Discontinuous Viscosity Solutions Research Articles

On anisotropic parabolic equation with nonstandard growth order

ABSTRACT In this paper, the existence and the uniqueness of an evolutionary anisotropic p i ( x ) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic p ( x ) − Laplace equation with convection term via L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

Backward reachability approach to state-constrained stochastic optimal control problem for jump-diffusion models

AbstractIn this paper, we consider the stochastic optimal control problem for jump-diffusion models with state constraints. In general, the value function of such problems is the discontinuous viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation since the regularity cannot be guaranteed at the boundary of the state constraint. By adapting the stochastic target theory, we obtain an equivalent representation of the original value function by means of the backward reachable set. We then show that this backward reachable can be characterized by the zero-level set of the auxiliary value function for the unconstrained stochastic control problem, which includes two additional unbounded controls as a consequence of the martingale representation theorem. We prove that the auxiliary value function is the unique continuous viscosity solution of the associated HJB equation, which is the second-order nonlinear partial integro-differential equation (PIDE). Our paper provides an explicit way to characterize the original (possibly discontinuous) value function as a zero-level set of the continuous solution of the auxiliary HJB equation. The proof of the existence and uniqueness requires a new technique due to the unbounded control sets and the presence of the singularity of the corresponding Lévy measure in the nonlocal operator of the HJB equation.

Viscosity solutions for doubly nonlinear evolution equations

We extend the theory of viscosity solutions to treat scalar-valued doubly nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e. we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g. we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop.

Discontinuous solutions of $U_t + H(U_x) = 0$ versus measure-valued solutions of $u_t + [H(u)]_x = 0$

Let H be a bounded Lipschitz continuous function. We discuss some recent results concerning discontinuous viscosity solutions of the Hamilton–Jacobi equation U_t + H(U_x) = 0 0, signed Radon measure-valued entropy solutions of the conservation law u_t + [H(u)]_x = 0 , and their connection.

Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations

We study the Cauchy problem for the simplest first-order Hamilton–Jacobi equation in one space dimension, with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. Uniqueness of discontinuous viscosity solutions is proven, if the initial data function has a finite number of jump discontinuities. Main ingredients of the proof are the barrier effect of spatial discontinuities of a solution (which is linked to the boundedness of the Hamiltonian), and a comparison theorem for semicontinuous viscosity subsolution and supersolution. These are defined in the spirit of the paper [H. Ishii, Perron’s method for Hamilton–Jacobi equations, Duke Math. J. 55 (1987) 368–384], yet using essential limits to introduce semicontinuous envelopes. The definition is shown to be compatible with Perron’s method for existence and is crucial in the uniqueness proof. We also describe some properties of the time evolution of spatial jump discontinuities of the solution, and obtain several results about singular Neumann problems which arise in connection with the above referred barrier effect.

A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

Global existence to a diagonal hyperbolic system for any BV initial data

In this paper, we study the existence of solutions for a diagonal hyperbolic system, that is not necessarily strictly hyperbolic, in one space dimension, considering discontinuous BV initial data without any restrictions on the size of its norm. This system appears naturally in various physical domains, particularly in isentropic gas dynamics and dislocation dynamics in materials. In the case of strictly hyperbolic systems, an existence and uniqueness of a discontinuous solution result is available for BV initial data with small norm, whereas several existence and uniqueness results have been presented for non-decreasing continuous solutions. In the present paper, we show the global in time existence of discontinuous viscosity solutions to a diagonal hyperbolic system for every initial data of bounded total variation, without the assumption that the system is strictly hyperbolic. Up to our knowledge, this is the first global existence result of large discontinuous solutions to this system.

Discontinuous Solutions of Hamilton–Jacobi Equations Versus Radon Measure-Valued Solutions of Scalar Conservation Laws: Disappearance of Singularities

Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton–Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton–Jacobi equation and signed singular measures in case of the conservation law.

$ BV $ solution for a non-linear Hamilton-Jacobi system

<p style='text-indent:20px;'>In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. For such kind of systems, previous results on the existence and uniqueness are available for quasi-monotone systems and other special systems in Lipschitz continuous space. It is worth mentioning that our system includes, in particular, the case of non-decreasing solution where some existence and uniqueness results arose for strictly hyperbolic systems with a small total variation. In the present paper, we consider initial data with unnecessarily small <inline-formula><tex-math id="M2">\begin{document}$ BV $\end{document}</tex-math></inline-formula> seminorm, and we use some <inline-formula><tex-math id="M3">\begin{document}$ BV $\end{document}</tex-math></inline-formula> bounds to prove a global-in-time existence result of this system in the framework of discontinuous viscosity solution.

Utility maximisation in a factor model with constant and proportional transaction costs

We study the problem of maximising expected utility of terminal wealth under constant and proportional transaction costs in a multidimensional market with prices driven by a factor process. We show that the value function is the unique viscosity solution of the associated quasi-variational inequalities and construct optimal strategies. While the value function turns out to be truly discontinuous, we are able to establish a comparison principle for discontinuous viscosity solutions which is strong enough to argue that the value function is unique, globally upper semicontinuous, and continuous if restricted to either borrowing or non-borrowing portfolios.

Fast-Marching Methods for Curvature Penalized Shortest Paths

We introduce numerical schemes for computing distances and shortest paths with respect to several planar paths models, featuring curvature penalization and data-driven velocity: the Dubins car, the Euler/Mumford elastica, and a two variants of the Reeds–Shepp car. For that purpose, we design monotone and causal discretizations of the associated Hamilton–Jacobi–Bellman PDEs, posed on the three-dimensional domain $${\mathbb R}^2 \times {\mathbb S}^1$$ . Our discretizations involve sparse, adaptive and anisotropic stencils on a cartesian grid, built using techniques from lattice geometry. A convergence proof is provided, in the setting of discontinuous viscosity solutions. The discretized problems are solvable in a single pass using a variant of the fast-marching algorithm. Numerical experiments illustrate the applications of our schemes in motion planning and image segmentation.

Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities

In this paper, we study a non-local coupled system arising in the modeling of the dynamics of dislocation densities in crystals. For this system, the global existence and uniqueness are available only for continuous viscosity solutions. In the present paper, we investigate the global time existence of this system by considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed of propagation property of this system, we show, in a particular setting, the global existence of discontinuous viscosity solutions of this problem.

Perron’s method for nonlocal fully nonlinear equations

This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perron's method. If the equation is uniformly elliptic in the sense of \cite{SS}, we prove the discontinuous viscosity solution is H\"older continuous and thus it is a viscosity solution.

Optimal Consumption and Investment with Fixed and Proportional Transaction Costs

The classical optimal investment and consumption problem with infinite horizon is studied in the presence of transaction costs. Both proportional and fixed costs as well as general utility functions are considered. Weak dynamic programming is proved in the general setting, and a comparison result for possibly discontinuous viscosity solutions of the dynamic programming equation is provided. Detailed numerical experiments illustrate several properties of the optimal investment strategies.

Motion Planning for Continuous-Time Stochastic Processes: A Dynamic Programming Approach

We study stochastic motion planning problems which involve a controlled process, with possibly discontinuous sample paths, visiting certain subsets of the state-space while avoiding others in a sequential fashion. For this purpose, we first introduce two basic notions of motion planning, and then establish a connection to a class of stochastic optimal control problems concerned with sequential stopping times. A weak dynamic programming principle (DPP) is then proposed, which characterizes the set of initial states that admit a control enabling the process to execute the desired maneuver with probability no less than some pre-specified value. The proposed DPP comprises auxiliary value functions defined in terms of discontinuous payoff functions. A concrete instance of the use of this novel DPP in the case of diffusion processes is also presented. In this case, we establish that the aforementioned set of initial states can be characterized as the level set of a discontinuous viscosity solution to a sequence of partial differential equations, for which the first one has a known boundary condition, while the boundary conditions of the subsequent ones are determined by the solutions to the preceding steps. Finally, the generality and flexibility of the theoretical results are illustrated on an example involving biological switches.

Utility maximization in an illiquid market in continuous time

A utility maximization problem in an illiquid market is studied. The financial market is assumed to have temporary price impact with finite resilience. After the formulation of this problem as a Markovian stochastic optimal control problem a dynamic programming approach is used for its analysis. In particular, the dynamic programming principle is proved and the value function is shown to be the unique discontinuous viscosity solution. This characterization is utilized to obtain numerical results for the optimal strategy and the loss due to illiquidity.

The stochastic reach-avoid problem and set characterization for diffusions

In this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective. Preceding approaches to solving these reachability problems are either confined to the deterministic setting or address almost-sure stochastic requirements. In contrast, we propose a methodology to tackle problems with less stringent requirements than almost sure. To this end, we first establish a connection between two distinct stochastic reach-avoid problems and three classes of stochastic optimal control problems involving discontinuous payoff functions. Subsequently, we focus on solutions of one of the classes of stochastic optimal control problems—the exit-time problem, which solves both the two reach-avoid problems mentioned above. We then derive a weak version of a dynamic programming principle (DPP) for the corresponding value function; in this direction our contribution compared to the existing literature is to develop techniques that admit discontinuous payoff functions. Moreover, based on our DPP, we provide an alternative characterization of the value function as a solution of a partial differential equation (PDE) in the sense of discontinuous viscosity solutions, along with boundary conditions both in Dirichlet and viscosity senses. Theoretical justifications are also discussed to pave the way for deployment of off-the-shelf PDE solvers for numerical computations. Finally, we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.

Global existence results for eikonal equation with BV initial data

In this paper, we study a local and a non-local eikonal equations in one dimensional space describing the evolution of interfaces moving with non-signed velocity. For these equations, the global existence and uniqueness are available only of Lipschitz continuous viscosity solutions in some particular cases. In the present paper, we are interested in the study of the global in time existence of these equations, considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed propagation property of these equations, we show, in a particular setting, global existence results of discontinuous viscosity solutions of this problem. An interesting application of these results is shown in the case of dislocation dynamics.

Existence result for a one-dimensional eikonal equation

In this Note, we consider the eikonal equation in one-dimensional space describing the evolution of interfaces moving with non-signed velocity. We prove a global existence result of discontinuous viscosity solutions in a weak sense by considering BV initial data.

Jump discontinuous viscosity solutions to second order degenerate elliptic equations

We prove existence and uniqueness of viscosity solutions to second order degenerate elliptic equations which are continuous or jump discontinuous on some interior surface. Moreover, we derive sharp conditions for Lipschitz regularity or gradient blow up of the viscosity solutions on such a surface.