53 research outputs found

    Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian

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    We study the long-time behavior of the unique viscosity solution uu of the viscous Hamilton-Jacobi Equation ut−Δu+∣Du∣m=fin Ω×(0,+∞)u_t-\Delta u + |Du|^m = f\hbox{in }\Omega\times (0,+\infty) with inhomogeneous Dirichlet boundary conditions, where Ω\Omega is a bounded domain of RN\mathbb{R}^N. We mainly focus on the superquadratic case (m>2m>2) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on f,f, 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

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    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

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    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 C0,αC^{0,\alpha}--regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications

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    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

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    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

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    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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