1,721,746 research outputs found
The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space
Full text linkWe study the asymptotic behavior of the solutions to a family of discounted Hamilton-Jacobi equations, posed in RN, when the discount factor goes to zero. The ambient space being noncompact, we introduce an assumption implying that the Aubry set is compact and there is no degeneracy at infinity. Our approach is to deal not with a single Hamiltonian and Lagrangian but with the whole space of generalized Lagrangians, and then to define via duality minimizing measures associated with both the corresponding ergodic and discounted equations. The asymptotic result follows from the convergence properties of these measures concerning the narrow topology. We use as duality tool a separation theorem in locally convex Hausdorff spaces, and we use the strict topology in the space of the bounded generalized Lagrangians as well
A class of stochastic optimal control problems with state constraint.
No full textWe consider optimal control problems with state constraint, where states X-t given as solutions of controlled stochastic differential equations are required to satisfy the constraint described either by the condition that Xt is an element of G ($) over bar for all t > 0 or by the condition that X-t is an element of G for all t > 0, with G being a given open subset of RN. Under suitable assumptions, we establish the unique existence of a continuous viscosity solution of the state constraint problem for the associated Hamilton-Jacobi-Bellman equation, which is fully nonlinear degenerate second order elliptic equation, Lipschitz and Holder regularity results for the viscosity solution of the state constraint problem, and that the value functions V associated with the constraint G, V, of the relaxed problem associated with the constraint G, and Vg associated with the constraint G, satisfy in the viscosity sense the state constraint problem and hence are identified with its unique viscosity solution
Relaxation in the Cauchy problem for Hamilton-Jacobi equations
No full textIn this note we study a little further the relaxation of Hamilton- Jacobi equations
Towards a reversed Faber–Krahn inequality for the truncated Laplacian
Full text linkWe consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator P1+ mapping a function u to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian
Flexible ureteroscopy and lasertripsy (FURSL) for paediatric renal calculi: results from a systematic review
No full textObjective: to understand the role, safety and efficacy of flexible ureterorenoscopy and lasertripsy (FURSL) for paediatric renal stones.Material and methods: a systematic review was conducted using studies identified by a literature search between January 1990 and March 2014. All English language articles reporting on a minimum of five patients ≤18-years old, treated with flexible ureteroscopy and lasertripsy for stone disease were included.Results: a total of six studies (282 patients) were reported, with a mean age of 7.3 years (range 0.25-C17 years). The stone sizes ranged from 1 to 30 mm. The mean stone-free rate across the three studies was 85.5% (range 58.0-C93.0%) after initial ureteroscopy, with a postoperative stent inserted in 81.8% (range 66.7-C98.0%). There were a total of 35 complications (12.4%), with the most severe complication being a Clavien class III (five ureteral injuries, one urinoma). There were no deaths in any of the studies.Conclusion: the present review shows that FURSL for management of renal calculi in the paediatric population is an effective and safe procedure. To ensure that outcomes keep on improving, these procedures should be undertaken by experienced surgeons who are familiar with the difficulties encountered in the paediatric population
Asymptotic Solutions of Viscous Hamilton-Jacobi Equations with Ornstein-Uhlenbeck Operators
No full textWe study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator in R^N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton–Jacobi equations due to Namah (1996), Namah and Roquejoffre (1999), Roquejoffre (1998), Fathi (1998), Barles and Souganidis (2000, 2001). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator
Existence through convexity for the truncated Laplacians
Full text linkWe study the Dirichlet problem on a bounded convex domain of RN, with zero boundary data, for truncated Laplacians Pk±, which are degenerate elliptic operators, for k< N, defined by the upper and respectively lower partial sum of k eigenvalues of the Hessian matrix. We establish a necessary and sufficient condition (Theorem 1) in terms of the “flatness” of domains for existence of a solution for general inhomogeneous term. This result, in particular, shows that the strict convexity of the domain is sufficient for the solvability of the Dirichlet problem. The result and related ideas are applied to the solvability of the Dirichlet problem for the operator Pk+ with lower order term when the domain is strictly convex and the existence of principal eigenfunctions for the operator P1+. An existence theorem is presented with regard to the principal eigenvalue for the Dirichlet problem with zero-th order term for the operator P1+. A nonexistence result is established for the operator Pk+ with first order term when the domain has a boundary portion which is nearly flat. Furthermore, when the domain is a ball, we study the Dirichlet problem, with a constant inhomogeneous term and a possibly sign-changing first order term, and the associated eigenvalue problem
Asymptotic solutions of Hamilton-Jacobi equations in Euclidean space.
No full textWe study the asymptotic behavior of the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation u_t+\alpha x Du +H(Du)=f(x) in R^n x [0;+\infty)$, where \alpha is a positive constant and H is a convex function on R^n, and establish a convergence result for the viscosity solution u(x,t) as t -> +infty
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