadratic, elliptic
equations by showing that viscosity subsolutions of such equations are locally
H\"older continuous, and even globally if the boundary of the domain is regular
enough. The aim of this paper is to provide a simplified proof of their
results, together with an interpretation of the regularity phenomena, some
extensions and various applications
ON UNBOUNDED SOLUTIONS OF ERGODIC PROBLEMS IN R^m FOR VISCOUS HAMILTON-JACOBI EQUATIONS
International audienceIn this article we study ergodic problems in the whole space R m for viscous Hamilton-Jacobi Equations in the case of locally Lips-chitz continuous and coercive right-hand sides. We prove in particular the existence of a critical value λ * for which (i) the ergodic problem has solutions for all λ ≤ λ * , (ii) bounded from below solutions exist and are associated to λ * , (iii) such solutions are unique (up to an additive constant). We obtain these properties without additional assumptions in the superquadratic case, while, in the subquadratic one, we assume the right-hand side to behave like a power. These results are slight generalizations of analogous results by N. Ichihara but they are proved in the present paper by partial differential equations methods, contrarily to N. Ichihara who is using a combination of pde technics with probabilistic arguments
Lipschitz regularity results for nonlinear strictly elliptic equations and applications
Most of lipschitz regularity results for nonlinear strictly elliptic
equations are obtained for a suitable growth power of the nonlinearity with
respect to the gradient variable (subquadratic for instance). For equations
with superquadratic growth power in gradient, one usually uses weak
Bernstein-type arguments which require regularity and/or convex-type
assumptions on the gradient nonlinearity. In this article, we obtain new
Lipschitz regularity results for a large class of nonlinear strictly elliptic
equations with possibly arbitrary growth power of the Hamiltonian with respect
to the gradient variable using some ideas coming from Ishii-Lions' method. We
use these bounds to solve an ergodic problem and to study the regularity and
the large time behavior of the solution of the evolution equation
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