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

The solvability of the nonlinear operator equationw=x+Bx, whereB is accretive in a general Banach spaceX is studied by means of discrete approximations. In particular, ifB is continuous and everywhere defined an algorithm is given for solving the equation.

On the range of accretive operators

It is known that solutions of -Δ∞u = -Σ n i,j=1 u xi u xj u xjxj = 0, that is, the ∞-harmonic functions, are exactly those functions having a comparison property with respect to the family of translates of the radial solutions G(x) = a|x|. We establish a more difficult linear result: a function in R n is harmonic if it has the comparison property with respect to sums of n translates of the radial harmonic functions G(x) = a|x| 2-n for n ¬= 2 and G(x) = b ln(|x|) for n = 2. An attempt to generalize these results for -Δ∞u = 0 (p = ∞) and -Δu = 0 (p = 2) to the general p-Laplacian leads to the fascinating discovery that certain sums of translates of radial p-superharmonic functions are again p-superharmonic. Mystery remains: the class of p-superharmonic functions so constructed for p ¬∈ {2,∞} does not suffice to characterize p-subharmonic functions.

/pdf/another-way-to-say-harmonic-2aed193ye8.pdf

Another way to say harmonic

One of the first studies of bifurcation is due to Euler (1744), who treated the buckling of a column subjected to axial compression (the so-called Elastica). He considered the following boundary-value problem 

$$\left\{ {\begin{array}{*{20}{c}} {\theta '' + \lambda \sin \theta = o} \\ {\theta '\left( o \right) = \theta '(1) = o} \\ \end{array} } \right.$$

(1)

where θ is the angle between the tangent to the column and the real axis, λ is the thrust applied and 1 is the length of the column: 
Open image in new window

Fig. 1
fig. 1

Obviously θ=0 is a solution of (0.l) for any real λ.This means that the column remains straight in equilibrium. But it is not difficult to see that as λ exceeds π2 a new family of solutions appears and the column buckles.

Mathematical Theory of Bifurcation

Note: With a historical overview by Elvira Mascolo. Reference CAA-BOOK-2008-007 Record created on 2008-11-25, modified on 2017-05-12

Calculus of Variations and Nonlinear Partial Differential Equations: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005

We study the function u ( x ) – v ( y ) − λ 2 ‖ x – y ‖ 2 to second order, when u is u. s. c. and v is l. s. c., near a point ( x ˆ , y ˆ ) where the maximum is attained. We obtain a sharpening of a result of P.-L. Lions and H. Ishii which implies comparison results for fully nonlinear elliptic equations of second order.

/pdf/semidifferentials-quadratic-forms-and-fully-nonlinear-4o3ejlbqtp.pdf

Michael G. Crandall

Papers

On the range of accretive operators

Another way to say harmonic

Mathematical Theory of Bifurcation

Calculus of Variations and Nonlinear Partial Differential Equations: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005

Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order