On viscosity solutions of irregular Hamilton-Jacobi equations

It is proved that the initial-value problem for admits a unique continuous viscosity solution under certain conditions which do not exclude that H(x, p) is discontinuous in x. Particular attention is devoted to the linear transport equation , where a may be discontinuous. Received: 21 October 2002     

Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing lagrangians,eikonal equations,and shape-from-shading

We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique bounded-from-below viscosity solution of the Hamilton-Jacobi equation that is null on the target. The result applies to problems with the property that all trajectories satisfying a certain integral condition must stay in a bounded set. We allow problems for which the Lagrangian is not uniformly bounded below by positive constants, in which the hypotheses of the known uniqueness results for Hamilton-Jacobi equations are not satisfied. We apply our theorems to eikonal equations from geometric optics, shape-from-shading equations from image processing, and variants of the Fuller Problem.     

Decay estimates for quasi-linear evolution equations

We consider global strong solutions of the quasi-linear evolution equations (1.1) and (1.2) below, corresponding to sufficiently small initial data, and prove some stability estimates, as t→+∞, that generalize the corresponding estimates in the linear case.     

A counterexample to uniqueness of generalized characteristics in Hamilton-Jacobi theory

The notion of generalized characteristics plays a pivotal role in the study of propagation of singularities for Hamilton-Jacobi equations. This note gives an example of nonuniqueness of forward generalized characteristics emanating from a given point.     

Hamilton-Jacobi equations having only action functions as solutions

Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution of the Hamilton-Jacobi equation

On W-solvability for special vectorial Hamilton-Jacobi systems

We study the solvability of special vectorial Hamilton-Jacobi systems of the form F(Du(x))=0 in a Sobolev space. In this paper we establish the general existence theorems for certain Dirichlet problems using suitable approximation schemes called W^1,p-reduction principles that generalize the similar reduction principle for Lipschitz solutions. Our approach, to a large extent, unifies the existing methods for the existence results of the special Hamilton-Jacobi systems under study. The method relies on a new Baire's category argument concerning the residual continuity of a Baire-one function. Some sufficient conditions for W^1,p-reduction are also given along with certain generalization of some known results and a specific application to the boundary value problem for special weakly quasiregular mappings.     

The relaxation-time limit in the compressible Euler–Maxwell equations

The aim of this paper is to study multidimensional Euler–Maxwell equations for plasmas with short momentum relaxation time. The convergence for the smooth solutions to the compressible Euler–Maxwell equations toward the solutions to the smooth solutions to the drift–diffusion equations is proved by means of the Maxwell iteration, as the relaxation time tends to zero. Meanwhile, the formal derivation of the latter from the former is justified.     

Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizersu_A(x)=U_A(|x|), having U_A(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and B_R is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?^∗∗:R×R→[0,∞] is just convex lsc and and ?^∗∗(s,⋅) is even; while ρ₁(⋅) and ρ₂(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?^∗∗(s,v)=∞ is freely allowed.     

Well posed cauchy problems for complex nonlinear equations must be semilinear

The first author was supported in part by CNPq     

Some homogenization results for non-coercive Hamilton–Jacobi equations

Recently, C. Imbert and R. Monneau study the homogenization of coercive Hamilton–Jacobi Equations with a u/ε-dependence: this unusual dependence leads to a non-standard cell problem and, in order to solve it, they introduce new ideas to obtain the estimates on the oscillations of the solutions. In this article, we use their ideas to provide new homogenization results for “standard” Hamilton–Jacobi Equations (i.e. without a u/ε-dependence) but in the case of non-coercive Hamiltonians. As a by-product, we obtain a simpler and more natural proof of the results of C. Imbert and R. Monneau, but under slightly more restrictive assumptions on the Hamiltonians.     

Strong trajectory attractors for dissipative Euler equations

The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R>0 are considered and the existence of a strong trajectory attractor in the space is established under the assumption that the external forces have bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method.     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.     

Harnack inequalities for Yamabe type equations

We give some a priori estimates of type sup×inf on Riemannian manifolds for Yamabe and prescribed curvature type equations. An application of those results is the uniqueness result for Δu+?u=uN−1 with ? small enough.     

On the continuation of solutions for elliptic equations in two variables

We consider nonlinear elliptic differential equations of second order in two variables

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. Supposing analyticity of F, we prove analyticity of the real solution z=z(x,y) in the open set Ω. Furthermore, we show that z may be continued as a real analytic solution for F=0 across the real analytic boundary arc Γ∂Ω, if z satisfies one of the boundary conditions z= or z_n=ψ(x,y,z,z_t) on Γ with real analytic functions and ψ, respectively (z_n denotes the derivative of z w.r.t. the outer normal n on Γ and z_t its derivative w.r.t. the tangent). The proof is based on ideas of H. Lewy combined with a uniformization method. Studying quasilinear equations, we get somewhat better results concerning the initial regularity of the given solution and a little more insight.     

Approximation of stationary solutions of Gaussian driven stochastic differential equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.     

Lipschitz continuity and local semiconcavity for exit time problems with state constraints

In the classical time optimal control problem, it is well known that the so-called Petrov condition is necessary and sufficient for the minimum time function to be locally Lipschitz continuous. In this paper, the same regularity result is obtained in the presence of nonsmooth state constraints. Moreover, for a special class of control systems we obtain a local semiconcavity result for the constrained minimum time function.     

Diffusive approximations for a class of nonlinear parabolic equations

We study a class of discrete velocity type approximations to nonlinear parabolic equations with source. After proving existence results and estimates on the solution to the relaxation system, we pass into the limit towards a weak solution, which is the unique entropy solution if the coefficients of the parabolic equation are constant.     

Hamilton-Jacobi methods for Vakonomic Mechanics

We extend the theory of Aubry-Mather measures to Hamiltonian systems that arise in vakonomic mechanics and sub-Riemannian geometry. We use these measures to study the asymptotic behavior of (vakonomic) action-minimizing curves, and prove a bootstrapping result to study the partial regularity of solutions of convex, but not strictly convex, Hamilton-Jacobi equations.      

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

This paper is devoted to studying a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under the suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

Blow-up behavior outside the origin for a semilinear wave equation in the radial case

We consider the semilinear wave equation in the radial case with conformal subcritical power nonlinearity. If we consider a blow-up point different from the origin, then we exhibit a new Lyapunov functional which is a perturbation of the one-dimensional case and extend all our previous results known in the one-dimensional case. In particular, we show that the blow-up set near non-zero non-characteristic points is of class C¹, and that the set of characteristic points is made of concentric spheres in finite number in for any R>1.