https://math.berkeley.edu/~sethian/Papers/sethian.osher.88.pdf

Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations

Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. >

/pdf/geodesic-active-contours-4wqlqxa13k.pdf

Geodesic active contours

https://apps.dtic.mil/sti/pdfs/ADA189392.pdf

Efficient implementation of essentially non-oscillatory shock-capturing schemes,II

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

https://www.ams.org/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf

User’s guide to viscosity solutions of second order partial differential equations

Publisher Summary This chapter analyzes viscosity solutions of Hamilton–Jacobi equations (HJEs) in Banach spaces. The chapter considers HJEs of the form H ( x , u , Du ) = 0 in Ω where Ω is an open subset of a Banach space V , V * is the dual of V , H ∈ C ( V × R × V *), and Du denotes the Frechet derivative of a function u : Ω → R . A function u is a classical solution of ( HJ ) in Ω if u is continuously Frechet differentiable on Ω and the equation is satisfied pointwise. The chapter describes that even if V = R n , the notion of a classical solution is too restrictive to admit the “solutions” of HJEs that are important in the areas in which they arise—in particular, the “value” functions of control theory, the calculus of variations, and differential games that are usually nonclassical solutions of HJEs. The chapter sketches the basic definitions and some existence and uniqueness theorems in Banach spaces. The stability property of the class of viscosity subsolutions in regular spaces is also discussed in the chapter.

Viscosity Solutions of Hamilton-Jacobi Equations in Banach Spaces

/pdf/on-nonuniqueness-of-viscosity-solutions-vbrvasmgrg.pdf

On nonuniqueness of viscosity solutions

This chapter concerns estimates on maxima of subsolutions of scalar fully nonlinear nondegenerate elliptic equations in an open domain, together with parabolic generalizations. It discusses a maximum principle of Aleksandrov-Bakelman-Pucci type.

A Note on Generalized Maximum Principles for Elliptic and Parabolic PDE

: results are given which apply to semilinear parabolic equations. Simple examples are provided by initial-value problems for semilinear equation like u sub t = delta u + f(u). As applied to this situation, the results show that for suitable f and t greater than 0 the solution u(t,phi) of the problem with the initial-value phi varies continuously together with its space derivatives as phi varied in a weaker sense - e.g. small perturbations of phi in some L sub P-norms cause small changes in u(t,phi) in the space on continuously differentiable functions. (Author)

A remark on semilinear perturbations of abstract parabolic equations

: A very substantial theory of quasilinear evolution equations, which applies to many problems of mathematical physics, has been developed by T. Kato. The theory obtains solutions of quasilinear problems via contraction mappings which are defined by means of a theory of linear evolution equations also developed by Kato. In the current work we show how existence theorems, etc. may be proved in the simplest of the settings considered by Kato via discretization in time. This method does not require an intervening linear theory and also may be viewed as giving results - admittedly crude - about numerical approximation of some of Kato's examples. Since the hypothesis of quasilinearity is not used explicitly herein, we employ the term quasinonlinear for the equations dealt with. (Author)

Michael G. Crandall

Papers

Viscosity Solutions of Hamilton-Jacobi Equations in Banach Spaces

On nonuniqueness of viscosity solutions

A Note on Generalized Maximum Principles for Elliptic and Parabolic PDE

A remark on semilinear perturbations of abstract parabolic equations

Quasinonlinear Evolution Equations.