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The Hamilton-Jacobi Equations
The theory of viscosity solutions was developed for certain types of first and second order partial differential equations. It has been particularly useful in describing the solutions of partial differential equations associated with deterministic and stochastic optimal control problems [16], [53]. In this thesis, we are interested in the theory of viscosity solutions of first-order Hamilton-Jacobi equation in its general form associated with boundary condition. The Hamilton-Jacobi equations play an important role in many fields of mathematics and physics, as for instance, calculus of variations [29], combustion [18], computer graphics [117], optimal control theory [93], differential games [55], image processing [122], quantum mechanics [114], and geometric optics [26]. For this reason, many theoretical and numerical studies have been devoted to solving the Hamilton-Jacobi equations. Let us remark that the equation (1.1.1) is global nonlinear equation. Classical approach to the study of the problem (1.1.1)-(1.1.2) is the method of characteristics. This technique gives an elementary way a local existence result for smooth solutions, and at the same time shows that no global smooth solution exists in general. By a classical solution of the problem (1.1.1)-(1.1.2), we mean a function u∈C1(Ω)∩C(Ω) satisfying (1.1.1) and (1.1.2). Nonlinear partial differential equations of the form (1.1.1) do not, in general, possess classical solutions as can be seen in the following example.
The Classical Theory of Viscosity Solutions
To solve uniqueness (and stability) question given in Section 1.1, in the early 1980s, M. G. Crandall and P.-L. Lions [38, 39] introduced a class of continuous generalized solutions of (1.1.1), called viscosity solutions (for reasons detailed below) which need not be differentiable anywhere, as the only regularity required in the definition is continuity. To motivate the definition of continuous viscosity solution of equation (1.1.1), let us consider the approximate equation for (1.1.1), namely, H(x,uε(x),Duε(x))−ε∇2uε(x) = 0, x∈Ω, (1.2.1) were ε > 0 is a small parameter. The equations (1.2.1) are quasilinear elliptic and have been studied for a long time (see in particular O. Ladyzenskaya and N. N. Uraltseva [87], D. Gilbarg and N. S. Trudinger [58], J. Serrin [121]). It is shown in [59, 85] that the equation (1.2.1) together with boundary condition has a unique classical solution. We hope that as ε→0 the solution uε ∈C2(Ω) of (1.2.1) will converge to some sort of weak solution of (1.1.1). This technique is the method of vanishing viscosity. It comes from a well known method in fluid dynamics where the coefficient ε represents physically the viscosity of the fluid and explains the name of solutions. The vanishing viscosity method works as follows. Suppose that the family of solutions of (1.2.1), namely, {uε}ε>0, is uniformly bounded and equicontinuous on compact set Ω. Consequently, the Arzela-Ascoli’s [79] compactness criterion, ensures that there exists a function.
1 Introduction
1.1 The Hamilton-Jacobi Equations
1.2 The Classical Theory of Viscosity Solutions
1.3 Discontinuous Viscosity Solutions
1.4 Objectives of this Thesis
1.5 Outline of this Thesis
1.6 Summary of Contributions
2 The Space of Hausdorff Continuous Interval Valued Functions
2.1 Introduction
2.2 Baire Operators and Graph Completion Operator
2.3 Hausdorff Continuous Functions
2.4 The Set H(X) is Dedekind Order Complete
2.5 Generalized Baire Operators and Graph Completion Operator
3 Hausdorff Continuous Viscosity Solutions of Hamilton-Jacobi Equations
3.1 Introduction
3.2 Hausdorff Continuous Viscosity Solution of Hamilton-Jacobi Equations
3.3 The Envelope Viscosity Solutions and Hausdorff Continuous Viscosity Solutions
3.4 Existence of Hausdorff Continuous Viscosity Solutions
3.5 Uniqueness of H-Continuous Viscosity Solution
3.6 Extending the Hamiltonian Operator over the Set H(Ω)
4 The Value Functions of Optimal Control Problem as Envelope Viscosity Solutions
4.1 Discounted Minimum Time Problem
4.2 The Value Function as an Envelope Viscosity Solution
4.3 Zermelo Navigation Problem
5 Nonstandard Finite Difference Methods for Solutions of Hamilton-Jacobi Equations and Conservation Laws
5.1 Introduction
5.2 A Monotone Scheme for Hamilton-Jacobi Equations via the Nonstandard Finite Difference Method
5.3 Total Variation Diminishing Nonstandard Finite Difference Schemes for Conservation Laws
6 Conclusion
Bibliography
