A Visit with the ∞-Laplace Equation

“…This highly degenerate nonlinear PDE arises as a variational equation in the "calculus of variations in the sup-norm" (Crandall [3], Aronsson et al [2]) and also appears in stochastic "tug-of-war" game theory (Peres et al [10]). A viscosity solution u is called an infinity harmonic function.…”

Section: Introductionmentioning

confidence: 99%

Everywhere differentiability of infinity harmonic functions

Evans

Smart

2011

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…We refer to [5,10,20] for more details. The operator Δ ∞ is called the infinity Laplacian and solutions of (2) are said to be ∞-harmonic.…”

Section: Motivationsmentioning

confidence: 99%

Extrapolation of Vector Fields Using the Infinity Laplacian and with Applications to Image Segmentation

Guillot

Guyader

2009

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. In this paper, we investigate a new Gradient-Vector-Flow (GVF) ( [38])-inspired static external force field for active contour models, deriving from the edge map of a given image and allowing to increase the capture range. Contrary to prior related works, we reduce the number of unknowns to a single one v by assuming that the expected vector field is the gradient field of a scalar function. The model is phrased in terms of a functional minimization problem comprising a data fidelity term and a regularizer based on the super norm of Dv. The minimization is achieved by solving a second order singular degenerate parabolic equation. A comparison principle as well as the existence/uniqueness of a viscosity solution together with regularity results are established. Experimental results for image segmentation with details of the algorithm are also presented.

show abstract

A Visit with the ∞-Laplace Equation

Cited by 65 publications

References 34 publications

Asymptotics for the best Sobolev constants and their extremal functions

Asymptotics for the best Sobolev constants and their extremal functions

Everywhere differentiability of infinity harmonic functions

Extrapolation of Vector Fields Using the Infinity Laplacian and with Applications to Image Segmentation

Contact Info

Product

Resources

About